8.7: Hypothesis Testing (1 of 5)
Learning Objectives
- When testing a claim, distinguish among situations involving one population mean, one population proportion, two population means, or two population proportions.
- Given a claim about a population, determine null and alternative hypotheses.
Introduction
In inference, we use a sample to draw a conclusion about a population. Two types of inference are the focus of our work in this course:
- Estimate a population parameter with a confidence interval.
- Test a claim about a population parameter with a hypothesis test.
We can also use samples from two populations to compare those populations. In this situation, the two types of inference focus on differences in the parameters.
- Estimate a difference in population parameters with a confidence interval.
- Test a claim about a difference in population parameters with a hypothesis test.
In “Estimating a Population Proportion,” we learned to estimate a population proportion using a confidence interval. For example, we estimated the proportion of all Tallahassee Community College students who are female and the proportion of all American adults who used the Internet to obtain medical information in the previous month. We will revisit confidence intervals in future modules.
Now we look more carefully at how to test a claim with a hypothesis test. Statistical investigations begin with research questions. We begin our discussion of hypothesis tests with research questions that require us to test a claim. Later we look at how a claim becomes a hypothesis.
Example
Research Questions about Testing Claims
Let’s revisit some of the research questions from examples in the module Types of Statistical Studies and Producing Data that involve testing a claim.
Is the average course load for community college students less than 12 semester hours? This question contains a claim about a population mean. The question contains information about the population, the variable, and the parameter. The population is all community college students. The variable is course load in semester hours . It is quantitative, so the parameter is a mean. The claim is, “The mean course load for all community college students is less than 12 semester hours.”
Do the majority of community college students qualify for federal student loans? This question contains a claim about a population proportion and information about the population, the variable, and the parameter. The population is all community college students. The variable is Qualify for federal student loan (yes or no). It is categorical, so the parameter is a proportion. The claim is, “The proportion of community college students who qualify is greater than 0.5” (a majority means more than half, or 0.5).
In community colleges, do female students and male students have different mean GPAs? This question contains a claim that compares two population means. Again, we see information about the populations, the variable, and the parameters. The two populations are female community college students and male community college students. The variable is GPA . It is quantitative, so the parameters are means. The claim is, “The mean GPA for female community college students is different from the mean GPA for male community college students.” Notice that the claim compares the two population means, but there is no claim about the numeric value of either mean.
Are college athletes more likely than nonathletes to receive academic advising? This question contains a claim that compares two population proportions: college athletes and college students who are not athletes. The variable is Receive academic advising (yes or no). The variable is categorical, so the parameters are proportions. The claim is, “The proportion of all college athletes who receive academic advising is greater than the proportion of all nonathletes in college who receive academic advising.” Notice that the claim compares two population proportions, but there is no claim about the numeric value of either proportion.
In the case of testing a claim about a single population parameter, we compare it to a numeric value. In the case of testing a claim about two population parameters, we compare them to each other.
Try It
Identify the type of claim in each research question below.
https://assessments.lumenlearning.co...sessments/3960
https://assessments.lumenlearning.co...sessments/3961
Next Steps: Forming Hypotheses
We already know that in inference we use a sample to draw a conclusion about a population. If the research question contains a claim about the population, we translate the claim into two related hypotheses.
The null hypothesis is a hypothesis about the value of the parameter. The null hypothesis relates to our work in Linking Probability to Statistical Inference where we drew a conclusion about a population parameter on the basis of the sampling distribution. We started with an assumption about the value of the parameter, then used a simulation to simulate the selection of random samples from a population with this parameter value. Or we used the parameter value in a mathematical model to describe the center and spread of the sampling distribution. The null hypothesis gives the value of the parameter that we will use to create the sampling distribution. In this way, the null hypothesis states what we assume to be true about the population.
The alternative hypothesis usually reflects the claim in the research question about the value of the parameter. The alternative hypothesis says the parameter is “greater than” or “less than” or “not equal to” the value we assume to true in the null hypothesis.
Example
Stating Hypotheses
Here are the hypotheses for the research questions from the previous example. The null hypothesis is abbreviated H 0 . The alternative hypothesis is abbreviated H a .
Is the average course load for community college students less than 12 semester hours?
- H 0 : The mean course load for community college students is equal to 12 semester hours.
- H a : The mean course load for community college students is less than 12 semester hours.
Do the majority of community college students qualify for federal student loans?
- H 0 : The proportion of community college students who qualify for federal student loans is 0.5.
- H a : The proportion of community college students who qualify for federal student loans is greater than 0.5.
When the research question contains a claim that compares two populations, the null hypothesis states that the parameters are equal. We will see in Modules 9 and 10 that we translate the null hypothesis into a statement about “no difference” in parameter values. We revisit this idea in more depth later.
In community colleges, do female students and male students have different mean GPAs?
- H 0 : In community colleges, female and male students have the same mean GPAs.
- H a : In community colleges, female and male students have different mean GPAs.
Are college athletes more likely than nonathletes to receive academic advising?
- H 0 : In colleges, the proportion of athletes who receive academic advising is equal to the proportion of nonathletes who receive academic advising.
- H a : In colleges, the proportion of athletes who receive academic advising is greater than the proportion of nonathletes who receive academic advising.
Comment
Here are some general observations about null and alternative hypotheses.
- The hypotheses are competing claims about the parameter or about the comparison of parameters.
- Both hypotheses are statements about the same population parameter or same two population parameters.
- The null hypothesis contains an equal sign.
- The alternative hypothesis is always an inequality statement. It contains a “less than” or a “greater than” or a “not equal to” symbol.
- In a statistical investigation, we determine the research question, and thus the hypotheses, before we collect data.
The process of forming hypotheses, collecting data, and using the data to draw a conclusion about the hypotheses is called hypothesis testing .
Contributors and Attributions
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


3.15: Scatterplots (2 of 5)
Learning Objectives
- Use a scatterplot to display the relationship between two quantitative variables. Describe the overall pattern (form, direction, and strength) and striking deviations from the pattern.
Interpreting the Scatterplot
How do we describe the relationship between two quantitative variables using a scatterplot? We describe the overall pattern and deviations from that pattern.
This is the same way we described the distribution of one quantitative variable using a dotplot or a histogram in Summarizing Data Graphically and Numerically . To describe the overall pattern of the distribution of one quantitative variable, we describe the shape, center, and spread. We also describe deviations from the pattern (outliers).
Similarly, in a scatterplot, we describe the overall pattern with descriptions of direction , form , and strength . Deviations from the pattern are still called outliers.
-
The direction of the relationship can be positive, negative, or neither:
A
positive (or increasing) relationship
means that an increase in one of the variables is associated with an increase in the other.
A negative (or decreasing) relationship means that an increase in one of the variables is associated with a decrease in the other.
Not all relationships can be classified as either positive or negative.
- The form of the relationship is its general shape. To identify the form, describe the shape of the data in the scatterplot. In practice, forms that we commonly use have mathematical equations. We look at a few of these equations in this course. For now, we simply describe the shape of the pattern in the scatterplot. Here are a couple of forms that are quite common: Linear form: The data points appear scattered about a line. We use a line to summarize the pattern in the data. We study the equation for a line in this module. Curvilinear form: The data points appear scattered about a smooth curve. We use a curve to summarize the pattern in the data. We study some specific types of curvilinear forms with their equations in Modules 4 and 12.
- The strength of the relationship is a description of how closely the data follow the form of the relationship. Let’s look, for example, at the following two scatterplots displaying positive, linear relationships: In the top scatterplot, the data points closely follow the linear pattern. This is an example of a strong linear relationship. In the bottom scatterplot, the data points also follow a linear pattern, but the points are not as close to the line. The data is more scattered about the line. This is an example of a weaker linear relationship.Labeling a relationship as strong or weak is not very precise. We develop a more precise way to measure the strength of a relationship shortly.
Outliers are points that deviate from the pattern of the relationship. In the scatterplot below, there is one outlier.
Try It
Fill in the letter of the description that matches each scatterplot.
Descriptions:
A: X = month (January = 1), Y = rainfall (inches) in Napa, CA in 2010 (Note: Napa has rain in the winter months and months with little to no rainfall in summer.)
B: X = month (January = 1), Y = average temperature in Boston MA in 2010 (Note: Boston has cold winters and hot summers.)
C: X = year (in five-year increments from 1970), Y = Medicare costs (in $) (Note: the yearly increase in Medicare costs has gotten bigger and bigger over time.)
D: X = average temperature in Boston MA (°F), Y = average temperature in Boston MA (°C) each month in 2010
E: X = chest girth (cm), Y = shoulder girth (cm) for a sample of men
F: X = engine displacement (liters), Y = city miles per gallon for a sample of cars (Note: engine displacement is roughly a measure of engine size. Large engines use more gas.)
Contributors and Attributions
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


2.12: Pilot Lights
Click play on the following audio player to listen along as you read this section.
Many gas-fired systems incorporate a pilot light to ignite the gas when the furnace is first turned on. A pilot light is a steady flame which is always present. It is fed by a normally-closed valve which is held open by a solenoid coil, and allows a small amount of gas through at all times. When the room thermostat engages the heating system and the main gas valves open, the in-rushing gas is instantly ignited by the pilot light flame.
If the pilot light was to ever extinguish, a danger might be presented if the flow of gas was allowed continue without combusting. It is for this reason that the valve is designed to automatically close if the flame should ever extinguish. To achieve this, a thermocouple is used to keep the valve open.
Spark ignition and electronic ignition, sometimes called a glow coil, are alternative methods of igniting the gas used for heating. All rely on the principle of instantly igniting the gas once the main gas valve opens.


12.6: Solve Exponential and Logarithmic Equations
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Learning Objectives
By the end of this section, you will be able to:
- Solve logarithmic equations using the properties of logarithms
- Solve exponential equations using logarithms
- Use exponential models in applications
Before you get started, take this readiness quiz.
