At the close of the year 1834, Mr. Freeland again hired me of my master, for the year 1835. But, by this time, I began to want to live upon free land as well as with Freeland; and I was no longer content, therefore, to live with him or any other slaveholder. I began, with the commencement of the year, to prepare myself for a final struggle, which should decide my fate one way or the other. My tendency was upward. I was fast approaching manhood, and year after year had passed, and I was still a slave. These thoughts roused me—I must do something. I therefore resolved that 1835 should not pass without witnessing an attempt, on my part, to secure my liberty. But I was not willing to cherish this determination alone. My fellow-slaves were dear to me. I was anxious to have them participate with me in this, my life-giving determination. I therefore, though with great prudence, commenced early to ascertain their views and feelings in regard to their condition, and to imbue their minds with thoughts of freedom. I bent myself to devising ways and means for our escape, and meanwhile strove, on all fitting occasions, to impress them with the gross fraud and inhumanity of slavery. I went first to Henry, next to John, then to the others. I found, in them all, warm hearts and noble spirits. They were ready to hear, and ready to act when a feasible plan should be proposed. This was what I wanted. I talked to them of our want of manhood, if we submitted to our enslavement without at least one noble effort to be free. We met often, and consulted frequently, and told our hopes and fears, recounted the difficulties, real and imagined, which we should be called on to meet. At times we were almost disposed to give up, and try to content ourselves with our wretched lot; at others, we were firm and unbending in our determination to go. Whenever we suggested any plan, there was shrinking—the odds were fearful. Our path was beset with the greatest obstacles; and if we succeeded in gaining the end of it, our right to be free was yet questionable—we were yet liable to be returned to bondage. We could see no spot, this side of the ocean, where we could be free. We knew nothing about Canada. Our knowledge of the north did not extend farther than New York; and to go there, and be forever harassed with the frightful liability of being returned to slavery—with the certainty of being treated tenfold worse than before—the thought was truly a horrible one, and one which it was not easy to overcome. The case sometimes stood thus: At every gate through which we were to pass, we saw a watchman—at every ferry a guard—on every bridge a sentinel—and in every wood a patrol. We were hemmed in upon every side. Here were the difficulties, real or imagined—the good to be sought, and the evil to be shunned.  On the other hand, away back in the dim distance, under the flickering light of the north star, behind some craggy hill or snow-covered mountain, stood a doubtful freedom—half frozen—beckoning us to come and share its hospitality. This in itself was sometimes enough to stagger us; but when we permitted ourselves to survey the road, we were frequently appalled. Upon either side we saw grim death, assuming the most horrid shapes. Now it was starvation, causing us to eat our own flesh;—now we were contending with the waves, and were drowned;—now we were overtaken, and torn to pieces by the fangs of the terrible bloodhound. We were stung by scorpions, chased by wild beasts, bitten by snakes, and finally, after having nearly reached the desired spot,—after swimming rivers, encountering wild beasts, sleeping in the woods, suffering hunger and nakedness,—we were overtaken by our pursuers, and, in our resistance, we were shot dead upon the spot! I say, this picture sometimes appalled us, and made us
“rather bear those ills we had,
Than fly to others, that we knew not of.”
In coming to a fixed determination to run away, we did more than Patrick Henry, when he resolved upon liberty or death. With us it was a doubtful liberty at most, and almost certain death if we failed. For my part, I should prefer death to hopeless bondage.
Sandy, one of our number, gave up the notion, but still encouraged us. Our company then consisted of Henry Harris, John Harris, Henry Bailey, Charles Roberts, and myself. Henry Bailey was my uncle, and belonged to my master. Charles married my aunt: he belonged to my master’s father-in-law, Mr. William Hamilton.
The plan we finally concluded upon was, to get a large canoe belonging to Mr. Hamilton, and upon the Saturday night previous to Easter holidays, paddle directly up the Chesapeake Bay. On our arrival at the head of the bay, a distance of seventy or eighty miles from where we lived, it was our purpose to turn our canoe adrift, and follow the guidance of the north star till we got beyond the limits of Maryland. Our reason for taking the water route was, that we were less liable to be suspected as runaways; we hoped to be regarded as fishermen; whereas, if we should take the land route, we should be subjected to interruptions of almost every kind. Any one having a white face, and being so disposed, could stop us, and subject us to examination.
The week before our intended start, I wrote several protections, one for each of us. As well as I can remember, they were in the following words, to wit:—
“This is to certify that I, the undersigned, have given the bearer, my
servant, full liberty to go to Baltimore, and spend the Easter holidays.
Written with mine own hand, &c., 1835.
“WILLIAM HAMILTON,
“Near St. Michael’s, in Talbot county, Maryland.”
We were not going to Baltimore; but, in going up the bay, we went toward Baltimore, and these protections were only intended to protect us while on the bay.
As the time drew near for our departure, our anxiety became more and more intense. It was truly a matter of life and death with us. The strength of our determination was about to be fully tested. At this time, I was very active in explaining every difficulty, removing every doubt, dispelling every fear, and inspiring all with the firmness indispensable to success in our undertaking; assuring them that half was gained the instant we made the move; we had talked long enough; we were now ready to move; if not now, we never should be; and if we did not intend to move now, we had as well fold our arms, sit down, and acknowledge ourselves fit only to be slaves. This, none of us were prepared to acknowledge. Every man stood firm; and at our last meeting, we pledged ourselves afresh, in the most solemn manner, that, at the time appointed, we would certainly start in pursuit of freedom. This was in the middle of the week, at the end of which we were to be off. We went, as usual, to our several fields of labor, but with bosoms highly agitated with thoughts of our truly hazardous undertaking. We tried to conceal our feelings as much as possible; and I think we succeeded very well.
After a painful waiting, the Saturday morning, whose night was to witness our departure, came. I hailed it with joy, bring what of sadness it might. Friday night was a sleepless one for me. I probably felt more anxious than the rest, because I was, by common consent, at the head of the whole affair. The responsibility of success or failure lay heavily upon me. The glory of the one, and the confusion of the other, were alike mine. The first two hours of that morning were such as I never experienced before, and hope never to again. Early in the morning, we went, as usual, to the field. We were spreading manure; and all at once, while thus engaged, I was overwhelmed with an indescribable feeling, in the fulness of which I turned to Sandy, who was near by, and said, “We are betrayed!” “Well,” said he, “that thought has this moment struck me.” We said no more. I was never more certain of any thing.
The horn was blown as usual, and we went up from the field to the house for breakfast. I went for the form, more than for want of any thing to eat that morning. Just as I got to the house, in looking out at the lane gate, I saw four white men, with two colored men. The white men were on horseback, and the colored ones were walking behind, as if tied. I watched them a few moments till they got up to our lane gate. Here they halted, and tied the colored men to the gate-post. I was not yet certain as to what the matter was. In a few moments, in rode Mr. Hamilton, with a speed betokening great excitement. He came to the door, and inquired if Master William was in. He was told he was at the barn. Mr. Hamilton, without dismounting, rode up to the barn with extraordinary speed. In a few moments, he and Mr. Freeland returned to the house. By this time, the three constables rode up, and in great haste dismounted, tied their horses, and met Master William and Mr. Hamilton returning from the barn; and after talking awhile, they all walked up to the kitchen door. There was no one in the kitchen but myself and John. Henry and Sandy were up at the barn. Mr. Freeland put his head in at the door, and called me by name, saying, there were some gentlemen at the door who wished to see me. I stepped to the door, and inquired what they wanted. They at once seized me, and, without giving me any satisfaction, tied me—lashing my hands closely together. I insisted upon knowing what the matter was. They at length said, that they had learned I had been in a “scrape,” and that I was to be examined before my master; and if their information proved false, I should not be hurt.
In a few moments, they succeeded in tying John. They then turned to Henry, who had by this time returned, and commanded him to cross his hands. “I won’t!” said Henry, in a firm tone, indicating his readiness to meet the consequences of his refusal. “Won’t you?” said Tom Graham, the constable. “No, I won’t!” said Henry, in a still stronger tone. With this, two of the constables pulled out their shining pistols, and swore, by their Creator, that they would make him cross his hands or kill him. Each cocked his pistol, and, with fingers on the trigger, walked up to Henry, saying, at the same time, if he did not cross his hands, they would blow his damned heart out.  Shoot, shoot,—and be damned! I won’t be tied!” This he said in a tone of loud defiance; and at the same time, with a motion as quick as lightning, he with one single stroke dashed the pistols from the hand of each constable. As he did this, all hands fell upon him, and, after beating him some time, they finally overpowered him, and got him tied.
During the scuffle, I managed, I know not how, to get my pass out, and, without being discovered, put it into the fire. We were all now tied; and just as we were to leave for Easton jail, Betsy Freeland, mother of William Freeland, came to the door with her hands full of biscuits, and divided them between Henry and John. She then delivered herself of a speech, to the following effect:—addressing herself to me, she said, “You devil! You yellow devil! it was you that put it into the heads of Henry and John to run away.  Henry nor John would never have thought of such a thing.” I made no reply, and was immediately hurried off towards St. Michael’s. Just a moment previous to the scuffle with Henry, Mr. Hamilton suggested the propriety of making a search for the protections which he had understood Frederick had written for himself and the rest. But, just at the moment he was about carrying his proposal into effect, his aid was needed in helping to tie Henry; and the excitement attending the scuffle caused them either to forget, or to deem it unsafe, under the circumstances, to search. So we were not yet convicted of the intention to run away.