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Solve: \(x^{2}=16\).
If you missed this problem, review Example 6.46. -
Solve: \(x^{2}−5x+6=0\).
If you missed this problem, review Example 6.45. -
Solve: \(x(x+6)=2x+5\).
If you missed this problem, review Example 6.47.
Solve Logarithmic Equations Using the Properties of Logarithms
In the section on logarithmic functions, we solved some equations by rewriting the equation in exponential form. Now that we have the properties of logarithms, we have additional methods we can use to solve logarithmic equations.
If our equation has two logarithms we can use a property that says that if \(\log _{a} M=\log _{a} N\) then it is true that \(M=N\). This is the One-to-One Property of Logarithmic Equations .
Definition \(\PageIndex{1}\)
One-to-One Property of Logarithmic Equations
For \(M>0,N>0,a>0\), and \(a≠1\) is any real number:
If \(\log _{a} M=\log _{a} N,\) then \(M=N\).
To use this property, we must be certain that both sides of the equation are written with the same base.
Remember that logarithms are defined only for positive real numbers. Check your results in the original equation. You may have obtained a result that gives a logarithm of zero or a negative number.
Example \(\PageIndex{1}\)
Solve: \(2 \log _{5} x=\log _{5} 81\).
Solution :
\(2 \log _{5} x=\log _{5} 81\)
Use the Power Property.
\(\log _{5} x^{2}=\log _{5} 81\)
Use the One-to-One Property, if \(\log _{a} M=\log _{a} N\), then \(M=N\).
\(x^{2}=81\)
Solve using the Square Root Property.
\(x=\pm 9\)
We eliminate \(x=-9\) as we cannot take the logarithm of a negative number.
\(x=9, \cancel{x=-9}\)
Check. \(x=9\)
\(\begin{aligned}2 \log _{5} x&=\log _{5} 81 \\ 2 \log _{5} 9 &\stackrel{?}{=} \log _{5} 81 \\ \log _{5} 9^{2} & \stackrel{?}{=}\log _{5} 81 \\ \log _{5} 81 & =\log _{5} 81\end{aligned}\)
Exercise \(\PageIndex{1}\)
Solve: \(2 \log _{3} x=\log _{3} 36\)
- Answer
-
\(x=6\)
Exercise \(\PageIndex{2}\)
Solve: \(3 \log x=\log 64\)
- Answer
-
\(x=4\)
Another strategy to use to solve logarithmic equations is to condense sums or differences into a single logarithm.
Example \(\PageIndex{2}\)
Solve: \(\log _{3} x+\log _{3}(x-8)=2\).
Solution :
\(\log _{3} x+\log _{3}(x-8)=2\)
Use the Product Property, \(\log _{a} M+\log _{a} N=\log _{a} M \cdot N\).
\(\log _{3} x(x-8)=2\)
Rewrite in exponential form.
\(3^{2}=x(x-8)\)
Simplify.
\(9=x^{2}-8 x\)
Subtract \(9\) from each side.
\(0=x^{2}-8 x-9\)
Factor.
\(0=(x-9)(x+1)\)
Use the Zero-Product-Property
\(x-9=0, \quad x+1=0\)
Solve each equation.
\(x=9, \quad \cancel{x=-1}\)
Check. \(x=-1\)
\(\begin{aligned} \log _{3} x+\log _{3}(x-8)&=2 \\ \log _{3}(-1)+\log _{3}(-1-8) &\stackrel{?}{=}2\end{aligned}\)
We cannot take the log of a negative number.
Check. \(x=9\)
\(\begin{aligned} \log _{3} x+\log _{3}(x-8) &=2 \\ \log _{3} 9+\log _{3}(9-8) & \stackrel{?}{=} 2 \\ 2+0 &\stackrel{?}{=}2 \\ 2 &=2 \end{aligned}\)
Exercise \(\PageIndex{3}\)
Solve: \(\log _{2} x+\log _{2}(x-2)=3\)
- Answer
-
\(x=4\)
Exercise \(\PageIndex{4}\)
Solve: \(\log _{2} x+\log _{2}(x-6)=4\)
- Answer
-
\(x=8\)
When there are logarithms on both sides, we condense each side into a single logarithm. Remember to use the Power Property as needed.
Example \(\PageIndex{3}\)
Solve: \(\log _{4}(x+6)-\log _{4}(2 x+5)=-\log _{4} x\).
Solution :
\(\log _{4}(x+6)-\log _{4}(2 x+5)=-\log _{4} x\)
Use the Quotient Property on the left side and the PowerProperty on the right.
\(\log _{4}\left(\frac{x+6}{2 x+5}\right)=\log _{4} x^{-1}\)
Rewrite \(x^{-1}=\frac{1}{x}\).
\(\log _{4}\left(\frac{x+6}{2 x+5}\right)=\log _{4} \frac{1}{x}\)
Use the One-to-One Property, if \(\log _{a} M=\log _{a} N\), then \(M=N\).
\(\frac{x+6}{2 x+5}=\frac{1}{x}\)
Solve the rational equation.
\(x(x+6)=2 x+5\)
Distribute.
\(x^{2}+6 x=2 x+5\)
Write in standard form.
\(x^{2}+4 x-5=0\)
Factor.
\((x+5)(x-1)=0\)
Use the Zero-Product-Property.
\(x+5=0, \quad x-1=0\)
Solve each equation.
\(\cancel{x=-5}, \quad x=1\)
Check.
We leave the check for you.
Exercise \(\PageIndex{5}\)
Solve: \(\log (x+2)-\log (4 x+3)=-\log x\).
- Answer
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\(x=3\)
Exercise \(\PageIndex{6}\)
Solve: \(\log (x-2)-\log (4 x+16)=\log \frac{1}{x}\).
- Answer
-
\(x=8\)
Example \(\PageIndex{4}\) Solve Exponential Equations Using Logarithms
Solve \(5^{x}=11\). Find the exact answer and then approximate it to three decimal places.
Solution :
\(5^{x}=11\)
Since the exponential is isolated, take the logarithm of both sides.
\(\log 5^{x}=\log 11\)
Use the Power Property to get the \(x\) as a factor, not an exponent.
\(x \log 5=\log 11\)
Solve for \(x\). Find the exact answer.
\(x=\frac{\log 11}{\log 5}\)
Approximate the answer.
\(x \approx 1.490\)
Since \(5^{1}=5\) and \(5^{2}=25\), does it makes sense that \(5^{1.490}≈11\)?
Exercise \(\PageIndex{7}\)
Solve \(7^{x}=43\). Find the exact answer and then approximate it to three decimal places.
- Answer
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\(x=\frac{\log 43}{\log 7} \approx 1.933\)
Exercise \(\PageIndex{8}\)
Solve \(8^{x}=98\). Find the exact answer and then approximate it to three decimal places.
- Answer
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\(x=\frac{\log 98}{\log 8} \approx 2.205\)
When we take the logarithm of both sides we will get the same result whether we use the common or the natural logarithm (try using the natural log in the last example. Did you get the same result?) When the exponential has base \(e\), we use the natural logarithm.
Example \(\PageIndex{5}\)
Solve \(3e^{x+2}=24\). Find the exact answer and then approximate it to three decimal places.
Solution :
\(3 e^{x+2}=24\)
Isolate the exponential by dividing both sides by \(3\).
\(e^{x+2}=8\)
Take the natural logarithm of both sides.
\(\ln e^{x+2}=\ln 8\)
Use the Power Property to get the \(x\) as a factor, not an exponent.
\((x+2) \ln e=\ln 8\)
Use the property \(\ln e=1\) to simplify.
\(x+2=\ln 8\)
Solve the equation. Find the exact answer.
\(x=\ln 8-2\)
Approximate the answer.
\(x \approx 0.079\)
Exercise \(\PageIndex{9}\)
Solve \(2e^{x−2}=18\). Find the exact answer and then approximate it to three decimal places.
- Answer
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\(x=\ln 9+2 \approx 4.197\)
Exercise \(\PageIndex{10}\)
Solve \(5e^{2x}=25\). Find the exact answer and then approximate it to three decimal places.
- Answer
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\(x=\frac{\ln 5}{2} \approx 0.805\)
Use Exponential Models in Applications
In previous sections we were able to solve some applications that were modeled with exponential equations. Now that we have so many more options to solve these equations, we are able to solve more applications.
We will again use the Compound Interest Formulas and so we list them here for reference.
Definition \(\PageIndex{2}\)
Compound Interest
For a principal, \(P\), invested at an interest rate, \(r\), for \(t\) years, the new balance, \(A\) is:
\(\begin{array}{ll}{A=P\left(1+\frac{r}{n}\right)^{n t}} & {\text { when compounded } n \text { times a year. }} \\ {A=P e^{r t}} & {\text { when compounded continuously. }}\end{array}\)
Example \(\PageIndex{6}\)
Jermael’s parents put $\(10,000\) in investments for his college expenses on his first birthday. They hope the investments will be worth $\(50,000\) when he turns \(18\). If the interest compounds continuously, approximately what rate of growth will they need to achieve their goal?
Solution :
Identify the variables in the formula.
\(\begin{aligned} A &=\$ 50,000 \\ P &=\$ 10,000 \\ r &=? \\ t &=17 \text { years } \\ A &=P e^{r t} \end{aligned}\)
Substitute the values into the formula.
\(50,000=10,000 e^{r \cdot 17}\)
Solve for \(r\). Divide each side by \(10,000\).
\(5=e^{17 r}\)
Take the natural log of each side.
\(\ln 5=\ln e^{17 r}\)
Use the Power Property.
\(\ln 5=17 r \ln e\)
Simplify.
\(\ln 5=17 r\)
Divide each side by \(17\).
\(\frac{\ln 5}{17}=r\)
Approximate the answer.
\(r \approx 0.095\)
Convert to a percentage.
\(r \approx 9.5 \%\)
They need the rate of growth to be approximately \(9.5\)%.
Exercise \(\PageIndex{11}\)
Hector invests $\(10,000\) at age \(21\). He hopes the investments will be worth $\(150,000\) when he turns \(50\). If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal?
- Answer
-
\(r \approx 9.3 \%\)
Exercise \(\PageIndex{12}\)
Rachel invests $\(15,000\) at age \(25\). She hopes the investments will be worth $\(90,000\) when she turns \(40\). If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal?
- Answer
-
\(r \approx 11.9 \%\)
We have seen that growth and decay are modeled by exponential functions. For growth and decay we use the formula \(A=A_{0} e^{k t}\). Exponential growth has a positive rate of growth or growth constant, \(k\), and exponential decay has a negative rate of growth or decay constant, \(k\).
Definition \(\PageIndex{3}\)
Exponential Growth and Decay
For an original amount, \(A_{0}\), that grows or decays at a rate, \(k\), for a certain time, \(t\), the final amount, \(A\), is:
\(A=A_{0} e^{k t}\)
We can now solve applications that give us enough information to determine the rate of growth. We can then use that rate of growth to predict other situations.
Example \(\PageIndex{7}\)
Researchers recorded that a certain bacteria population grew from \(100\) to \(300\) in \(3\) hours. At this rate of growth, how many bacteria will there be \(24\) hours from the start of the experiment?
Solution :
This problem requires two main steps. First we must find the unknown rate, \(k\). Then we use that value of \(k\) to help us find the unknown number of bacteria.
Identify the variables in the formula.
\(\begin{aligned} A &=300 \\ A_{0} &=100 \\ k &=? \\ t &=3 \text { hours } \\ A &=A_{0} e^{k t} \end{aligned}\)
Substitute the values in the formula.
\(300=100 e^{k \cdot 3}\)
Solve for \(k\). Divide each side by \(100\).
\(3=e^{3 k}\)
Take the natural log of each side.
\(\ln 3=\ln e^{3 k}\)
Use the Power Property.
\(\ln 3=3 k \ln e\)
Simplify.
\(\ln 3=3 k\)
Divide each side by \(3\).
\(\frac{\ln 3}{3}=k\)
Approximate the answer.
\(k \approx 0.366\)
We use this rate of growth to predict the number of bacteria there will be in \(24\) hours.
\(\begin{aligned} A &=? \\ A_{0} &=100 \\ k &=\frac{\ln 3}{3} \\ t &=24 \text { hours } \\ A &=A_{0} e^{k t} \end{aligned}\)
Substitute in the values.
\(A=100 e^{\frac{\ln 3}{3} \cdot 24}\)
Evaluate.
\(A \approx 656,100\)
At this rate of growth, they can expect \(656,100\) bacteria.
Exercise \(\PageIndex{13}\)
Researchers recorded that a certain bacteria population grew from \(100\) to \(500\) in \(6\) hours. At this rate of growth, how many bacteria will there be \(24\) hours from the start of the experiment?
- Answer
-
There will be \(62,500\) bacteria.
Exercise \(\PageIndex{14}\)
Researchers recorded that a certain bacteria population declined from \(700,000\) to \(400,000\) in \(5\) hours after the administration of medication. At this rate of decay, how many bacteria will there be \(24\) hours from the start of the experiment?
- Answer
-
There will be \(5,870,061\) bacteria.
Radioactive substances decay or decompose according to the exponential decay formula. The amount of time it takes for the substance to decay to half of its original amount is called the half-life of the substance.
Similar to the previous example, we can use the given information to determine the constant of decay, and then use that constant to answer other questions.
Example \(\PageIndex{8}\)
The half-life of radium-226 is \(1,590\) years. How much of a \(100\) mg sample will be left in \(500\) years?
Solution :
This problem requires two main steps. First we must find the decay constant \(k\). If we start with \(100\)-mg, at the half-life there will be \(50\)-mg remaining. We will use this information to find \(k\). Then we use that value of \(k\) to help us find the amount of sample that will be left in \(500\) years.
Identify the variables in the formula.
\(\begin{aligned} A &=50 \\ A_{0} &=100 \\ k &=? \\ t &=1590 \text { years } \\ A &=A_{0} e^{k t} \end{aligned}\)
Substitute the values in the formula.
\(50=100 e^{k \cdot 1590}\)
Solve for \(k\). Divide each side by \(100\).
\(0.5=e^{1590 k}\)
Take the natural log of each side.
\(\ln 0.5=\ln e^{1590 k}\)
Use the Power Property.
\(\ln 0.5=1590 k \ln e\)
Simplify.
\(\ln 0.5=1590 k\)
Divide each side by \(1590\).
\(\frac{\ln 0.5}{1590}=k\) exact answer
We use this rate of growth to predict the amount that will be left in \(500\) years.
\(\begin{aligned} A &=? \\ A_{0} &=100 \\ k &=\frac{\ln 0.5}{1590} \\ t &=500\: \mathrm{years} \\ A &=A_{0} e^{k t} \end{aligned}\)
Substitute in the values.
\(A=100 e^{\frac{1 \mathrm{n} 0.5}{1500} \cdot 500}\)
Evaluate.
\(A \approx 80.4 \mathrm{mg}\)
In \(500\) years there would be approximately \(80.4\) mg remaining.
Exercise \(\PageIndex{15}\)
The half-life of magnesium-27 is \(9.45\) minutes. How much of a \(10\)-mg sample will be left in \(6\) minutes?
- Answer
-
There will be \(6.43\) mg left.
Exercise \(\PageIndex{16}\)
The half-life of radioactive iodine is \(60\) days. How much of a \(50\)-mg sample will be left in \(40\) days?
- Answer
-
There will be \(31.5\) mg left.
Access these online resources for additional instruction and practice with solving exponential and logarithmic equations.
Key Concepts
-
One-to-One Property of Logarithmic Equations:
For \(M>0, N>0, a>0\), and \(a≠1\) is any real number:
If \(\log _{a} M=\log _{a} N,\) then \(M=N\)
-
Compound Interest:
For a principal, \(P\), invested at an interest rate, \(r\), for \(t\) years, the new balance, \(A\), is:\(\begin{array}{ll}{A} & {=P\left(1+\frac{r}{n}\right)^{n t}} & {\text { when compounded } n \text { times a year. }} \\ {A} & {=P e^{r t}} & {\text { when compounded continuously. }}\end{array}\)
- Exponential Growth and Decay: For an original amount, \(A_{0}\) that grows or decays at a rate, \(r\), for a certain time \(t\), the final amount, \(A\), is \(A=A_{0} e^{r t}\).


28.4: Practice Exercises
28.4 Practice Exercises
1.
Analyze the chords with Roman numerals in the blanks below the staff, then add the specified non-chord tones, specifying suspensions with the appropriate interval numbers.
Answer.
2.
Analyze the figured bass below to write Roman numerals in the blanks below the staff. Then add the following non-chord tones: (1) 3 suspensions, (2) one passing tone, (3) one neighbor tone, and (4) one anticipation.
Answer.
3.
Voice lead the following Roman numeral progression then add the following non-chord tones: (1) a neighbor tone during beat one, (2) a neighbor tone during beat two, (3) an appoggiatura on beat 3, (4) a suspension on beat four, (5) a suspension on beat 5, (6) an anticipation during beat six, and (7) a suspension on beat 7.
Answer.
Click here to download the homework assignment for this chapter.


15.14B: Cystitis
- Differentiate among the distinct types of cystitis: traumatic, interstitial, eosinophilic, hemorrhagic cystitis, and cystitis cystica, recognizing their causes and risk factors
A urinary tract infection (UTI), a bacterial infection that affects the lower urinary tract, is also known as a simple cystitis (a bladder infection). Symptoms from a lower urinary tract infection include painful urination and either frequent urination or the urge to urinate (or both). Cystitis is a urinary bladder inflammation that can result from any one of a number of distinct syndromes. It is most commonly caused by a bacterial infection in which case it is referred to as a urinary tract infection.
- Pressure in the lower pelvis
- Painful urination (dysuria)
- Frequent urination (polyuria) or urgent need to urinate (urinary urgency)
- Need to urinate at night (nocturia)
- Urine that contains traces of blood (haematuria)
- Dark, cloudy or strong-smelling urine
- Pain above the pubic bone, or in the lower back or abdomen
- Feeling unwell, weak, or feverish
There are several medically distinct types of cystitis, each having a unique etiology and therapeutic approach:
- Traumatic cystitis is probably the most common form of cystitis in the female. It is due to bruising of the bladder, usually by abnormally forceful sexual intercourse. This is often followed by bacterial cystitis, frequently by coliform bacteria being transferred from the bowel through the urethra into the bladder.
- Interstitial cystitis (IC) is considered more of an injury to the bladder resulting in constant irritation and rarely involves the presence of infection. IC patients are often misdiagnosed with UTI/cystitis for years before they are told that their urine cultures are negative. Antibiotics are not used to treatment of IC. The cause of IC is unknown, although some suspect it may be autoimmune where the immune system attacks the bladder. Several therapies are now available.
- Eosinophilic cystitis (EC) is a rare form of cystitis that is diagnosed via biopsy. In these cases, the bladder wall is infiltrated with a high number of eosinophils. The cause of EC may be attributed to infection by Schistosoma haematobium or by certain medications in afflicted children. Some consider it a form of interstitial cystitis.
- Hemorrhagic cystitis can occur as a side effect of cyclophosphamide, ifosfamide, and radiation therapy. Radiation cystitis, one form of hemorrhagic cystitis is a rare consequence of patients undergoing radiation therapy for the treatment of cancer. Several adenovirus serotypes have been associated with an acute, self-limited hemorrhagic cystitis, which occurs primarily in boys. It is characterized by hematuria, and virus can usually be recovered from the urine. In sexually active women the most common cause of urinary tract infection is from E. coli and Staphylococcus saprophyticus.