When we got about half way to St. Michael’s, while the constables having us in charge were looking ahead, Henry inquired of me what he should do with his pass. I told him to eat it with his biscuit, and own nothing; and we passed the word around, “Own nothing;” and “Own nothing!” said we all. Our confidence in each other was unshaken. We were resolved to succeed or fail together, after the calamity had befallen us as much as before. We were now prepared for any thing. We were to be dragged that morning fifteen miles behind horses, and then to be placed in the Easton jail. When we reached St. Michael’s, we underwent a sort of examination. We all denied that we ever intended to run away. We did this more to bring out the evidence against us, than from any hope of getting clear of being sold; for, as I have said, we were ready for that. The fact was, we cared but little where we went, so we went together. Our greatest concern was about separation. We dreaded that more than any thing this side of death. We found the evidence against us to be the testimony of one person; our master would not tell who it was; but we came to a unanimous decision among ourselves as to who their informant was. We were sent off to the jail at Easton. When we got there, we were delivered up to the sheriff, Mr. Joseph Graham, and by him placed in jail. Henry, John, and myself, were placed in one room together—Charles, and Henry Bailey, in another. Their object in separating us was to hinder concert.
 Such a set of beings I never saw before! I felt myself surrounded by so many fiends from perdition. A band of pirates never looked more like their father, the devil. They laughed and grinned over us, saying, “Ah, my boys! we have got you, haven’t we?” And after taunting us in various ways, they one by one went into an examination of us, with intent to ascertain our value. They would impudently ask us if we would not like to have them for our masters. We would make them no answer, and leave them to find out as best they could. Then they would curse and swear at us, telling us that they could take the devil out of us in a very little while, if we were only in their hands.
While in jail, we found ourselves in much more comfortable quarters than we expected when we went there. We did not get much to eat, nor that which was very good; but we had a good clean room, from the windows of which we could see what was going on in the street, which was very much better than though we had been placed in one of the dark, damp cells. Upon the whole, we got along very well, so far as the jail and its keeper were concerned. Immediately after the holidays were over, contrary to all our expectations, Mr. Hamilton and Mr. Freeland came up to Easton, and took Charles, the two Henrys, and John, out of jail, and carried them home, leaving me alone. I regarded this separation as a final one. It caused me more pain than any thing else in the whole transaction. I was ready for any thing rather than separation. I supposed that they had consulted together, and had decided that, as I was the whole cause of the intention of the others to run away, it was hard to make the innocent suffer with the guilty; and that they had, therefore, concluded to take the others home, and sell me, as a warning to the others that remained. It is due to the noble Henry to say, he seemed almost as reluctant at leaving the prison as at leaving home to come to the prison. But we knew we should, in all probability, be separated, if we were sold; and since he was in their hands, he concluded to go peaceably home.
I was now left to my fate. I was all alone, and within the walls of a stone prison.  I expected to have been safe in a land of freedom; but now I was covered with gloom, sunk down to the utmost despair. I thought the possibility of freedom was gone. I was kept in this way about one week, at the end of which, Captain Auld, my master, to my surprise and utter astonishment, came up, and took me out, with the intention of sending me, with a gentleman of his acquaintance, into Alabama. But, from some cause or other, he did not send me to Alabama, but concluded to send me back to Baltimore, to live again with his brother Hugh, and to learn a trade.
Thus, after an absence of three years and one month, I was once more permitted to return to my old home at Baltimore. My master sent me away, because there existed against me a very great prejudice in the community, and he feared I might be killed.
In a few weeks after I went to Baltimore, Master Hugh hired me to Mr. William Gardner, an extensive ship-builder, on Fell’s Point. I was put there to learn how to calk. It, however, proved a very unfavorable place for the accomplishment of this object. Mr. Gardner was engaged that spring in building two large man-of-war brigs, professedly for the Mexican government. The vessels were to be launched in the July of that year, and in failure thereof, Mr. Gardner was to lose a considerable sum; so that when I entered, all was hurry. There was no time to learn any thing. Every man had to do that which he knew how to do. In entering the shipyard, my orders from Mr. Gardner were, to do whatever the carpenters commanded me to do. This was placing me at the beck and call of about seventy-five men. I was to regard all these as masters. Their word was to be my law. My situation was a most trying one. At times I needed a dozen pair of hands. I was called a dozen ways in the space of a single minute. Three or four voices would strike my ear at the same moment. It was—”Fred., come help me to cant this timber here.”—”Fred., come carry this timber yonder.”—”Fred., bring that roller here.”—”Fred., go get a fresh can of water.”—”Fred., come help saw off the end of this timber.”—”Fred., go quick, and get the crowbar.”—”Fred., hold on the end of this fall.”—”Fred., go to the blacksmith’s shop, and get a new punch.”— ”Hurra, Fred! run and bring me a cold chisel.”—”I say, Fred., bear a hand, and get up a fire as quick as lightning under that steam-box.”—”Halloo, nigger! come, turn this grindstone.”—”Come, come! move, move! and bowse this timber forward.”—”I say, darky, blast your eyes, why don’t you heat up some pitch?”—”Halloo! halloo! halloo!” (Three voices at the same time.) “Come here!—Go there!—Hold on where you are!
This was my school for eight months; and I might have remained there longer, but for a most horrid fight I had with four of the white apprentices, in which my left eye was nearly knocked out, and I was horribly mangled in other respects. The facts in the case were these: Until a very little while after I went there, white and black ship-carpenters worked side by side, and no one seemed to see any impropriety in it. All hands seemed to be very well satisfied. Many of the black carpenters were freemen. Things seemed to be going on very well. All at once, the white carpenters knocked off, and said they would not work with free colored workmen. Their reason for this, as alleged, was, that if free colored carpenters were encouraged, they would soon take the trade into their own hands, and poor white men would be thrown out of employment. They therefore felt called upon at once to put a stop to it.  Now, though this did not extend to me in form, it did reach me in fact. My fellow-apprentices very soon began to feel it degrading to them to work with me. They began to put on airs, and talk about the “niggers” taking the country, saying we all ought to be killed; and, being encouraged by the journeymen, they commenced making my condition as hard as they could, by hectoring me around, and sometimes striking me. I, of course, kept the vow I made after the fight with Mr. Covey, and struck back again, regardless of consequences; and while I kept them from combining, I succeeded very well; for I could whip the whole of them, taking them separately. They, however, at length combined, and came upon me, armed with sticks, stones, and heavy handspikes.  There was one at each side of me, and one behind me. While I was attending to those in front, and on either side, the one behind ran up with the handspike, and struck me a heavy blow upon the head. It stunned me. I fell, and with this they all ran upon me, and fell to beating me with their fists. I let them lay on for a while, gathering strength. In an instant, I gave a sudden surge, and rose to my hands and knees. Just as I did that, one of their number gave me, with his heavy boot, a powerful kick in the left eye. My eyeball seemed to have burst. When they saw my eye closed, and badly swollen, they left me. With this I seized the handspike, and for a time pursued them. But here the carpenters interfered, and I thought I might as well give it up. It was impossible to stand my hand against so many. All this took place in sight of not less than fifty white ship-carpenters, and not one interposed a friendly word; but some cried, “Kill the damned nigger!  He struck a white person.” I found my only chance for life was in flight. I succeeded in getting away without an additional blow, and barely so; for to strike a white man is death by Lynch law,—and that was the law in Mr. Gardner’s ship-yard; nor is there much of any other out of Mr. Gardner’s ship-yard.
I went directly home, and told the story of my wrongs to Master Hugh; and I am happy to say of him, irreligious as he was, his conduct was heavenly, compared with that of his brother Thomas under similar circumstances. He listened attentively to my narration of the circumstances leading to the savage outrage, and gave many proofs of his strong indignation at it. The heart of my once overkind mistress was again melted into pity. My puffed-out eye and blood-covered face moved her to tears.  It was almost compensation for my suffering to witness, once more, a manifestation of kindness from this, my once affectionate old mistress. Master Hugh was very much enraged. He gave expression to his feelings by pouring out curses upon the heads of those who did the deed. As soon as I got a little the better of my bruises, he took me with him to Esquire Watson’s, on Bond Street, to see what could be done about the matter. Mr. Watson inquired who saw the assault committed. Master Hugh told him it was done in Mr. Gardner’s ship-yard at midday, where there were a large company of men at work. “As to that,” he said, “the deed was done, and there was no question as to who did it.” His answer was, he could do nothing in the case, unless some white man would come forward and testify. He could issue no warrant on my word. If I had been killed in the presence of a thousand colored people, their testimony combined would have been insufficient to have arrested one of the murderers. Master Hugh, for once, was compelled to say this state of things was too bad. Of course, it was impossible to get any white man to volunteer his testimony in my behalf, and against the white young men. Even those who may have sympathized with me were not prepared to do this. It required a degree of courage unknown to them to do so; for just at that time, the slightest manifestation of humanity toward a colored person was denounced as abolitionism, and that name subjected its bearer to frightful liabilities. The watchwords of the bloody-minded in that region, and in those days, were, “Damn the abolitionists!” and “Damn the niggers!” There was nothing done, and probably nothing would have been done if I had been killed. Such was, and such remains, the state of things in the Christian city of Baltimore.
Master Hugh, finding he could get no redress, refused to let me go back again to Mr. Gardner. He kept me himself, and his wife dressed my wound till I was again restored to health. He then took me into the ship-yard of which he was foreman, in the employment of Mr. Walter Price. There I was immediately set to calking, and very soon learned the art of using my mallet and irons. In the course of one year from the time I left Mr. Gardner’s, I was able to command the highest wages given to the most experienced calkers. I was now of some importance to my master. I was bringing him from six to seven dollars per week. I sometimes brought him nine dollars per week: my wages were a dollar and a half a day. After learning how to calk, I sought my own employment, made my own contracts, and collected the money which I earned. My pathway became much more smooth than before; my condition was now much more comfortable. When I could get no calking to do, I did nothing. During these leisure times, those old notions about freedom would steal over me again. When in Mr. Gardner’s employment, I was kept in such a perpetual whirl of excitement, I could think of nothing, scarcely, but my life; and in thinking of my life, I almost forgot my liberty. I have observed this in my experience of slavery,—that whenever my condition was improved, instead of its increasing my contentment, it only increased my desire to be free, and set me to thinking of plans to gain my freedom. I have found that, to make a contented slave, it is necessary to make a thoughtless one. It is necessary to darken his moral and mental vision, and, as far as possible, to annihilate the power of reason. He must be able to detect no inconsistencies in slavery; he must be made to feel that slavery is right; and he can be brought to that only when he ceases to be a man.