- Cystitis cystica is a chronic cystitis glandularis accompanied by the formation of cysts. This disease can cause chronic urinary tract infections. It appears as small cysts filled with fluid and lined by one or more layers of epithelial cells. These are due to hydropic degeneration in center of Brunn’s nests
Key Points
- Cystitis is a urinary bladder inflammation that is most commonly caused by a bacterial infection in which case it is referred to as a urinary tract infection. Symptoms from a lower urinary tract include painful urination and either frequent urination or urge to urinate (or both), but neither may be present.
- Traumatic cystitis is probably the most common form of cystitis in the female. It is due to bruising of the bladder, usually by abnormally forceful sexual intercourse.
- Interstitial cystitis (IC) is considered more of an injury to the bladder resulting in constant irritation and rarely involves the presence of infection.
Key Terms
- cystitis : Cystitis is a urinary bladder inflammation that can result from any one of a number of distinct syndromes. It is most commonly caused by a bacterial infection in which case it is referred to as a urinary tract infection.


28.1: Introduction to Diabetes
By the end of this section, you should be able to:
- 28.1.1 Describe the pathophysiology of type 1 and type 2 diabetes.
- 28.1.2 Identify clinical manifestations of diabetes.
- 28.1.3 Identify the etiology and diagnostic studies related to diabetes.
The islets of Langerhans are clusters of cells located in the pancreas. They are composed of alpha and beta cells, which are crucial for glucose homeostasis. Alpha (α) cells produce glucagon, which breaks down glycogen into glucose in the liver. Beta (β) cells secrete insulin, which regulates glucose metabolism and facilitates absorption of glucose from the bloodstream into cells (see Figure 28.2). Diabetes mellitus is a metabolic disease that is characterized by hyperglycemia associated with alterations in carbohydrate, protein, and fat metabolism (Dilworth et al., 2021).
After reviewing the client’s symptoms, the nurse anticipates which diagnosis by the health care provider?
- Answer
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Type 2 diabetes
To determine the client’s blood glucose level over the last 3 months, which diagnostic test does the nurse anticipate the health care provider will order?
- Answer
-
Glycosylated hemoglobin (hemoglobin A1c)


10.1: Perfect Competition
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Definition of Perfect Competition
Perfect competition is a market structure that leads to the Pareto-efficient allocation of economic resources.
Learning Objectives
- Describe degrees of competition in different market structures
Market structure is determined by the number and size distribution of firms in a market, entry conditions, and the extent of product differentiation. The major types of market structure include the following:
- Monopoly: An industry structure where a single firm produces a product for which there are no close substitutes. Monopolists are price makers. Barriers to entry and exit exist, and, in order to ensure profits, a monopoly will attempt to maintain them.
- Monopolistic competition: A market structure in which there is a large number of firms, each having a small portion of the market share and slightly differentiated products. There are close substitutes for the product of any given firm, so competitors have slight control over price. There are relatively insignificant barriers to entry or exit, and success invites new competitors into the industry.
- Oligopoly: An industry structure in which there are a few firms producing products that range from slightly differentiated to highly differentiated. Each firm is large enough to influence the industry. Barriers to entry exist.
- Perfect competition: An industry structure in which there are many firms, none large enough to influence the industry, producing homogeneous products. Firms are price takers. There are no barriers to entry. Agriculture comes close to being perfectly competitive.
Perfect competition leads to the Pareto-efficient allocation of economic resources. Because of this it serves as a natural benchmark against which to contrast other market structures. However, in practice, very few industries can be described as perfectly competitive. Nevertheless, it is used because it provides important insights.
A perfectly competitive market has several important characteristics:
- All producers contribute insignificantly to the market. Their own production levels do not change the supply curve.
- All producers are price takers. They cannot influence the market. If a firm tries to raise its price consumers would buy from a competitor with a lower price instead.
- Products are homogeneous. The characteristics of a good or service do not vary between suppliers.
- Producers enter and exit the market freely.
- Both buyers and sellers have perfect information about the price, utility, quality, and production methods of products.
- There are no transaction costs. Buyers and sellers do not incur costs in making an exchange of goods in a perfectly competitive market.
- Producers earn zero economic profits in the long run.
Conditions of Perfect Competition
A firm in a perfectly competitive market may generate a profit in the short-run, but in the long-run it will have economic profits of zero.
Learning Objectives
- Calculate total revenue, average revenue, and marginal revenue for a firm in a perfectly competitive market
The concept of perfect competition applies when there are many producers and consumers in the market and no single company can influence the pricing. A perfectly competitive market has the following characteristics:
- There are many buyers and sellers in the market.
- Each company makes a similar product.
- Buyers and sellers have access to perfect information about price.
- There are no transaction costs.
- There are no barriers to entry into or exit from the market.
All goods in a perfectly competitive market are considered perfect substitutes, and the demand curve is perfectly elastic for each of the small, individual firms that participate in the market. These firms are price takers–if one firm tries to raise its price, there would be no demand for that firm’s product. Consumers would buy from another firm at a lower price instead.
Firm Revenues
A firm in a competitive market wants to maximize profits just like any other firm. The profit is the difference between a firm’s total revenue and its total cost. For a firm operating in a perfectly competitive market, the revenue is calculated as follows:
- Total Revenue = Price * Quantity
- AR (Average Revenue) = Total Revenue / Quantity
- MR (Marginal Revenue) = Change in Total Revenue / Change in Quantity
The average revenue (AR) is the amount of revenue a firm receives for each unit of output. The marginal revenue (MR) is the change in total revenue from an additional unit of output sold. For all firms in a competitive market, both AR and MR will be equal to the price.
Profit Maximization
In order to maximize profits in a perfectly competitive market, firms set marginal revenue equal to marginal cost (MR=MC). MR is the slope of the revenue curve, which is also equal to the demand curve (D) and price (P). In the short-term, it is possible for economic profits to be positive, zero, or negative. When price is greater than average total cost, the firm is making a profit. When price is less than average total cost, the firm is making a loss in the market.
Perfect Competition in the Short Run : In the short run, it is possible for an individual firm to make an economic profit. This scenario is shown in this diagram, as the price or average revenue, denoted by P, is above the average cost denoted by C.
Over the long-run, if firms in a perfectly competitive market are earning positive economic profits, more firms will enter the market, which will shift the supply curve to the right. As the supply curve shifts to the right, the equilibrium price will go down. As the price goes down, economic profits will decrease until they become zero.
When price is less than average total cost, firms are making a loss. Over the long-run, if firms in a perfectly competitive market are earning negative economic profits, more firms will leave the market, which will shift the supply curve left. As the supply curve shifts left, the price will go up. As the price goes up, economic profits will increase until they become zero.
In sum, in the long-run, companies that are engaged in a perfectly competitive market earn zero economic profits. The long-run equilibrium point for a perfectly competitive market occurs where the demand curve (price) intersects the marginal cost (MC) curve and the minimum point of the average cost (AC) curve.
Perfect Competition in the Long Run : In the long-run, economic profit cannot be sustained. The arrival of new firms in the market causes the demand curve of each individual firm to shift downward, bringing down the price, the average revenue and marginal revenue curve. In the long-run, the firm will make zero economic profit. Its horizontal demand curve will touch its average total cost curve at its lowest point.
The Demand Curve in Perfect Competition
A perfectly competitive firm faces a demand curve is a horizontal line equal to the equilibrium price of the entire market.
Learning Objectives
- Describe the demand for goods in perfectly competitive markets
In a perfectly competitive market the market demand curve is a downward sloping line, reflecting the fact that as the price of an ordinary good increases, the quantity demanded of that good decreases. Price is determined by the intersection of market demand and market supply; individual firms do not have any influence on the market price in perfect competition. Once the market price has been determined by market supply and demand forces, individual firms become price takers. Individual firms are forced to charge the equilibrium price of the market or consumers will purchase the product from the numerous other firms in the market charging a lower price (keep in mind the key conditions of perfect competition). The demand curve for an individual firm is thus equal to the equilibrium price of the market.
Demand Curve for a Firm in a Perfectly Competitive Market : The demand curve for an individual firm is equal to the equilibrium price of the market. The market demand curve is downward-sloping.
The demand curve for a firm in a perfectly competitive market varies significantly from that of the entire market.The market demand curve slopes downward, while the perfectly competitive firm’s demand curve is a horizontal line equal to the equilibrium price of the entire market. The horizontal demand curve indicates that the elasticity of demand for the good is perfectly elastic. This means that if any individual firm charged a price slightly above market price, it would not sell any products.
A strategy often used to increase market share is to offer a firm’s product at a lower price than the competitors. In a perfectly competitive market, firms cannot decrease their product price without making a negative profit. Instead, assuming that the firm is a profit-maximizer, it will sell its goods at the market price.
Key Points
- The major types of market structure include monopoly, monopolistic competition, oligopoly, and perfect competition.
- Perfect competition is an industry structure in which there are many firms producing homogeneous products. None of the firms are large enough to influence the industry.
- The characteristics of a perfectly competitive market include insignificant contributions from the producers, homogenous products, perfect information about products, no transaction costs, and no long-term economic profits.
- In practice, very few industries can be described as perfectly competitive, though agriculture comes close.
- A perfectly competitive market is characterized by many buyers and sellers, undifferentiated products, no transaction costs, no barriers to entry and exit, and perfect information about the price of a good.
- The total revenue for a firm in a perfectly competitive market is the product of price and quantity (TR = P * Q). The average revenue is calculated by dividing total revenue by quantity. Marginal revenue is calculated by dividing the change in total revenue by change in quantity.
- A firm in a competitive market tries to maximize profits. In the short-run, it is possible for a firm’s economic profits to be positive, negative, or zero. Economic profits will be zero in the long-run.
- In the short-run, if a firm has a negative economic profit, it should continue to operate if its price exceeds its average variable cost. It should shut down if its price is below its average variable cost.
- In a perfectly competitive market individual firms are price takers. The price is determined by the intersection of the market supply and demand curves.
- The demand curve for an individual firm is different from a market demand curve. The market demand curve slopes downward, while the firm’s demand curve is a horizontal line.
- The firm’s horizontal demand curve indicates a price elasticity of demand that is perfectly elastic.
Key Terms
- monopoly : A situation, by legal privilege or other agreement, in which solely one party (company, cartel etc. ) exclusively provides a particular product or service, dominating that market and generally exerting powerful control over it.
- Monopolistic competition : A market structure in which there is a large number of firms, each having a small proportion of the market share and slightly differentiated products.
- oligopoly : An economic condition in which a small number of sellers exert control over the market of a commodity.
- economic profit : The difference between the total revenue received by the firm from its sales and the total opportunity costs of all the resources used by the firm.
- Perfectly elastic : Describes a situation when any increase in the price, no matter how small, will cause demand for a good to drop to zero.
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8.3.4: Sampling in a Fair Way
Lesson
Let's explore ways to get representative samples.
Exercise \(\PageIndex{1}\): Ages of Moviegoers
A survey was taken at a movie theater to estimate the average age of moviegoers.
Here is a dot plot showing the ages of the first 20 people surveyed.
- What questions do you have about the data from survey?
- What assumptions would you make based on these results?
Exercise \(\PageIndex{2}\): Comparing Methods for Selecting Samples
Take turns with your partner reading each option aloud. For each situation, discuss:
- Would the different methods for selecting a sample lead to different conclusions about the population?
- What are the benefits of each method?
- What might each method overlook?
- Which of the methods listed would be the most likely to produce samples that are representative of the population being studied?
- Can you think of a better way to select a sample for this situation?
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Lin is running in an election to be president of the seventh grade. She wants to predict her chances of winning. She has the following ideas for surveying a sample of the students who will be voting:
- Ask everyone on her basketball team who they are voting for.
- Ask every third girl waiting in the lunch line who they are voting for.
- Ask the first 15 students to arrive at school one morning who they are voting for.
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A nutritionist wants to collect data on how much caffeine the average American drinks per day. She has the following ideas for how she could obtain a sample:
- Ask the first 20 adults who arrive at a grocery store after 10:00 a.m. about the average amount of caffeine they consume each day.
- Every 30 minutes, ask the first adult who comes into a coffee shop about the average amount of caffeine they consume each day.
Exercise \(\PageIndex{3}\): That's the First Straw
Your teacher will have some students draw straws from a bag.
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As each straw is taken out and measured, record its length (in inches) in the table.
straw 1 straw 2 straw 3 straw 4 straw5 sample 1 sample 2 Table \(\PageIndex{1}\) -
Estimate the mean length of all the straws in the bag based on:
- the mean of the first sample.
- the mean of the second sample.
- Were your two estimates the same? Did the mean length of all the straws in the bag change in between selecting the two samples? Explain your reasoning.
- The actual mean length of all of the straws in the bag is about 2.37 inches. How do your estimates compare to this mean length?
- If you repeated the same process again but you selected a larger sample (such as 10 or 20 straws, instead of just 5), would your estimate be more accurate? Explain your reasoning.
Exercise \(\PageIndex{4}\): That's the Last Straw
There were a total of 35 straws in the bag. Suppose we put the straws in order from shortest to longest and then assigned each straw a number from 1 to 35. For each of these methods, decide whether it would be fair way to select a sample of 5 straws. Explain your reasoning.
- Select the straws numbered 1 through 5.
- Write the numbers 1 through 35 on pieces of paper that are all the same size. Put the papers into a bag. Without looking, select five papers from the bag. Use the straws with those numbers for your sample.
- Using the same bag as the previous question, select one paper from the bag. Use the number on that paper to select the first straw for your sample. Then use the next 4 numbers in order to complete your sample. (For example, if you select number 17, then you also use straws 18, 19, 20, and 21 for your sample.)
- Create a spinner with 35 sections that are all the same size, and number them 1 through 35. Spin the spinner 5 times and use the straws with those numbers for your sample.
Are you ready for more?
Computers accept inputs, follow instructions, and produce outputs, so they cannot produce truly random numbers. If you knew the input, you could predict the output by following the same instructions the computer is following. When truly random numbers are needed, scientists measure natural phenomena such as radioactive decay or temperature variations. Before such measurements were possible, statisticians used random number tables, like this:
Use this table to select a sample of 5 straws. Pick a starting point at random in the table. If the number is between 01 and 35, include that number straw in your sample. If the number has already been selected, or is not between 01 and 35, ignore it, and move on to the next number.
Summary
A sample is selected at random from a population if it has an equal chance of being selected as every other sample of the same size. For example, if there are 25 students in a class, then we can write each of the students' names on a slip of paper and select 5 papers from a bag to get a sample of 5 students selected at random from the class.
Other methods of selecting a sample from a population are likely to be biased . This means that it is less likely that the sample will be representative of the population as a whole. For example, if we select the first 5 students who walk in the door, that will not give us a random sample because students who typically come late are not likely to be selected. A sample that is selected at random may not always be a representative sample, but it is more likely to be representative than using other methods.
It is not always possible to select a sample at random. For example, if we want to know the average length of wild salmon, it is not possible to identify each one individually, select a few at random from the list, and then capture and measure those exact fish. When a sample cannot be selected at random, it is important to try to reduce bias as much as possible when selecting the sample.
Glossary Entries
Definition: Mean
The mean is one way to measure the center of a data set. We can think of it as a balance point. For example, for the data set 7, 9, 12, 13, 14, the mean is 11.
To find the mean, add up all the numbers in the data set. Then, divide by how many numbers there are. \(7+9+12+13+14=55\) and \(55\div 5=11\).
Definition: Mean Absolute Deviation (MAD)
The mean absolute deviation is one way to measure how spread out a data set is. Sometimes we call this the MAD. For example, for the data set 7, 9, 12, 13, 14, the MAD is 2.4. This tells us that these travel times are typically 2.4 minutes away from the mean, which is 11.
To find the MAD, add up the distance between each data point and the mean. Then, divide by how many numbers there are.
\(4+2+1+2+3=12\) and \(12\div 5=2.4\)
Definition: Median
The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order.
For the data set 7, 9, 12, 13, 14, the median is 12.
For the data set 3, 5, 6, 8, 11, 12, there are two numbers in the middle. The median is the average of these two numbers. \(6+8=14\) and \(14\div 2=7\).
Definition: Population
A population is a set of people or things that we want to study.
For example, if we want to study the heights of people on different sports teams, the population would be all the people on the teams.
Definition: Representative
A sample is representative of a population if its distribution resembles the population's distribution in center, shape, and spread.
For example, this dot plot represents a population.
This dot plot shows a sample that is representative of the population.
Definition: Sample
A sample is part of a population. For example, a population could be all the seventh grade students at one school. One sample of that population is all the seventh grade students who are in band.
Practice
Exercise \(\PageIndex{5}\)
The meat department manager at a grocery store is worried some of the packages of ground beef labeled as having one pound of meat may be under-filled. He decides to take a sample of 5 packages from a shipment containing 100 packages of ground beef. The packages were numbered as they were put in the box, so each one has a different number between 1 and 100.
Describe how the manager can select a fair sample of 5 packages.
Exercise \(\PageIndex{6}\)
Select all the reasons why random samples are preferred over other methods of getting a sample.
- If you select a random sample, you can determine how many people you want in the sample.
- A random sample is always the easiest way to select a sample from a population.
- A random sample is likely to give you a sample that is representative of the population.
- A random sample is a fair way to select a sample, because each person in the population has an equal chance of being selected.
- If you use a random sample, the sample mean will always be the same as the population mean.
Exercise \(\PageIndex{7}\)
Jada is using a computer’s random number generator to produce 6 random whole numbers between 1 and 100 so she can use a random sample. The computer produces the numbers: 1, 2, 3, 4, 5, and 6. Should she use these numbers or have the computer generate a new set of random numbers? Explain your reasoning.
Exercise \(\PageIndex{8}\)
A group of 100 people is divided into 5 groups with 20 people in each. One person’s name is chosen, and everyone in their group wins a prize. Noah simulates this situation by writing 100 different names on papers and putting them in a bag, then drawing one out. Kiran suggests there is a way to do it with fewer paper slips. Explain a method that would simulate this situation with fewer than 100 slips of paper.
(From Unit 8.1.6)
Exercise \(\PageIndex{9}\)
Data collected from a survey of American teenagers aged 13 to 17 was used to estimate that 29% of teens believe in ghosts. This estimate was based on data from 510 American teenagers. What is the population that people carrying out the survey were interested in?
- All people in the United States.
- The 510 teens that were surveyed.
- All American teens who are between the ages of 13 and 17.
- The 29% of the teens surveyed who said they believe in ghosts.
(From Unit 8.3.2)
Exercise \(\PageIndex{10}\)
A computer simulates flipping a coin 100 times, then counts the longest string of heads in a row.
Based on these results, estimate the probability that there will be at least 15 heads in a row.
| trial | most heads in a row |
|---|---|
| \(1\) | \(8\) |
| \(2\) | \(6\) |
| \(3\) | \(5\) |
| \(4\) | \(11\) |
| \(5\) | \(13\) |
(From Unit 8.2.1)


2.3: Drug Administration Routes, Preparation, and Administration
By the end of this section, you should be able to:
- 2.3.1 Identify the different routes of drug administration.
- 2.3.2 Discuss sites for parenteral therapy.
- 2.3.3 Analyze nursing interventions related to drug administration.
- 2.3.4 Explain equipment and techniques for drug administration.
This section will discuss the different routes for medication administration, how to prepare for administration, and the various methods for administering drugs to the client. Both enteral and parenteral sites will be reviewed, along with the equipment needed for each type of drug administration. Techniques for drug administration will be described.