I was now getting, as I have said, one dollar and fifty cents per day. I contracted for it; I earned it; it was paid to me; it was rightfully my own; yet, upon each returning Saturday night, I was compelled to deliver every cent of that money to Master Hugh. And why? Not because he earned it,—not because he had any hand in earning it,—not because I owed it to him,—nor because he possessed the slightest shadow of a right to it; but solely because he had the power to compel me to give it up. The right of the grim-visaged pirate upon the high seas is exactly the same.
Chapter XI
I now come to that part of my life during which I planned, and finally succeeded in making, my escape from slavery. But before narrating any of the peculiar circumstances, I deem it proper to make known my intention not to state all the facts connected with the transaction. My reasons for pursuing this course may be understood from the following: First, were I to give a minute statement of all the facts, it is not only possible, but quite probable, that others would thereby be involved in the most embarrassing difficulties. Secondly, such a statement would most undoubtedly induce greater vigilance on the part of slaveholders than has existed heretofore among them; which would, of course, be the means of guarding a door whereby some dear brother bondman might escape his galling chains. I deeply regret the necessity that impels me to suppress any thing of importance connected with my experience in slavery. It would afford me great pleasure indeed, as well as materially add to the interest of my narrative, were I at liberty to gratify a curiosity, which I know exists in the minds of many, by an accurate statement of all the facts pertaining to my most fortunate escape. But I must deprive myself of this pleasure, and the curious of the gratification which such a statement would afford. I would allow myself to suffer under the greatest imputations which evil-minded men might suggest, rather than exculpate myself, and thereby run the hazard of closing the slightest avenue by which a brother slave might clear himself of the chains and fetters of slavery.
I have never approved of the very public manner in which some of our western friends have conducted what they call the underground railroad, but which I think, by their open declarations, has been made most emphatically the upperground railroad. I honor those good men and women for their noble daring, and applaud them for willingly subjecting themselves to bloody persecution, by openly avowing their participation in the escape of slaves. I, however, can see very little good resulting from such a course, either to themselves or the slaves escaping; while, upon the other hand, I see and feel assured that those open declarations are a positive evil to the slaves remaining, who are seeking to escape. They do nothing towards enlightening the slave, whilst they do much towards enlightening the master. They stimulate him to greater watchfulness, and enhance his power to capture his slave. We owe something to the slave south of the line as well as to those north of it; and in aiding the latter on their way to freedom, we should be careful to do nothing which would be likely to hinder the former from escaping from slavery. I would keep the merciless slaveholder profoundly ignorant of the means of flight adopted by the slave. I would leave him to imagine himself surrounded by myriads of invisible tormentors, ever ready to snatch from his infernal grasp his trembling prey. Let him be left to feel his way in the dark; let darkness commensurate with his crime hover over him; and let him feel that at every step he takes, in pursuit of the flying bondman, he is running the frightful risk of having his hot brains dashed out by an invisible agency. Let us render the tyrant no aid; let us not hold the light by which he can trace the footprints of our flying brother. But enough of this. I will now proceed to the statement of those facts, connected with my escape, for which I am alone responsible, and for which no one can be made to suffer but myself.
In the early part of the year 1838, I became quite restless. I could see no reason why I should, at the end of each week, pour the reward of my toil into the purse of my master. When I carried to him my weekly wages, he would, after counting the money, look me in the face with a robber-like fierceness, and ask, “Is this all?” He was satisfied with nothing less than the last cent. He would, however, when I made him six dollars, sometimes give me six cents, to encourage me. It had the opposite effect. I regarded it as a sort of admission of my right to the whole. The fact that he gave me any part of my wages was proof, to my mind, that he believed me entitled to the whole of them. I always felt worse for having received any thing; for I feared that the giving me a few cents would ease his conscience, and make him feel himself to be a pretty honorable sort of robber. My discontent grew upon me. I was ever on the look-out for means of escape; and, finding no direct means, I determined to try to hire my time, with a view of getting money with which to make my escape. In the spring of 1838, when Master Thomas came to Baltimore to purchase his spring goods, I got an opportunity, and applied to him to allow me to hire my time. He unhesitatingly refused my request, and told me this was another stratagem by which to escape. He told me I could go nowhere but that he could get me; and that, in the event of my running away, he should spare no pains in his efforts to catch me. He exhorted me to content myself, and be obedient. He told me, if I would be happy, I must lay out no plans for the future. He said, if I behaved myself properly, he would take care of me. Indeed, he advised me to complete thoughtlessness of the future, and taught me to depend solely upon him for happiness. He seemed to see fully the pressing necessity of setting aside my intellectual nature, in order to contentment in slavery. But in spite of him, and even in spite of myself, I continued to think, and to think about the injustice of my enslavement, and the means of escape.
About two months after this, I applied to Master Hugh for the privilege of hiring my time. He was not acquainted with the fact that I had applied to Master Thomas, and had been refused. He too, at first, seemed disposed to refuse; but, after some reflection, he granted me the privilege, and proposed the following terms: I was to be allowed all my time, make all contracts with those for whom I worked, and find my own employment; and, in return for this liberty, I was to pay him three dollars at the end of each week; find myself in calking tools, and in board and clothing. My board was two dollars and a half per week. This, with the wear and tear of clothing and calking tools, made my regular expenses about six dollars per week. This amount I was compelled to make up, or relinquish the privilege of hiring my time. Rain or shine, work or no work, at the end of each week the money must be forthcoming, or I must give up my privilege. This arrangement, it will be perceived, was decidedly in my master’s favor. It relieved him of all need of looking after me. His money was sure. He received all the benefits of slaveholding without its evils; while I endured all the evils of a slave, and suffered all the care and anxiety of a freeman. I found it a hard bargain. But, hard as it was, I thought it better than the old mode of getting along. It was a step towards freedom to be allowed to bear the responsibilities of a freeman, and I was determined to hold on upon it. I bent myself to the work of making money. I was ready to work at night as well as day, and by the most untiring perseverance and industry, I made enough to meet my expenses, and lay up a little money every week. I went on thus from May till August.  The ground for his refusal was a failure on my part, one Saturday night, to pay him for my week’s time. This failure was occasioned by my attending a camp meeting about ten miles from Baltimore. During the week, I had entered into an engagement with a number of young friends to start from Baltimore to the camp ground early Saturday evening; and being detained by my employer, I was unable to get down to Master Hugh’s without disappointing the company. I knew that Master Hugh was in no special need of the money that night. I therefore decided to go to camp meeting, and upon my return pay him the three dollars. I staid at the camp meeting one day longer than I intended when I left. But as soon as I returned, I called upon him to pay him what he considered his due. I found him very angry; he could scarce restrain his wrath. He said he had a great mind to give me a severe whipping. He wished to know how I dared go out of the city without asking his permission. I told him I hired my time and while I paid him the price which he asked for it, I did not know that I was bound to ask him when and where I should go. This reply troubled him; and, after reflecting a few moments, he turned to me, and said I should hire my time no longer; that the next thing he should know of, I would be running away. Upon the same plea, he told me to bring my tools and clothing home forthwith. I did so; but instead of seeking work, as I had been accustomed to do previously to hiring my time, I spent the whole week without the performance of a single stroke of work. I did this in retaliation. Saturday night, he called upon me as usual for my week’s wages. I told him I had no wages; I had done no work that week. Here we were upon the point of coming to blows. He raved, and swore his determination to get hold of me. I did not allow myself a single word; but was resolved, if he laid the weight of his hand upon me, it should be blow for blow. He did not strike me, but told me that he would find me in constant employment in future. I thought the matter over during the next day, Sunday, and finally resolved upon the third day of September, as the day upon which I would make a second attempt to secure my freedom. I now had three weeks during which to prepare for my journey. Early on Monday morning, before Master Hugh had time to make any engagement for me, I went out and got employment of Mr. Butler, at his ship-yard near the drawbridge, upon what is called the City Block, thus making it unnecessary for him to seek employment for me. At the end of the week, I brought him between eight and nine dollars. He seemed very well pleased, and asked why I did not do the same the week before. He little knew what my plans were. My object in working steadily was to remove any suspicion he might entertain of my intent to run away; and in this I succeeded admirably. I suppose he thought I was never better satisfied with my condition than at the very time during which I was planning my escape. The second week passed, and again I carried him my full wages; and so well pleased was he, that he gave me twenty-five cents, (quite a large sum for a slaveholder to give a slave,) and bade me to make a good use of it. I told him I would.
Things went on without very smoothly indeed, but within there was trouble. It is impossible for me to describe my feelings as the time of my contemplated start drew near. I had a number of warmhearted friends in Baltimore,—friends that I loved almost as I did my life,—and the thought of being separated from them forever was painful beyond expression. It is my opinion that thousands would escape from slavery, who now remain, but for the strong cords of affection that bind them to their friends. The thought of leaving my friends was decidedly the most painful thought with which I had to contend. The love of them was my tender point, and shook my decision more than all things else. Besides the pain of separation, the dread and apprehension of a failure exceeded what I had experienced at my first attempt. The appalling defeat I then sustained returned to torment me. I felt assured that, if I failed in this attempt, my case would be a hopeless one—it would seal my fate as a slave forever. I could not hope to get off with any thing less than the severest punishment, and being placed beyond the means of escape. It required no very vivid imagination to depict the most frightful scenes through which I should have to pass, in case I failed. The wretchedness of slavery, and the blessedness of freedom, were perpetually before me. It was life and death with me. But I remained firm, and, according to my resolution, on the third day of September, 1838, I left my chains, and succeeded in reaching New York without the slightest interruption of any kind. How I did so,—what means I adopted,—what direction I travelled, and by what mode of conveyance,—I must leave unexplained, for the reasons before mentioned.