Forms and Routes of Drug Administration
There are many different forms of medication : liquid, suspensions, tablets, capsules, lotions, and ointment, to name a few. There are also many routes through which medications can be given and absorbed into the body. The routes of medication administration are broadly categorized as follows:
- Enteral administration : “Enteral” means “pertaining to the intestines.” Most enteral medications are absorbed in the intestines. The primary routes for enteral administration are oral and, to a lesser extent, rectal. Some clients have tubes placed directly into the gastrointestinal tract (e.g., nasogastric tubes or percutaneous endoscopic gastrostomy [PEG] tubes). Absorption will vary, but all will be affected by the first-pass effect.
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Parenteral administration
:
“Parenteral” refers to any drug that is administered outside of the GI tract; however, it most commonly refers to injectable drugs administered via the subcutaneous, intramuscular, or intravenous routes. Drugs administered via these routes have improved bioavailability because they bypass the first-pass effect, making absorption and onset of action more rapid.
- Percutaneous administration : Some sources will define percutaneous administration as a separate category or a subcategory of parenteral routes. The percutaneous route refers to topical drugs absorbed through the skin—lotions, ointments, creams, or patches.
The following sections describe the equipment needed for the administration of medications. The various techniques of each route are detailed, along with their pertinent advantages and disadvantages. Nursing implications are also covered in relation to each route of administration.
Safety Alert
Medication Safety
The following are some tips for medication safety:
- It is best practice to prepare medications for only one client at a time. This safety practice reduces the risk of inadvertently administering medications to the incorrect client.
- Medications that require a focused assessment or monitoring should be kept separate from other medications. For example, if administering a medication that lowers blood pressure and heart rate, vital signs should be assessed before giving the drug. Because opioids may cause respiratory depression, respiratory rate and oxygen saturation should be assessed before and after administration of the drug.
- All unit-dose medications should be opened at the bedside rather than in the medication room.
- Never leave medications unattended at the bedside unless specifically ordered. Remain with the client until all medications have been administered.
Product (Drug) Labeling
Each prescription drug includes a package insert that provides clients with information about the drug. Many package inserts are developed by the manufacturer and approved by the FDA for use by clients and caregivers (FDA, 2023a). Some of the information contained in the inserts includes generic and trade names, routes, instructions for taking the drug, and how to store and dispose of the drug. Any side effects, especially if the drug has serious side effects, are listed, as are directions about what to do if adverse effects occur. General information about the safe use of the drug, how to report side effects, and ingredients are also listed. These package inserts are often one of the best resources for free information for the client.
Oral Medications
Oral administration encompasses several different drug forms. Liquids, elixirs, suspensions, tablets, capsules, and caplets may all be given orally. Oral administration is usually quick, easy, and convenient, but the onset of action is longer and unpredictable due to the first-pass effect, and not all drugs can be administered this way. Table 2.1 lists the advantages and disadvantages of oral administration.
| Advantages | Disadvantages |
|---|---|
|
|
Steps to administering an oral medication:
-
Assemble the appropriate equipment:
- Drinking cup
- Straw
- Disposable medication cup (souffle cup or calibrated plastic medication cup for liquids)
- Assess the client to determine if the drug is safe and appropriate to give.
- Check the medication, dose, and expiration date.
- Check NPO status and ensure the client does not have nausea or vomiting. (NPO is a Latin term meaning nil per os , or nothing by mouth. Sometimes this will include medications.)
-
Follow the seven rights of medication administration
throughout
the procedure (at least three times or according to institutional policy).
- During verification of the contents of the medication administration record and the orders
- When preparing the medication
- At the bedside
- Wait to open blister packs or oral unit doses until at the client’s bedside.
- Perform hand hygiene.
- Don gloves if you anticipate touching the pill or the client’s mouth during administration.
-
Identify the client, and verify allergies and reaction. (If the institution uses barcode scanning, scan the client’s ID band and the barcode on the blister pack using the protocol recommended by the institution.)
- Perform the third medication check at the bedside. This check is completed by verifying that the medication name, dosage, route, and time match the medication administration record (this is the last opportunity to prevent an error from occurring). Most institutions now have barcode scanning at the bedside as an additional layer of security.
-
Explain the medication to the client:
- Name (brand and generic)
- Dosage
- Indication, rationale, or reason for the drug to be given
- Frequency
- Route
- Adverse effects
- Position the client in an upright position or on side as condition allows.
- Assess the client’s ability to swallow and the gag reflex by offering a sip of water.
- Ask the client if they prefer all medications at once or one or two at a time.
- Give the client the medication with a cup of water (approximately 8 ounces unless the client is on a fluid restriction).
- Document administration within the medication administration record (MAR).
- Perform hand hygiene.
- Evaluate the client’s response to the drug(s) within the appropriate time frame.
Nursing Implications for Oral Medications
The nurse should do the following for clients who are taking oral medications:
- If a tablet needs to be split, split only tablets that are scored. If a client has difficulty swallowing a tablet or capsule, consult a pharmacist for advice about the technique of administration because some capsules may be opened and emptied into a food or liquid. Timed-release capsules or tablets should not be crushed or chewed because this may affect the rate of absorption and toxicity may occur. For this reason, timed-release capsules should not be opened and emptied into food for ease of swallowing.
- Use a hospital-approved device to split the tablet. (Some health systems split the tablets in the pharmacy and send them to the unit in unit-dosed packaging for safety purposes.)
-
Discard any unused portion according to institutional policy.
- If the drug is a controlled substance, document the waste with another nurse in the medication room.
-
If a tablet needs to be crushed:
- Ensure that it can be crushed.
- Never crush sustained-release, extended-release, or enteric-coated tablets.
- If crushing more than one tablet, keep them separate; do not combine them.
- When filling a calibrated plastic cup with liquids, fill at eye level.
- Always remain with the client until all medications are taken; do not leave drugs at the bedside unattended.
Sublingual and Buccal Administration
Absorption of sublingual medications occurs in the area under the tongue, whereas buccal medications are absorbed in the oral mucosa, generally between the cheek and gums. These are vascular areas, and medications administered here are absorbed rapidly because they do not undergo the first-pass effect. Table 2.2 lists the advantages and disadvantages of sublingual and buccal administration .
| Advantages | Disadvantages |
|---|---|
|
|
Steps to administering a sublingual or buccal medication:
-
Assemble the appropriate equipment:
- Disposable medication cup (souffle cup)
- Drinking cup
- Straw
- Assess the client to determine if the drug is safe and appropriate to give.
- Check the medication, dose, and expiration date.
-
Follow the seven rights of medication administration
throughout
the procedure (at least three times or according to institutional policy).
- During medication reconciliation
- When preparing the medication
- At the bedside
- Perform hand hygiene.
- Don gloves if you anticipate touching the pill or the client’s mouth during administration.
- Identify the client, and verify allergies and reaction. (If the institution uses barcode scanning, scan the client’s ID band and the barcode on the blister pack, using the protocol recommended by the institution.)
- Perform the third medication check at the bedside.
-
Explain the medication to the client:
- Name (brand and generic)
- Dosage
- Indication, rationale, or reason for the drug to be given
- Frequency
- Route
- Adverse effects
- Offer sips of water to moisten the oral cavity.
- Assist the client in placing the medication sublingually (or between the cheek and gum for buccal drugs).
- Instruct the client to allow the medication to dissolve completely. Discuss the importance of not swallowing or chewing the pill.
- Educate the client about the importance of abstaining from food, drinking, or smoking until after the medication has dissolved.
- Document administration within the MAR.
- Perform hand hygiene.
- Evaluate the client’s response to the drug(s) within the appropriate time frame.
Nursing Implications for Sublingual or Buccal Medications
The nurse should do the following for clients who are taking sublingual or buccal medications:
-
Always remain with the client until all medications are taken; do not leave drugs at the bedside unattended unless the provider has ordered the medication to be left at the bedside.
- Exception: Sublingual nitroglycerin tablets or sprays are often ordered to be left at the bedside so that a client may take them as needed in the event of chest pain.
Nasal Spray Administration
Nasal sprays can be rapidly absorbed into the mucous membranes of the nasal cavity. Table 2.3 lists the advantages and disadvantages of nasal sprays.
| Advantages | Disadvantages |
|---|---|
|
|
Steps to administering a nasal spray:
-
Assemble the appropriate equipment:
- Clean gloves
- Tissue
- Medication
- Assess the client to determine if the drug is safe and appropriate to give.
- Check medication, dose, and expiration date.
-
Follow the seven rights of medication administration
throughout
the procedure (at least three times or according to institutional policy).
- During medication reconciliation
- When preparing the medication
- At the bedside
-
Educate the client.
- Explain the method for administering the medication. (This route may be self-administered in the future; however, the nurse should observe this in order to provide appropriate documentation in the MAR.)
- Inform the client that they may experience a burning or stinging sensation with administration.
- Instruct the client to gently blow their nose (unless it is contraindicated for the client).
- Assess the nostrils for erythema, edema, drainage, or tenderness.
- Client should be upright in a sitting position with their head tilted back.
- Block one nostril.
- Hold the medication bottle upright and shake.
- Immediately insert the tip of the applicator into the nostril.
- Ask the client to inhale while simultaneously squeezing a spray into the nostril.
- Once the bottle has been squeezed to deliver the medication, do not release the squeeze until the spray bottle has been removed from the nares. Ensure that the nozzle of the nasal spray does not touch the nasal turbinates or septum because pain or injury could occur.
- Repeat the process in the other nostril if indicated.
- Have tissue available if needed to blot the nostril. The client should avoid blowing their nose immediately.
- Wipe the spray applicator with a clean, dry cloth or tissue.
- Remove gloves and perform hand hygiene.
- Document administration within the MAR.
- Evaluate the client’s response to the drug(s) within the appropriate time frame.
Be aware that some nasal medications may vary from this procedure; it is important to consult the product labeling to confirm the appropriate administration technique.
Nursing Implications for Nasal Sprays
The nurse should do the following for clients who are taking nasal sprays:
- Do not readminister the drug if the client sneezes following the administration of the nasal spray because there is no way to assess how much of the drug has been absorbed.
Removing Parenteral Medication from a Vial
When administering a parenteral medication , such as a subcutaneous or intramuscular injection, it is important to remember that this is an invasive procedure (a needle is inserted into the client). The medication may come in a prefilled syringe; however, it is usually drawn up by the nurse from a vial of medication. The nurse should be very alert during the process of drawing up and administering the medication to keep the needle and contents sterile.
Steps to withdrawing medication from a vial:
- Perform hand hygiene and don clean gloves (not sterile).
- Inspect and verify the medication, dose, volume, and expiration date.
- Verify the dosage calculation.
- Remove the plastic cap from the top of the unused vial with a flick of the thumb.
-
Wipe the rubber stopper or port with an alcohol swab and allow it to air dry for approximately 10 seconds.
- The cap does not keep the top of the port sterile. Dust and microbial contaminants can collect under the cap, so it is important to cleanse with alcohol.
- Insulin and tuberculin syringes have preattached needles. If drawing up insulin, insulin syringes have preattached needles with orange caps (see Figure 2.6). Insulin syringes are marked in unit measures rather than in milliliters (mL). The needles on these syringes are fragile and bend very easily, so it is important to be careful when inserting and withdrawing the needles from the vial.
-
When drawing up medications into syringes that are not insulin or tuberculin syringes:
- Attach a blunt-tipped needle to the syringe of choice. The syringe choice should be large enough to hold the dose of medication, but the smallest syringe closest to that measurement (i.e., if administering 4 mL, draw it up in a 5 mL syringe rather than a 10 mL, 20 mL, etc.).
- Remove the needle cap and draw air into the empty syringe to the volume of medication to be given (e.g., if giving 2 mL of medication, then draw up 2 mL of air).
-
Insert the air into the vial of medication through the center of the rubber port at the top of the vial.
- Ensuring that the tip of the needle is above the fluid level of the vial will help avoid the presence of bubbles. (It prevents agitation of the drug.)
- Be sure to maintain the sterility of the needle.
- Do not touch the needle.
- Be careful not to bend the needle.
- Inject the air into the vial.
-
Invert the vial and hold it at eye level to slowly withdraw the desired volume of medication.
- If the medication is withdrawn too quickly, air bubbles may enter the syringe.
- Important tip: Ensure that the tip of the needle is below the fluid level in the vial so that no air is drawn into the syringe.
- Withdraw slightly more of the medication than needed.
- Express any air bubbles and the excess medication back into the vial until the desired amount of medication is in the syringe.
- Withdraw the needle from the vial, being careful not to bend the needle.
- Exchange the blunt needle for a regular needle prior to administration.
Steps to administering a subcutaneous medication:
-
Assemble the appropriate equipment:
- Medication
- Sterile syringe (1–3 mL)
- Small-gauge needles (3/8–5/8 inch) (tuberculin and insulin syringes have preattached needles)
- Alcohol swabs
- Gloves (clean gloves, not sterile)
- Assess the client to determine if the drug is safe and appropriate to give.
- Check the medication, dose, volume, and expiration date.
-
Follow the seven rights of medication administration
throughout
the procedure (at least three times or according to institutional policy).
- During medication reconciliation
- When drawing up the medication
- At the bedside
- Perform hand hygiene.
- Don gloves.
- Identify the client, and verify allergies and reaction. (If the institution uses barcode scanning, scan the client’s ID band and the barcode on the vial or unit dose, using the protocol recommended by the institution.)
- Perform the third medication check at the bedside.
-
Explain the medication to the client:
- Name (brand and generic)
- Dosage
- Indication, rationale, or reason for the drug to be given
- Frequency
- Route
- Adverse effects
- Prepare medication using the correct needle length (3/8–5/8 inches), gauge (25–29 gauge), and syringe (usually no more than 1.5 mL can be given via this route). For clients with little adipose tissue, use the smaller needle.
-
Select an injection site with an adequate fat pad.
- Avoid bruises, rashes, inflammation, or areas of injury.
- Ensure that the injection site is a minimum of 2 inches away from the umbilicus, a stoma, or an incision.
- Preferred sites include the abdomen, upper arms, and anterior thighs.
- Assist the client into a position in which the site or extremity can be relaxed.
- Cleanse the area with an alcohol swab using a circular motion by starting at the center and working outward in a widening circle to about 2–3 inches. Allow to air dry.
- Grasp the skinfold between your thumb and index or third finger of the nondominant hand. (A new alcohol swab or gauze can be placed between the fourth and fifth fingers of this hand to use after injection.)
- Remove the needle cap carefully and dispose.
- Instruct the client that they will feel a “pinch.”
- Quickly and smoothly insert the needle into the skin and adipose tissue at a 45- to 90-degree angle. The anticoagulant enoxaparin is a subcutaneous injection that should be given at a 90-degree angle.
- Inject the medication with the dominant hand depressing the plunger with slow and even pressure, while holding the barrel of the syringe steady with the nondominant hand.
- Do not aspirate for subcutaneous injections.
- Withdraw the needle smoothly at the same angle that it was inserted to prevent trauma at the injection site.
- Apply gentle pressure with the alcohol swab, but do not massage the site, especially if the medication given was an anticoagulant such as heparin or enoxaparin, because this may cause extensive bruising.
- Activate the safety device on the syringe and dispose of the syringe in the sharps container or a puncture-resistant needle disposal container according to institutional policy. Never throw it into the trash.
- Do not recap the needle! This is a safety hazard for the nurse. Recapping needles may lead to needle sticks and exposure to pathogens.
- Remove gloves and perform hand hygiene.
- Document in the MAR. When documenting a subcutaneous injection, be sure to document the site the medication was administered to allow for the rotation of sites.
- Evaluate the client’s response to the drug(s) within the appropriate time frame.
Nursing Implications for Subcutaneous Administration
The nurse should do the following for clients receiving a subcutaneous injection:
- For heparins and insulins: Both are high-alert medications. A second nurse will need to verify the dose. Do not draw up the dose until a witness is available to verify.
-
Never draw heparin up into an insulin syringe. (Fortunately, many heparins come in prefilled syringes for safety reasons.)
- Insulin, and only insulin, should be drawn up into an insulin syringe. Never draw up insulin into a regular syringe with milliliter (mL) markings because this will cause an overdose of insulin.
- Administer subcutaneous injections at a 45- to 90-degree angle depending upon the body habitus of the individual. For extremely thin individuals and children, ensure that the angle is shallow enough that the medication is not given intramuscularly
Link to Learning
ISMP Guidelines for Safe Subcutaneous Insulin Use
Review the ISMP Guidelines for Optimizing Safe Subcutaneous Insulin Use in Adults . According to the Institute for Safe Medication Practices (ISMP), insulin is associated with more medication errors than any other type or class of drugs.
Intramuscular Injections
Intramuscular injections (IM) are administered deep into the muscular tissue beneath the dermis and subcutaneous layers (see Figure 2.9). The most common sites for IM injections are the ventrogluteal and deltoid areas. Vastus lateralis landmarks are preferred for infants and children under age 2. Table 2.5 lists the advantages and disadvantages of intramuscular administration.
| Advantages | Disadvantages |
|---|---|
|
|
Steps to administering an intramuscular medication:
-
Assemble the appropriate equipment:
- Medication
- Syringe (3 mL)
- Needles (unless preattached)
- Alcohol swabs
- Gloves (clean gloves, not sterile)
- Assess the client to determine if the drug is safe and appropriate to give.
- Check the medication, dose, volume, and expiration date.
-
Follow the seven rights of medication administration
throughout
the procedure (at least three times or according to institutional policy).
- During medication reconciliation
- When drawing up the medication
- At the bedside
- Perform hand hygiene.
- Don gloves.
- Identify the client, and verify allergies and reaction. (If the institution uses barcode scanning, scan the client’s ID band and the barcode on the vial or package, using the protocol recommended by the institution.)
- Perform the third medication check at the bedside.
-
Explain the medication to the client.
- Name (brand and generic)
- Dosage
- Indication, rationale, or reason for the drug to be given
- Frequency
- Route
- Adverse effects
- Prepare medication using the correct needle length (1–1.5 inches), gauge (18–27 gauge), and syringe (no more than 3 mL can be given via the ventrogluteal route, and no more than 1 mL can be given in the deltoid). (See Figure 2.9.)
-
Select an injection site.
- Avoid areas of hardness, bruising, rashes, inflammation, injury, or infection.
- Assess muscle size and integrity.
- Assist the client into a position in which the site or extremity can be relaxed.
- Cleanse the area with an alcohol swab using a circular motion by starting at the center and working outward in a widening circle to about 2 inches. Allow to air dry for 10 seconds.
- Remove the needle cap carefully.
- Instruct the client that they will feel a “stick.”
- Grasp the syringe like a dart with the dominant hand.
- Quickly and smoothly insert the needle through the skin and adipose tissue to the muscular layer at a 90-degree angle.
- Inject the medication with the dominant hand depressing the plunger with slow and even pressure, while holding the barrel of the syringe near the hub steady with the nondominant hand.
- Withdraw the needle smoothly at the same angle that it was inserted.
- Apply gentle pressure with the alcohol swab, but do not massage the site.
- Activate the safety device on the syringe and dispose of the syringe in the sharps container or in a puncture-resistant needle disposal container according to institutional policy. Never throw it into the trash.
- Do not recap the needle! This is a safety hazard for the nurse. Recapping needles may lead to needle sticks and exposure to pathogens.
- Remove gloves and perform hand hygiene.
- Document in the MAR. When documenting an IM injection, be sure to document the site the medication was administered to allow for the rotation of sites if other doses are necessary.
- Evaluate the client’s response to the drug(s) within the appropriate time frame.