I have been frequently asked how I felt when I found myself in a free State. I have never been able to answer the question with any satisfaction to myself. It was a moment of the highest excitement I ever experienced. I suppose I felt as one may imagine the unarmed mariner to feel when he is rescued by a friendly man-of-war from the pursuit of a pirate. In writing to a dear friend, immediately after my arrival at New York, I said I felt like one who had escaped a den of hungry lions. This state of mind, however, very soon subsided; and I was again seized with a feeling of great insecurity and loneliness. I was yet liable to be taken back, and subjected to all the tortures of slavery. This in itself was enough to damp the ardor of my enthusiasm. But the loneliness overcame me. There I was in the midst of thousands, and yet a perfect stranger; without home and without friends, in the midst of thousands of my own brethren—children of a common Father, and yet I dared not to unfold to any one of them my sad condition. I was afraid to speak to any one for fear of speaking to the wrong one, and thereby falling into the hands of money-loving kidnappers, whose business it was to lie in wait for the panting fugitive, as the ferocious beasts of the forest lie in wait for their prey. The motto which I adopted when I started from slavery was this—”Trust no man!” I saw in every white man an enemy, and in almost every colored man cause for distrust. It was a most painful situation; and, to understand it, one must needs experience it, or imagine himself in similar circumstances. Let him be a fugitive slave in a strange land—a land given up to be the hunting-ground for slaveholders—whose inhabitants are legalized kidnappers—where he is every moment subjected to the terrible liability of being seized upon by his fellowmen, as the hideous crocodile seizes upon his prey!—I say, let him place himself in my situation—without home or friends—without money or credit—wanting shelter, and no one to give it— wanting bread, and no money to buy it,—and at the same time let him feel that he is pursued by merciless men-hunters, and in total darkness as to what to do, where to go, or where to stay,—perfectly helpless both as to the means of defence and means of escape,—in the midst of plenty, yet suffering the terrible gnawings of hunger,— in the midst of houses, yet having no home,—among fellow-men, yet feeling as if in the midst of wild beasts, whose greediness to swallow up the trembling and half-famished fugitive is only equalled by that with which the monsters of the deep swallow up the helpless fish upon which they subsist,—I say, let him be placed in this most trying situation,—the situation in which I was placed,—then, and not till then, will he fully appreciate the hardships of, and know how to sympathize with, the toil-worn and whip-scarred fugitive slave.
Thank Heaven, I remained but a short time in this distressed situation. I was relieved from it by the humane hand of Mr. David Ruggles, whose vigilance, kindness, and perseverance, I shall never forget. I am glad of an opportunity to express, as far as words can, the love and gratitude I bear him. Mr. Ruggles is now afflicted with blindness, and is himself in need of the same kind offices which he was once so forward in the performance of toward others. I had been in New York but a few days, when Mr. Ruggles sought me out, and very kindly took me to his boarding-house at the corner of Church and Lespenard Streets. Mr. Ruggles was then very deeply engaged in the memorable Darg case, as well as attending to a number of other fugitive slaves, devising ways and means for their successful escape; and, though watched and hemmed in on almost every side, he seemed to be more than a match for his enemies.
Very soon after I went to Mr. Ruggles, he wished to know of me where I wanted to go; as he deemed it unsafe for me to remain in New York. I told him I was a calker, and should like to go where I could get work. I thought of going to Canada; but he decided against it, and in favor of my going to New Bedford, thinking I should be able to get work there at my trade. At this time, Anna, my intended wife, came on; for I wrote to her immediately after my arrival at New York, (notwithstanding my homeless, houseless, and helpless condition,) informing her of my successful flight, and wishing her to come on forthwith. In a few days after her arrival, Mr. Ruggles called in the Rev. J. W. C. Pennington, who, in the presence of Mr. Ruggles, Mrs. Michaels, and two or three others, performed the marriage ceremony, and gave us a certificate, of which the following is an exact copy:—
“This may certify, that I joined together in holy matrimony Frederick
Johnson and Anna Murray, as man and wife, in the presence of Mr. David
Ruggles and Mrs. Michaels.
“JAMES W. C. PENNINGTON
“New York, Sept. 15, 1838”
Upon receiving this certificate, and a five-dollar bill from Mr. Ruggles, I shouldered one part of our baggage, and Anna took up the other, and we set out forthwith to take passage on board of the steamboat John W. Richmond for Newport, on our way to New Bedford. Mr. Ruggles gave me a letter to a Mr. Shaw in Newport, and told me, in case my money did not serve me to New Bedford, to stop in Newport and obtain further assistance; but upon our arrival at Newport, we were so anxious to get to a place of safety, that, notwithstanding we lacked the necessary money to pay our fare, we decided to take seats in the stage, and promise to pay when we got to New Bedford. We were encouraged to do this by two excellent gentlemen, residents of New Bedford, whose names I afterward ascertained to be Joseph Ricketson and William C. Taber. They seemed at once to understand our circumstances, and gave us such assurance of their friendliness as put us fully at ease in their presence.
It was good indeed to meet with such friends, at such a time. Upon reaching New Bedford, we were directed to the house of Mr. Nathan Johnson, by whom we were kindly received, and hospitably provided for. Both Mr. and Mrs. Johnson took a deep and lively interest in our welfare. They proved themselves quite worthy of the name of abolitionists. When the stage-driver found us unable to pay our fare, he held on upon our baggage as security for the debt. I had but to mention the fact to Mr. Johnson, and he forthwith advanced the money.
We now began to feel a degree of safety, and to prepare ourselves for the duties and responsibilities of a life of freedom. On the morning after our arrival at New Bedford, while at the breakfast-table, the question arose as to what name I should be called by. The name given me by my mother was, “Frederick Augustus Washington Bailey.” I, however, had dispensed with the two middle names long before I left Maryland so that I was generally known by the name of “Frederick Bailey.” I started from Baltimore bearing the name of “Stanley.” When I got to New York, I again changed my name to “Frederick Johnson,” and thought that would be the last change. But when I got to New Bedford, I found it necessary again to change my name. The reason of this necessity was, that there were so many Johnsons in New Bedford, it was already quite difficult to distinguish between them. I gave Mr. Johnson the privilege of choosing me a name, but told him he must not take from me the name of “Frederick.” I must hold on to that, to preserve a sense of my identity. Mr. Johnson had just been reading the “Lady of the Lake,” and at once suggested that my name be “Douglass.” From that time until now I have been called “Frederick Douglass;” and as I am more widely known by that name than by either of the others, I shall continue to use it as my own.
I was quite disappointed at the general appearance of things in New Bedford. The impression which I had received respecting the character and condition of the people of the north, I found to be singularly erroneous. I had very strangely supposed, while in slavery, that few of the comforts, and scarcely any of the luxuries, of life were enjoyed at the north, compared with what were enjoyed by the slaveholders of the south. I probably came to this conclusion from the fact that northern people owned no slaves. I supposed that they were about upon a level with the non-slaveholding population of the south. I knew they were exceedingly poor, and I had been accustomed to regard their poverty as the necessary consequence of their being non-slaveholders. I had somehow imbibed the opinion that, in the absence of slaves, there could be no wealth, and very little refinement. And upon coming to the north, I expected to meet with a rough, hard-handed, and uncultivated population, living in the most Spartan-like simplicity, knowing nothing of the ease, luxury, pomp, and grandeur of southern slaveholders. Such being my conjectures, any one acquainted with the appearance of New Bedford may very readily infer how palpably I must have seen my mistake.
In the afternoon of the day when I reached New Bedford, I visited the wharves, to take a view of the shipping. Here I found myself surrounded with the strongest proofs of wealth. Lying at the wharves, and riding in the stream, I saw many ships of the finest model, in the best order, and of the largest size. Upon the right and left, I was walled in by granite warehouses of the widest dimensions, stowed to their utmost capacity with the necessaries and comforts of life. Added to this, almost every body seemed to be at work, but noiselessly so, compared with what I had been accustomed to in Baltimore. There were no loud songs heard from those engaged in loading and unloading ships. I heard no deep oaths or horrid curses on the laborer. I saw no whipping of men; but all seemed to go smoothly on. Every man appeared to understand his work, and went at it with a sober, yet cheerful earnestness, which betokened the deep interest which he felt in what he was doing, as well as a sense of his own dignity as a man. To me this looked exceedingly strange. From the wharves I strolled around and over the town, gazing with wonder and admiration at the splendid churches, beautiful dwellings, and finely-cultivated gardens; evincing an amount of wealth, comfort, taste, and refinement, such as I had never seen in any part of slaveholding Maryland.
Every thing looked clean, new, and beautiful. I saw few or no dilapidated houses, with poverty-stricken inmates; no half-naked children and barefooted women, such as I had been accustomed to see in Hillsborough, Easton, St. Michael’s, and Baltimore. The people looked more able, stronger, healthier, and happier, than those of Maryland. I was for once made glad by a view of extreme wealth, without being saddened by seeing extreme poverty. But the most astonishing as well as the most interesting thing to me was the condition of the colored people, a great many of whom, like myself, had escaped thither as a refuge from the hunters of men. I found many, who had not been seven years out of their chains, living in finer houses, and evidently enjoying more of the comforts of life, than the average of slaveholders in Maryland. I will venture to assert, that my friend Mr. Nathan Johnson (of whom I can say with a grateful heart, “I was hungry, and he gave me meat; I was thirsty, and he gave me drink; I was a stranger, and he took me in”) lived in a neater house; dined at a better table; took, paid for, and read, more newspapers; better understood the moral, religious, and political character of the nation,—than nine tenths of the slaveholders in Talbot county Maryland. Yet Mr. Johnson was a working man. His hands were hardened by toil, and not his alone, but those also of Mrs. Johnson. I found the colored people much more spirited than I had supposed they would be. I found among them a determination to protect each other from the bloodthirsty kidnapper, at all hazards. Soon after my arrival, I was told of a circumstance which illustrated their spirit. A colored man and a fugitive slave were on unfriendly terms. The former was heard to threaten the latter with informing his master of his whereabouts. Straightway a meeting was called among the colored people, under the stereotyped notice, “Business of importance!” The betrayer was invited to attend. The people came at the appointed hour, and organized the meeting by appointing a very religious old gentleman as president, who, I believe, made a prayer, after which he addressed the meeting as follows: “Friends, we have got him here, and I would recommend that you young men just take him outside the door, and kill him!” With this, a number of them bolted at him; but they were intercepted by some more timid than themselves, and the betrayer escaped their vengeance, and has not been seen in New Bedford since. I believe there have been no more such threats, and should there be hereafter, I doubt not that death would be the consequence.
I found employment, the third day after my arrival, in stowing a sloop with a load of oil. It was new, dirty, and hard work for me; but I went at it with a glad heart and a willing hand. I was now my own master. It was a happy moment, the rapture of which can be understood only by those who have been slaves. It was the first work, the reward of which was to be entirely my own. There was no Master Hugh standing ready, the moment I earned the money, to rob me of it. I worked that day with a pleasure I had never before experienced. I was at work for myself and newly-married wife. It was to me the starting-point of a new existence. When I got through with that job, I went in pursuit of a job of calking; but such was the strength of prejudice against color, among the white calkers, that they refused to work with me, and of course I could get no employment.
Finding my trade of no immediate benefit, I threw off my calking habiliments, and prepared myself to do any kind of work I could get to do. Mr. Johnson kindly let me have his wood-horse and saw, and I very soon found myself a plenty of work. There was no work too hard—none too dirty. I was ready to saw wood, shovel coal, carry wood, sweep the chimney, or roll oil casks,—all of which I did for nearly three years in New Bedford, before I became known to the anti-slavery world.
In about four months after I went to New Bedford, there came a young man to me, and inquired if I did not wish to take the “Liberator.” I told him I did; but, just having made my escape from slavery, I remarked that I was unable to pay for it then. I, however, finally became a subscriber to it. The paper came, and I read it from week to week with such feelings as it would be quite idle for me to attempt to describe. The paper became my meat and my drink. My soul was set all on fire. Its sympathy for my brethren in bonds—its scathing denunciations of slaveholders— its faithful exposures of slavery—and its powerful attacks upon the upholders of the institution—sent a thrill of joy through my soul, such as I had never felt before!
I had not long been a reader of the “Liberator,” before I got a pretty correct idea of the principles, measures and spirit of the anti-slavery reform. I took right hold of the cause. I could do but little; but what I could, I did with a joyful heart, and never felt happier than when in an anti-slavery meeting. I seldom had much to say at the meetings, because what I wanted to say was said so much better by others. But, while attending an anti-slavery convention at Nantucket, on the 11th of August, 1841, I felt strongly moved to speak, and was at the same time much urged to do so by Mr. William C. Coffin, a gentleman who had heard me speak in the colored people’s meeting at New Bedford. It was a severe cross, and I took it up reluctantly. The truth was, I felt myself a slave, and the idea of speaking to white people weighed me down. I spoke but a few moments, when I felt a degree of freedom, and said what I desired with considerable ease. From that time until now, I have been engaged in pleading the cause of my brethren—with what success, and with what devotion, I leave those acquainted with my labors to decide.


7.2: Claims on Population Means
- Test claims on population means both when \(\sigma\) is known and when \(\sigma\) is unknown
- Generalize the three forms of hypothesis tests
- Introduce, motivate, and utilize test statistics in computing \(p\)-values
Section 7.2 Excel File: (contains all of the data sets for this section)
Means and Hypothesis Testing
Now that we have been introduced to the general logic of hypothesis testing, we will begin to address the particulars found within hypothesis testing based on the parameters of interest. We begin with testing hypotheses about the population mean. Just like our considerations of confidence intervals for means, we will have two cases to consider based on whether the population standard deviation is known or unknown. The latter case is the more frequently occurring case as we discussed in the chapter on confidence intervals, but we again begin with the case that the population standard deviation is known because of pedagogical considerations.
Recall the general process of hypothesis testing. A claim is made that warrants testing. This hypothesis may be derived from simple observation, past experimental data, a person, or an institution, and it tends to be the alternative hypothesis. A competing hypothesis, which tends to be the null hypothesis, is constructed. The statement of the alternative hypothesis determines the type of test to be conducted (either a one-tailed test or a two-tailed test). At this point, researchers generally settle on the level of the test (the probability of a type I error given that the null hypothesis is true). The next step in the process is designing the experiment to ensure that the test can actually be conducted. The calculation of the \(p\)-value depends on the sampling distribution of the sample statistics used to estimate the population parameter, assuming the null hypothesis is true. We need to ensure certain conditions are met. When testing claims on population means, we utilize the sampling distribution of sample means, which is normal when the underlying population is normal, and approximately normal for the most common distributions (recall our previous discussion on sampling distributions) when the sample size is larger than \(30.\) We also need to ensure that our sample is randomly selected. Once we have designed the experiment, it must be conducted and analyzed. We then compute the \(p\)-value, the probability that at least as extreme as what was observed happens. If the \(p\)-value \(<\) \(\alpha\) value, then we conclude that there is sufficient evidence to reject the null hypothesis. If the \(p\)-value \(\ge\) \(\alpha\) value, then we conclude that there is not sufficient evidence to reject the null and hence fail to reject the null hypothesis. When sharing the results of the hypothesis test, include the \(p\)-value so others can also assess at the desired level of significance. With this succinct review, let us enter into testing claims on population means when \(\sigma\) is known.
Claims on Population Means (\(\sigma\) known)
A bottling company is responsible for bottling \(2\) liter bottles of Dr. Pepper. The company policy regarding quality assurance is required to randomly sample \(100\) bottles each week to assess how well the bottles are being filled. The company assesses the test at a significance level of \(0.01.\) If the company that built the bottling equipment guarantees that the machinery operates with a standard deviation of \(0.1\) liters and the last sample of \(100\) randomly chosen bottles had a sample mean of \(1.98\) liters, determine the hypotheses, make a conclusion regarding the test, and interpret the meaning within the context of the problem.
- Answer
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Since we are dealing with the amount of Dr. Pepper filled in the \(2\) liter bottles, we are dealing with population means. Each bottle has a certain amount of Dr. Pepper. Each bottle is supposed to have \(2\) liters of soda. So the population mean should be \(2\) liters. One hypothesis would be \(\mu\) \(=2\) \(\text{liters}.\) The bottling company wants to make sure that it is not overfilling (it does not want to shrink its profit margin) nor underfilling (it does not want to upset its customers and reface any false advertising lawsuits). We conclude that the other hypothesis would be \(\mu\) \(\ne 2\) \(\text{liters}.\)
We now need to determine which hypothesis is to be the null hypothesis. If the company assumes that the machinery is not filling properly from the outset, it would automatically recalibrate the machinery every time; the collecting of evidence would be unnecessary. If the company acts as if the machines are not working properly when they really are, the company would unnecessarily waste production time. On the other hand, if the company acts as if the machines are filling properly when they really are not, the company would produce bottles without the proper amount in them. If it was enough that customers would notice, the employees would likely notice before shipping them out. We, therefore, set the hypotheses as follows.\[\begin{align*}H_0&:\mu=2\text{ liters}\\H_1&:\mu\ne 2\text{ liters}\end{align*}\]Since we had a random sample of \(100\) bottles, we have that the sampling distribution of sample means is approximately normal. Since we assume that the null hypothesis is true for the computation of the \(p\)-value, we have that the mean of the sampling distribution of sample means is \(\mu_{\bar{x}}\) \(=2\) liters. We also have that the standard deviation of the sampling distribution of sample means is \(\sigma_{\bar{x}}=\frac{0.1}{\sqrt{100}}\) \(=\frac{0.1}{10}\) \(=0.01.\) Since our alternative hypothesis is that the mean is not equal to \(2\) liters, we are looking for evidence in two directions; we have a two-tailed test. Recall that the sample mean was \(1.98\) liters, which is less than \(2\) liters. We need to find the value that would be just as extreme except in the opposite direction. Since \(1.98\) liters is \(0.02\) liters below the hypothesized value, the other value we are looking for is \(0.02\) liters above the hypothesized value, namely \(2.02\) liters. To find the \(p\)-value, we compute the area in the left tail ending at \(1.98\) liters and the right tail starting at \(2.02\) liters. See the figure below for a visual.
Figure \(\PageIndex{1}\): Sampling distribution of sample means
We compute the \(p\)-value using technology. Note that since the sampling distribution of sample means is approximately normal, the area in the two tails is equal due to symmetry. This eases the calculation. For this first exercise, we compute it both ways.\[\begin{align*}p\text{-value}&=\text{NORM.DIST}(1.98,2,0.01,1)+(1-\text{NORM.DIST}(2.02,2,0.01,1))\\&\approx 0.02275+(1-0.97725)\\&\approx0.0455\\p\text{-value}&=2\cdot\text{NORM.DIST}(1.98,2,0.01,1)\\&\approx 2\cdot 0.02275\\&\approx0.0455\end{align*}\]
Since \(0.0455\) is not less than \(0.01,\) we do not have sufficient evidence to reject the null hypothesis that the machines are filling the \(2\) liter bottles properly. The machines, therefore, pass the weekly test for quality assurance.
In \(2024,\) researchers at the University of Maryland edited the genes of poplar trees to reduce the amount of lignin naturally present in the tree. This is desirable because the process of strengthening wood involves heat and compression, and the more lignin present in the tree, the harder the tree is to compress. The amount of lignin present in a poplar tree is often reported as a percent of the tree's dry weight. The average poplar tree has \(27\%\) of its dry weight due to the weight of lignin.
Suppose that independent researchers want to verify the claim. They request to randomly sample \(45\) of the many thousands of genetically altered poplar trees growing across the various stations the University of Maryland researchers utilize. For the sake of open and honest scientific research, their requests are granted. If it is known that the standard deviation of the percent of weight of lignin in all poplars, including this genetically altered poplar, is \(4.2\%\) and the sample mean was \(25.3\%,\) conduct the hypothesis test at the \(0.005\) level of significance.
- Answer
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The measure of interest for each tree is the percent of dry weight that is due to the presence of lignin. Despite the fact that this measurement returns a percent, we are not considering proportions as our parameter of interest. We are interested in the population mean of the percent weight due to lignin measurements in the genetically altered poplar trees. It is known that the typical poplar tree has \(27\%\) of its dry weight due to the weight of lignin, and the researchers at the University of Maryland think that they have reduced the amount of lignin naturally present in the genetically altered poplar tree. We naturally obtain the hypothesis that \(\mu\) \(<27\) \(\text{percent}.\) From here, we identify the opposing hypothesis that \(\mu\) \(\ge 27\) \(\text{percent}.\) Note that this is a one-tailed test. We conduct a one-tailed test because regardless of whether the genetically altered poplar trees have the same amount of lignin or more lignin than the regular poplar trees, interest in these genetically altered trees would fade. There is no need to distinguish between no change and a change for the worse.
To select which hypothesis is to be considered the null hypothesis, we note that we are conducting the experiment to test that the claims of the University of Maryland researchers are true; so, we do not want to assume their conclusion from the beginning. We will set the null hypothesis to say that the genetically altered poplars have at least as much lignin present as a percent of their dry weights as regular poplar trees.\[\begin{align*}H_0&:\mu\ge 27\text{ percent}\\H_1&:\mu< 27\text{ percent}\end{align*}\]We now begin to look at the sampling distribution of sample means given the size of the random sample and the assumption that the null hypothesis is true. Since the sample size is \(45,\) we expect the sampling distribution of sample means to be approximately normal. Since there are many possible population means under the assumption that the null hypothesis is true, we conduct our study with the value of \(\mu\) that will produce the largest \(p\)-value. This occurs when \(\mu\) \(=27\) \(\text{percent}.\)
We have \(\mu_{\bar{x}}\) \(=27\) \(\text{percent}\) and \(\sigma_{\bar{x}}=\frac{4.2}{\sqrt{45}}\) \(\approx 0.6261\) \(\text{percent}\). We need to determine what would be considered at least as extreme as the evidence from the sample which produced a sample mean of \(25.3\) \(\text{percent}\). In this case, the more extreme would be smaller and smaller percentages. So we are looking for the area in the tail on the left side of the sampling distribution that ends at \(25.3\) \(\text{percent}\). Notice how the direction of the tail matches the direction of the inequality in the alternative hypothesis; we call this a left-tailed test. See the figure below for a visualization.
Figure \(\PageIndex{2}\): Sampling distribution of sample means
We thus compute the \(p\)-value using technology.\[p\text{-value}=\text{NORM.DIST}(25.3,27,\frac{4.2}{\sqrt{45}},1)\approx 0.0033\nonumber\]Since \(0.0033\) \(<0.005,\) there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis that the genetically altered popular trees have less lignin naturally present. There is sufficient evidence in support of the claims of the researchers at the University of Maryland.
At this stage, we have conducted both types of one-tailed tests: right-tailed (recall the last section with corn yields) and left-tailed (this section with lignin). When we operated under the assumption of the truth of the null hypothesis, we had to decide which value of \(\mu\) to use in computing the \(p\)-value. We chose the value that would be the hardest to find sufficient evidence (the value that would produce the largest \(p\)-value). This turned out to be when \(\mu\) equaled the common or accepted value from the problem (the \(70\) bushels per acre of the commonly used corn and the \(27\%\) of the regular poplar trees). This will always be the case and contributes to some textbooks making the pedagogical choice to say that the null hypothesis is always of the form: the parameter equals the standard value. We do not adopt this simplification for the form of the null hypothesis, but we do emphasize that when the parameter equals the standard value, the \(p\)-value produced will be the largest, which is essential for our analyses.
We also want to emphasize that there are three tests to consider: left-tailed tests, right-tailed tests, and two-tailed tests. These three tests correspond to the three possible forms of hypotheses. Consider looking at the alternative hypotheses. In this last text exercise about poplar trees, the alternative hypothesis was \(\mu\) \(<27\) \(\text{percent},\) and we computed the \(p\)-value by calculating the area in the left tail of the distribution. In the text exercise about Dr. Pepper, the alternative hypothesis was \(\mu\) \(\ne 2\) \(\text{liters},\) and we considered the area in both tails. And finally, in the exercise about corn yields, the alternative hypothesis was \(\mu\) \(>70\) \(\text{bushels}\) \(\text{per}\) \(\text{acre},\) and we looked only at the area in the right tail. Notice that the direction of the tail matches the direction of the inequality sign in the alternative hypothesis. All of this is provided in the figure below within the context of means but applies to any parameter (\(\mu_0\) indicates the common or accepted value of the population mean). \begin{array}{|c|c|c|} \hline \text{left-tailed test}&\text{two-tailed test}&\text{right-tailed test} \\ \hline H_0: \mu\ge\mu_0 &H_0: \mu=\mu_0 &H_0: \mu\le\mu_0 \\ H_1: \mu<\mu_0 &H_1: \mu\ne\mu_0 &H_1: \mu>\mu_0 \\ \hline \text{use } \mu=\mu_0 \text{ to compute } p\text{-value} &\text{use } \mu=\mu_0 \text{ to compute } p\text{-value}&\text{use } \mu=\mu_0 \text{ to compute } p\text{-value} \\ \hline \nonumber \end{array}
Test Statistics
Recall that we can transform any normal distribution into the standard normal distribution and that this transformation preserves area. An implication of these facts is that the computation of \(p\)-values can be done within the context of the standard normal distribution once we transform the particular sample mean. We call this transformed value of the calculated sample statistic the test statistic . As discussed, different transformations can change sampling distributions of particular sample statistics to particular common distributions. For now, understand that the basic idea of the test statistic is that it is a value that represents the value of the sample statistic computed from the actual sample collected which facilitates computing the \(p\)-value.
Recall that the \(z\)-score transformation sends any normal distribution with a mean \(\mu\) and a standard deviation \(\sigma\) to the standard normal distribution given by the formula below.\[z=\frac{x-\mu}{\sigma}\nonumber\]
Repeat the hypothesis tests from Text Exercises \(7.2.1\) and \(7.2.2\) using test statistics to compute the \(p\)-values. Verify that the same \(p\)-values are computed which in turn yield the same conclusions as before.
- The first text exercise considered filling \(2\) liter bottles of Dr. Pepper. A sample of \(100\) \(2\) liters was randomly chosen which produced a sample mean of \(1.98\) liters. The population standard deviation was \(0.1\) liters. The hypothesis test was to be conducted on the hypotheses below at the \(\alpha=0.01\) level of significance.\[\begin{align*}H_0&:\mu=2\text{ liters}\\H_1&:\mu\ne 2\text{ liters}\end{align*}\]
- Answer
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All conditions to conduct a hypothesis test are met; for details, review Text Exercise \(7.2.1.\) We need to compute our test statistic. We assume that the null hypothesis is true and, therefore, know that the sampling distribution is approximately normal with \(\mu_{\bar{x}}\) \(=2\) liters and \(\sigma_{\bar{x}}\) \(=\frac{0.1}{\sqrt{100}}\) \(=0.01\) liters. Since we are transforming the sampling distribution of sample means with \(\sigma\) known into the standard normal distribution our \(z\)-transformation takes on the form below.\[z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}=\frac{\bar{x}-\mu_0}{\frac{\sigma}{\sqrt{n}}}\nonumber\]We can insert the values from our particular context to arrive at the following.\[z=\frac{1.98-2}{\frac{0.1}{\sqrt{100}}}=\frac{-0.02}{0.01}=-2\nonumber\] Understand the \(-2\) value to mean that under the assumption of the truth of the null hypothesis the evidence that we collected from the sample mean is \(2\) standard deviations below the hypothesized population mean.
The alternative hypothesis contains the \(\ne\) sign implying a two-tailed test. We need to consider what is equally extreme in the opposite direction. Since the standard normal distribution is centered at \(0,\) all we have to do is take the value equal in magnitude and opposite in sign, \(2.\) Two values are equally extreme if they are the same number of standard deviations away from the population mean. We have the following visualization for computing the \(p\)-value.
Figure \(\PageIndex{3}\): Standard normal distribution
\[\begin{align*}p\text{-value}&=\text{NORM.S.DIST}(-2,1)+(1-\text{NORM.S.DIST}(2,1))\\&\approx 0.02275+(1-0.97725)\\&\approx0.0455\\p\text{-value}&=2\cdot\text{NORM.S.DIST}(-2,1)\\&\approx 2\cdot 0.02275\\&\approx0.0455\end{align*}\]We arrive at the same \(p\)-value as before and since \(0.0455\) is not less than \(0.01,\) we fail to reject the null hypothesis.
- The second text exercise considered the percent weight of poplar trees due to lignin. A sample of \(45\) genetically altered poplar trees was randomly chosen, producing a sample mean of \(25.3\) percent. The population standard deviation was \(4.2\) percent. The hypothesis test was conducted on the hypotheses below at the \(\alpha=0.005\) significance level.\[\begin{align*}H_0&:\mu\ge 27\text{ percent}\\H_1&:\mu< 27\text{ percent}\end{align*}\]
- Answer
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All of the conditions to conduct a hypothesis test are met; for details, review Text Exercise \(7.2.2.\) We need to compute our test statistic. We assume that the null hypothesis is true and, therefore, know that the sampling distribution is approximately normal with \(\mu_{\bar{x}}\) \(=27\) percent and \(\sigma_{\bar{x}}\) \(=\frac{4.2}{\sqrt{45}}\) \(\approx 0.6261\) percent. We can compute our test statistic.\[z=\frac{25.3-27}{\frac{4.2}{\sqrt{45}}}\approx\frac{-1.7}{0.6261}\approx-2.7152\nonumber\]We can understand the \(-2.7152\) value to mean that assuming the truth of the null hypothesis the evidence that we collected from the sample mean is \(2.7152\) standard deviations below the hypothesized population mean.
Since the alternative hypothesis contains the \(<\) sign, this hypothesis test is a one-tailed test, and more extreme values would be values to the left, resulting in a left-tailed test. We have the following visualization for computing the \(p\)-value.
Figure \(\PageIndex{4}\): Standard normal distribution
\[p\text{-value}=\text{NORM.S.DIST}\left(\frac{-1.7}{\frac{4.2}{\sqrt{45}}},1\right)\approx \text{NORM.S.DIST}\left(-2.7152,1\right)\approx 0.0033\nonumber\]We again produce the same \(p\)-value which yields that there is sufficient evidence to reject the null hypothesis in favor of concluding the alternative hypothesis: the genetically altered popular trees have less lignin naturally present.
Claims on Population Means (\(\sigma\) unknown)
We are now prepared to move to the more common situation: testing hypotheses about population means when the population standard deviation is unknown. The added complication comes from the fact that we do not know the standard deviation of the sampling distribution. We can estimate the population standard deviation using the sample standard deviation from our collected sample, but using this estimate has ramifications.
Recall constructing confidence intervals for population means when \(\sigma\) was unknown, we considered what happens under the \(t\)-transformation.\[t=\frac{\bar{x}-\mu_{\bar{x}}}{\frac{s}{\sqrt{n}}}\nonumber\]We concluded that the \(t\) variable followed a particular distribution, the Student's \(t\)-distribution with \(n-1\) degrees of freedom. Notice how similar the formula for the variable \(t\) is to the formula for calculating the test statistic when \(\sigma\) is known. The only difference is that one formula has an \(s\) while the other has a \(\sigma\). Just as the \(z\)-score transformation provided the formula to calculate the test statistic when \(\sigma\) is known, the \(t\)-transformation provides the formula we use to compute the test statistic when \(\sigma\) is unknown. We, therefore, know the distribution of test statistics when \(\sigma\) is unknown is the \(t\)-distribution with \(n-1\) degrees of freedom. We use this fact to compute the \(p\)-value for testing claims on population means when \(\sigma\) is unknown. For a refresher on the \(t\)-distribution see Section \(6.4.\) The two processes for testing hypotheses about population means are very similar. The main difference is the distribution in which we calculate the \(p\)-value. When \(\sigma\) is known, we use the standard normal distribution. When \(\sigma\) is unknown, we use the \(t\)-distribution with \(n-1\) degrees of freedom.
Before we test hypotheses about population means with \(\sigma\) unknown, let us review the overall process of hypothesis testing to reinforce the procedure and highlight the distinctions between various situations.
- Use natural observation, previous experimental results, or the claims of others to formulate a hypothesis that warrants testing. Within the context of means, each observation must admit some quantitative fact that can be measured and averaged. This excludes considerations of whether or not observations have a particular quality; that will be studied in the section on claims on population proportions.
- Identify a competing hypothesis and consider the ramifications of acting as if one of the hypotheses is true when, in fact, it is not. Name the hypothesis with the less drastic ramifications as the null hypothesis. The novel or claimed hypothesis is generally the alternative hypothesis. See note in the previous section regarding other ways to help distinguish between null and alternative hypotheses.
- Determine the methodology of collecting evidence against the null hypothesis and determine what constitutes sufficient evidence by setting the level of significance. Make sure the design meets the requirements of the test intended to be conducted. For claims on population means, ensure that the sample is randomly selected and that either the underlying population is normally distributed or that the sample is large enough that the sampling distribution of sample means is approximately normal. In most cases, \(n>30\) will be sufficient.
- Conduct the experiment and collect the evidence.
- Compute the test statistic. Be sure to make the distinction between sample and population standard deviations. The most common situation is that we only have access to the sample standard deviation \(s\) and, therefore, must use the \(t\)-transformation to compute our test statistic. We often denote the test statistic based on which transformation is used. If the population standard deviation is known, the \(z\)-score transformation is used, and the test statistic is denoted with a \(z.\) If the population standard deviation is unknown, the \(t\)-transformation is used and the test statistic is denoted with a \(t.\)
- Use the hypotheses to determine whether a test is a left-tailed, right-tailed, or two-tailed test. Note that the directions match the sign in the alternative hypothesis.
- Determine the \(p\)-value by considering the test statistic, the appropriate distribution, and the type of test and then using technology to make an appropriate calculation.
- Compare the \(p\)-value to the \(\alpha\) value. If the \(p\)-value \(<\alpha\) value, then we reject the null hypothesis in favor of the alternative hypothesis. If the \(p\)-value \(\ge\alpha\) value, we fail to reject the null hypothesis.
A guest speaker at a local library presented on the change in human physical characteristics over the last two centuries in the United States. The presenter claimed that male height has consistently increased over that time and will continue to do so. The last evidence cited in this regard was in \(1970,\) stating that the average height of adult males was \(176\) centimeters. Given the span of over \(50\) years, we decide to test the hypothesis that the average height of adult males in \(2024\) has increased since \(1970.\)
Suppose we collect a sample of \(16\) adult males randomly selected and measure their heights in centimeters. The data is presented below. We decide to conduct the hypothesis test at an \(\alpha\) value of \(0.05.\)\[169,171,171.5,173,173,174.25,174.5,175,176,177,177.75,178,178,179,179.5,180\nonumber\]
- Answer
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We must confirm that our circumstances enable a hypothesis test to be conducted; we need a random sample and reasonable confidence that the shape of the sampling distribution of sampling means is approximately normal. The first component, the random sample, is easily confirmed. The sample size chosen is \(16\), which does not meet the typical threshold of more than \(30.\). We must recall that male and female adult height are normally distributed. We, therefore, have that the sampling distribution of sample means is normally distributed. We can conduct the test. Note that no population standard deviations are given. We must utilize the recently discussed method that involves the \(t\)-distribution!
The guest speaker made the claim that the average height of adult males has increased over time. The population mean was \(176\) centimeters back in \(1970;\) so, we identify one of the hypotheses as \(\mu>176\) centimeters. The opposite hypothesis would thus be \(\mu\leq 176\) centimeters. This is a one-tailed test because both the average being the same or smaller than before are equally detrimental to the claims of the guest speaker. In both cases, the ramifications of acting as if one hypothesis is true when it is false seem to be mild. So, we pick the null hypothesis to be the one contrary to the claimed hypothesis. We settle on the hypotheses as follows. \[\begin{align*}H_0&:\mu\leq 176\text{ centimeters}\\H_1&:\mu>176\text{ centimeters}\end{align*}\]
Given the hypotheses, we have a right-tailed test. We compute our test statistic.\[t=\frac{\bar{x}-\mu_{0}}{\frac{s}{\sqrt{n}}}\nonumber\]To do this, we must compute the sample mean and sample standard deviation using the values collected from our random sample. We produce the following results: \(n=16,\) \(\bar{x}\approx 175.4063\) centimeters, and \(s\approx 3.2887\) centimeters.
At this stage, we notice that our sample mean is less than the hypothesized population mean. Since we have a right-tailed test, we are looking for evidence against the null hypothesis in the form of sample means larger than the hypothesized value. We can thus immediately conclude that there is not sufficient evidence to reject the null hypothesis. We will show the remainder of the computation to solidify the process and strengthen our conclusion.\[t\approx\frac{175.4063-176}{\frac{3.2887}{\sqrt{16}}}\approx-0.7222\nonumber\]
Figure \(\PageIndex{5}\): Right-tailed test with \(t=-0.7222\) using \(t\)-distribution with \(15\) degrees of freedom
As we can see, the shaded area is over half of the area due to the symmetry of the \(t\)-distribution. We now compute the \(p\)-value using technology.\[ p\text{-value} \approx 1-\text{T.DIST}(-0.7222,15,1)\approx 1-0.2406\approx 0.7594\nonumber\]The \(p\)-value is larger than the \(\alpha\) value of \(0.05\). We conclude the test by failing to reject the null hypothesis. There is not sufficient evidence to support the guest speaker's claim.
It is still possible that the average height of adult males has increased, and something rare occurred in the act of sampling. It is also possible that the average height is the same as it was. Based on the evidence, we may also be open to the idea that the average height may be smaller. Reviewing the guest speaker's reasoning behind why the heights have been increasing may be prudent. Is there a faulty assumption? Is there an explanation as to why it may have been increasing and now possibly decreasing? Perhaps we can conduct another random experiment to test whether the mean is now less than it was before. Hypothesis tests that fail to reject the null hypothesis can still inform further research and inquiry.
A study published in \(2021,\) concluded that the average weekly recreational screen time of \(18-29\) year olds (emerging adults) increased from \(2018\) to \(2020\) during the pandemic estimating the average weekly recreational screen time with the confidence interval \(28.5\pm11.6\) hours. Recreational screen time does not include screen time associated with work or school.
The pandemic is now behind us, but the effects of the pandemic are still playing out. A researcher is still interested in the weekly recreational screen time of emerging adults and conducts a study on \(56\) randomly selected emerging adults with a sample mean of \(30.4\) hours and a sample standard deviation of \(6.3\) hours. The researcher adopts the following hypotheses to be tested at the \(0.05\) level of significance. Conduct the test.\[\begin{align*}H_0&:\mu= 28.5\text{ hours}\\H_1&:\mu\ne28.5\text{ hours}\end{align*}\]Note that this sample data is completely fabricated.
- Answer
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Having been given the hypotheses, we note that we are conducting a two-tailed test. The researcher adopted the central value of confidence interval from the study to compare the current data. We do not know the population standard deviation and thus operate within the realm of the \(t\)-transformation and \(t\)-distribution.\[t=\frac{30.4-28.5}{\frac{6.3}{\sqrt{56}}}\approx2.2569\nonumber\]Since we have a two-tailed test, we determine the test-statistic that is equally as extreme as the computed test statistic in the opposite direction. Again due to symmetry, this is the value of equal magnitude but opposite sign. We have the following visualization for computing the \(p\)-value.
Figure \(\PageIndex{6}\): Two-tailed test with \(t=\pm 2.2569\) using \(t\)-distribution with \(55\) degrees of freedom
\[p\text{-value} =2\cdot\text{T.DIST}(-2.2569,55,1)\approx 2\cdot0.0140\approx 0.0280\nonumber\]We again used symmetry, noting that the boundaries of the tails are equidistant from \(0,\) so that we can double the area found in one of the tails. The left tail can be computed directly using the negative value of the two test statistics.
We compare \(0.0280\) with \(0.05\) and find that \(0.0280\) \(<0.05.\) We have sufficient evidence to reject the null hypothesis that the average weekly recreational screen time for emerging adults in \(2024\) is \(28.5\) hours and conclude that the average weekly recreational screen time for emerging adults in \(2024\) is not \(28.5\) hours. The evidence indicates that the average may actually be higher, but to reach such a conclusion, another study must be conducted.


Illustration
Learning Objectives
- Determine the purpose and structure of the illustration essay.
- Understand how to write an illustration essay.
THE PURPOSE OF ILLUSTRATION IN WRITING
To illustrate means to show or demonstrate something clearly. An effective illustration essay clearly demonstrates and supports a point through the use of evidence.
The controlling idea of an essay is called a thesis. A writer can use different types of evidence to support his or her thesis. Using scientific studies, experts in a particular field, statistics, historical events, current events, analogies, and personal anecdotes are all ways in which a writer can illustrate a thesis. Ultimately, you want the evidence to help the reader “see” your point, as one would see a good illustration in a magazine or on a website. The stronger your evidence is, the more clearly the reader will consider your point.
Using evidence effectively can be challenging, though. The evidence you choose will usually depend on your subject and who your reader is (your audience). When writing an illustration essay, keep in mind the following:
- Use evidence that is appropriate to your topic as well as appropriate for your audience.
- Assess how much evidence you need to adequately explain your point depending on the complexity of the subject and the knowledge of your audience regarding that subject.
For example, if you were writing about a new communication software and your audience was a group of English-major undergrads, you might want to use an analogy or a personal story to illustrate how the software worked. You might also choose to add a few more pieces of evidence to make sure the audience understands your point. However, if you were writing about the same subject and you audience members were information technology (IT) specialists, you would likely use more technical evidence because they would be familiar with the subject.
Keeping in mind your subject in relation to your audience will increase your chances of effectively illustrating your point.
tip
You never want to insult your readers’ intelligence by overexplaining concepts the audience members may already be familiar with, but it may be necessary to clearly articulate your point. When in doubt, add an extra example to illustrate your idea.
Exercise \(\PageIndex{1}\)
On a separate piece of paper, form a thesis based on each of the following three topics. Then list the types of evidence that would best explain your point for each of the two audiences.
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Topic: Combat and mental health
Audience: family members of veterans, doctors -
Topic: Video games and teen violence
Audience: parents, children -
Topic: Architecture and earthquakes
Audience: engineers, local townspeople
THE STRUCTURE OF AN ILLUSTRATION ESSAY
The controlling idea, or thesis, belongs at the beginning of the essay. Evidence is then presented in the essay’s body paragraphs to support the thesis. You can start supporting your main point with your strongest evidence first, or you can start with evidence of lesser importance and have the essay build to increasingly stronger evidence. This type of organization—order of importance—you learned about in Chapter 8 “The Writing Process: How Do I Begin?” and Chapter 9 “Writing Essays: From Start to Finish”.
The time transition words listed in Table 10.1 “Transition Words and Phrases for Expressing Time” are also helpful in ordering the presentation of evidence. Words like first, second, third, currently, next, and finally all help orient the reader and sequence evidence clearly. Because an illustration essay uses so many examples, it is also helpful to have a list of words and phrases to present each piece of evidence. Table 10.2 “Phrases of Illustration” provides a list of phrases for illustration.
| case in point |
for instance |
specifically |
to illustrate |
|
for example |
in particular |
in this case |
one example/ another example |
tip
Vary the phrases of illustration you use. Do not rely on just one. Variety in choice of words and phrasing is critical when trying to keep readers engaged in your writing and your ideas.
WRITING AT WORK
In the workplace, it is often helpful to keep the phrases of illustration in mind as a way to incorporate them whenever you can. Whether you are writing out directives that colleagues will have to follow or requesting a new product or service from another company, making a conscious effort to incorporate a phrase of illustration will force you to provide examples of what you mean.
Exercise \(\PageIndex{2}\)
On a separate sheet of paper, form a thesis based on one of the following topics. Then support that thesis with three pieces of evidence. Make sure to use a different phrase of illustration to introduce each piece of evidence you choose.
- Cooking
- Baseball
- Work hours
- Exercise
- Traffic
Collaboration
Please share with a classmate and compare your answers. Discuss which topic you like the best or would like to learn more about. Indicate which thesis statement you perceive as the most effective.
WRITING AN ILLUSTRATION ESSAY
First, decide on a topic that you feel interested in writing about. Then create an interesting introduction to engage the reader. The main point, or thesis, should be stated at the end of the introduction.
Gather evidence that is appropriate to both your subject and your audience. You can order the evidence in terms of importance, either from least important to most important or from most important to least important. Be sure to fully explain all of your examples using strong, clear supporting details. See Chapter 15 “Readings: Examples of Essays” to read a sample illustration essay.
Exercise \(\PageIndex{3}\)
On a separate sheet of paper, write a five-paragraph illustration essay. You can choose one of the topics from “Exercise 1” or “Exercise 2”, or you can choose your own.
key takeaways
- An illustration essay clearly explains a main point using evidence.
- When choosing evidence, always gauge whether the evidence is appropriate for the subject as well as the audience.
- Organize the evidence in terms of importance, either from least important to most important or from most important to least important.
- Use time transitions to order evidence.
- Use phrases of illustration to call out examples.
References
- This section was originally from Writing for Success, found at the University of Minnesota open textbook project. Full license information: This is a derivative of Writing for Success by a publisher who has requested that they and the original author not receive attribution, originally released and is used under CC BY-NC-SA. This work, unless otherwise expressly stated, is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.


10.S: Organohalides (Summary)
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Concepts & Vocabulary
10.1 Introduction to Organohalides
- Alkyl halides (and allyl and benzyl halides) are more reactive than vinyl and aryl halides.
10.2 Names and Properties of Alkyl Halides
- Reactivity of alkyl halides is often related to the substitution of the carbon atom the halogen is attached to.
- Alkyl halides are categorized by the number of bonds to other alkyl groups ( primary , secondary , and tertiary ).
- Carbon-halogen bonds are polarized with partial positive charges on carbon and partial negative charges on the halogen.
- Fluorine is the most electronegative of the halogens while iodine is the least electronegative.
- Iodine is the largest of the halogens yielding the longest/weakest bonds to carbon of the halogens.
- Since haloalkanes have dipole-dipole interactions, they have greater intermolecular forces than similar sized alkanes and therefore higher boiling points.
- Alkyl halides are either slightly soluble or insoluble in water, but are soluble in organic solvents.
10.3 Preparing Alkyl Halides from Alkanes - Radical Halogenation
- Halogenation of alkanes is exothermic, so it is energetically favorable.
- Radical chain mechanisms consist of three steps: initiation , propagation and termination .
- Hydrogens on more substituted carbon atoms are more reactive to radical halogenation.
10.4 Preparing Alkyl Halides from Alkenes - Allylic Bromination
- More substituted radicals and radicals with resonance structures are more stable than other radicals.
- Radical substitution can be carried out at the allylic or benzylic carbon by reacting with NBS.
10.5 Stability of the Allyl Radical - Resonance Revisited
- Allyl cations, anions and radicals have resonance structures. To draw these resonance structures non-bonded and pi-bond electrons can be moved.
- Resonance hybrids are used to show the combination of all resonance structures for a molecule or ion.
10.6 Preparing Alkyl Halides form Alcohols
- Alcohols can be reacted with hydrohalogen acids or a mixture of halogen salts and a stronger acid (to form hydrohalogen acids in situ ).
- Alcohols will also react with thionyl chloride or with phosphorus halides to from haloalkanes.
10.7 Reactions of Alkyl Halides - Grignard Reactions
- Organometallic reagents can be formed from alkyl halides and reactive metals (such as lithium and magnesium).
- Alkyl magnesium halide compounds are callled Grignard reagents.
- Grignard reagents react as bases where the alkyl group gets protonated and the metal complexes to the conjugate base of the reacting acid.
10.8 Organometallic Coupling Reactions
- Lithium dialkyl copper compounds are called Gilman reagents.
- Gilman reagents have different reactivity from the other organometallics (lithium and Grignard reagents).
- Organometallics can be reacted with alkyl halides to join to alkyl groups (coupling reactions).
10.9 Oxidation and Reduction in Organic Chemistry
- Gaining bonds to hydrogen for organic molecules is reduction.
- Losing bonds to hydrogen for organic molecules is oxidation.
Skills to Master
- Skill 10.1 Differentiate between types of halides (alkyl, allyl, aryl, benzyl, and vinyl).
- Skill 10.2 Differentiate between substitution of alkyl halides (primary, secondary, and tertiary).
- Skill 10.3 Identify relative reactivity of carbon-hydrogen bonds to radical halogenation.
- Skill 10.4 Draw resonance structures for radical compounds.
- Skill 10.5 Draw mechanisms for radical halogenation of alkanes (initiation, propagation and termination).
- Skill 10.6 Calculate the enthalpy change of a reaction using bond dissociation energies of reactants and products.
- Skill 10.7 Determine products for allylic bromination reactions.
- Skill 10.8 Draw resonance structures for allylic and other similar compounds and ions.
- Skill 10.9 Draw products of reactions of alcohols to form alkyl halides.
- Skill 10.10 Write equations to form Grignard reagents from alkyl halides.
- Skill 10.11 Draw reaction products for Grignard reagents acting as bases.
- Skill 10.12 Write equations for the formation of Gilman reagents.
- Skill 10.13 Draw reaction products of organometallic coupling reactions.
- Skill 10.14 Explain oxidation and reduction in organic molecules.