Steps to administering an IVP medication:
-
Assemble the appropriate equipment:
- Medication
- Syringe with a needleless device
- Needles
- Normal saline flushes
- Diluent, if needed
- Alcohol swabs
- Gloves
- Assess the client to determine if the drug is safe and appropriate to give.
- Check the medication, dose, volume, and expiration date.
- Check the compatibility of the medication with the IV fluids that are hanging.
- Double-check dosage calculations.
-
Follow the seven rights of medication administration
throughout
the procedure (at least three times or according to institutional policy).
- During medication reconciliation
- When drawing up the medication
- At the bedside
- Perform hand hygiene.
- Don gloves.
-
In the medication room (or possibly the client’s room in some institutions), use aseptic technique to draw the medication into the syringe as described in the previous section.
- Use a syringe size closest to the amount of the drug needed (e.g., a 3 mL syringe to draw up 1–3 mL or a 5 mL syringe to draw up 4–5 mL).
- Double-check the rate of administration.
- Label the medication syringe with the client’s name, date of birth, medication name, dosage and volume, time, and initials. (This may vary slightly between institutions.)
- Once in the client’s room, identify the client, and verify allergies and reaction. (If the institution uses barcode scanning, scan the client’s ID band and the barcode on the vial or package, using the protocol recommended by the institution.)
- Perform the third medication check at the bedside.
-
Explain the medication to the client:
- Name (brand and generic)
- Dosage
- Indication, rationale, or reason for the drug to be given
- Frequency
- Route
- Adverse effects
- Assess the IV site. Check for redness, swelling, or tenderness. Assess local skin temperature for warmth.
- Unclamp the saline lock and expel air bubbles from a saline flush. Then remove the disinfecting cap from the port.
- Scrub the hub of the port and the threads with an alcohol swab or the institution’s preferred cleanser for 15 seconds.
- Remove the tip from the flush (see Figure 2.12) and insert the saline flush by twisting and pushing to the right. Once the flush is engaged with the saline lock, gently aspirate for blood return to assess for patency of IV.
Steps to administering a transdermal medication:
- Assess the client to determine if the drug is safe and appropriate to give.
- Check the medication, dose, volume, and expiration date.
-
Follow the seven rights of medication administration
throughout
the procedure (at least three times or according to institutional policy).
- During medication reconciliation
- When preparing the medication
- At the bedside
- Wait to open the patch or disk until at the client’s bedside.
- Perform hand hygiene.
- Don clean gloves for the administration of any patch or ointment. Never apply a patch or ointment with the bare hand because the medication can be transferred to you in the process.
- Identify the client and verify allergies and reaction. (If the institution uses barcode scanning, scan the client’s ID band and the barcode on the blister pack, tube, or package, using the protocol recommended by the institution.)
- Perform the third medication check at the bedside.
-
Explain the medication to the client:
- Name (brand and generic)
- Dosage
- Indication, rationale, or reason for the drug to be given
- Frequency
- Route
- Adverse effects
-
Remove any old patches that remain on the skin.
- Many patches are small, clear, transparent disks, so some are difficult to find.
- Assess the skin for irritation at the site of the old patch.
- Cleanse the skin with soap and water and allow it to dry before applying patches or ointments.
- Ensure that the site of the new patch is free of irritation, scrapes, open sores, or bruises. It is best if it is located on an area with little to no hair.
- Rotate sites each time a new patch is placed.
- Label the patch prior to placing it on the client with the nurse’s initials, the date, and the time administered.
- Perform hand hygiene.
- Document in the MAR.
- Evaluate the client’s response to the drug(s) within the appropriate time frame.
Nursing Implications for Transdermal Administration
The nurse should do the following for clients receiving a transdermal patch:
- Educate the client to administer the patch at the same time each day.
- Administer after a shower or bath.
- Always remain with the client until all medications are taken; do not leave drugs at the bedside unattended.
- Educate the client to develop a schedule for rotating the sites of application.
- Never cut a patch in half (unless allowed per the drug product’s labeling) because this may release all of the medication at once, resulting in an overdose. Patches are developed with special technology to release the medication slowly over a long period of time. Some patches may only need to be replaced once each week.
- Don gloves to remove a patch and dispose of according to institutional policy. Never dispose of a patch in the trash. Children have removed them from the trash thinking they were stickers, and this resulted in harm to the child. Pets also have eaten them.
- Educate the client that the patch’s effects may last for many hours following its removal (up to 72 hours).
Cutaneous Administration
Ointments and lotions are medications that can be applied to the skin. Some are used for local therapy (e.g., hydrocortisone lotion applied to a rash), whereas some are used for systemic absorption. A common medication that is delivered via an ointment for systemic absorption is nitroglycerin (for angina). This medication will be discussed specifically due to its unique formulation.
Steps to administering nitroglycerin ointment (nitroglycerin paste):
-
Assemble the appropriate equipment:
- Gloves
- Nitroglycerin ointment and application paper
- Paper tape
- Assess the client to determine if the drug is safe and appropriate to give.
- Check the medication, dose, volume, and expiration date.
-
Follow the seven rights of medication administration
throughout
the procedure (at least three times or according to institutional policy).
- During medication reconciliation
- When measuring the medication
- At the bedside
- Perform hand hygiene.
- It is important to don clean gloves for the administration of ointment. Never apply ointment with bare hands because the medication can be transferred to you in the process.
- Identify the client and verify allergies and reaction. (If the institution uses barcode scanning, scan the client’s ID band and the barcode on the tube of ointment, using the protocol recommended by the institution.)
- Perform the third medication check at the bedside.
-
Explain the medication to the client:
- Name (brand and generic)
- Dosage
- Indication, rationale, or reason for the drug to be given
- Frequency
- Route
- Adverse effects
-
Remove any old nitroglycerin doses that remain on the skin.
- Assess the skin for irritation at the site of the old nitroglycerin applicator paper.
- Cleanse the skin with soap and water and allow it to dry before applying the ointment.
- Ensure that the site of the new dose is free of irritation, open sores, scrapes, or bruises. It is best if it is located on an area with little to no hair.
- Rotate sites each time ointment is applied.
- To administer the drug, lay the applicator paper down on the counter with the print side facing down.
- Measure the amount of ointment to be used on the applicator paper, which is marked in a 2-inch strip with marks every half inch. The ointment is in a tube similar to that of toothpaste and should be gently squeezed so that a strip of ointment is placed on the applicator paper in the appropriate measurement. For example, the provider may order “nitroglycerin ointment 1 inch every 6 hours.” So, a one-inch ribbon of ointment would then be placed on the paper (see Figure 2.13).
Steps to administering vaginal medications:
-
Assemble the appropriate equipment
- Clean gloves
- Water-soluble lubricant for vaginal suppositories
- Vaginal applicator
- Perineal pad
- Medication
- Assess the client to determine if the drug is safe and appropriate to give.
- Check the medication, dose, and expiration date.
-
Follow the seven rights of medication administration
throughout
the procedure (at least three times or according to institutional policy).
- During medication reconciliation
- When preparing the medication
- At the bedside
- Perform hand hygiene.
- Don clean gloves for the administration of vaginal suppositories.
- Request that the client void prior to inserting the medication.
- Identify the client and verify allergies and reaction. (If the institution uses barcode scanning, scan the client’s ID band and the barcode on the blister pack, container, tube, or package, using the protocol recommended by the institution.)
- Perform the third medication check at the bedside.
-
Explain the medication to the client:
- Name (brand and generic)
- Dosage
- Indication, rationale, or reason for the drug to be given
- Frequency
- Route
- Adverse effects
- Provide privacy and drape the client with a sheet.
- Position the client supine, hips elevated, knees bent, with the feet flat on the bed near the hips.
- Provide perineal care as necessary.
- Fill the applicator with the prescribed medication.
- Lubricate with water-soluble lubricant.
- Spread the labia, using the nondominant hand, and expose the vagina. Gently insert the applicator into the vagina approximately 2 inches using the dominant hand.
- Push the plunger to deposit the medication into the vagina.
- Remove the applicator and wrap it in a paper towel for cleaning later or disposal.
- For suppositories, remove the wrapping and lubricate the room-temperature suppository with water-soluble jelly. Use an applicator if available; otherwise, use a finger on the dominant hand to insert the suppository about 3–4 inches into the vagina along its posterior wall. For creams or foams, insert 2–3 inches.
- Client should remain in position for 10 minutes.
- Apply a perineal pad if the client wishes.
- Wash the applicator after each use.
- Remove gloves.
- Perform hand hygiene.
- Document in the medication administration record.
- Evaluate the client’s response to the drug(s) within the appropriate time frame.
Nursing Implications for Vaginal Applications
The nurse should do the following for clients who are taking vaginal medications:
- Administer at bedtime, when possible, to allow the medication to remain in place for as long as possible.
- Assess for vaginal discharge and any other symptoms.
- Be clear in your teaching of the process because vaginal medications may be administered by the client.
- Educate the client to refrain from using douches and abstain from sexual intercourse after inserting medication.
Rectal Administration
Several medications can be given via the rectal route. This route can be used if clients are suffering from nausea and vomiting, especially if no IV is in place. This route has both a mixed first-pass effect and a non-first-pass effect. There are capillaries in the rectum that feed the portal circulation, which causes some of the medication to undergo first-pass effect; however, some of the medication will also be absorbed into the perirectal tissues locally.
Suppositories are medications that are solid at room temperature but soften and dissolve once in the rectal cavity. These medications are wrapped in foil or plastic packaging (see Figure 2.14), and it is important to remove the packaging prior to inserting the suppository in the client.
Table 2.9 lists the advantages and disadvantages of rectal administration .
| Advantages | Disadvantages |
|---|---|
|
|
Steps to administering rectal suppositories:
-
Assemble the appropriate equipment:
- Gloves
- Water-soluble lubricant
- Medication
- Bedpan, if client is on bed rest
- Assess the client to determine if the drug is safe and appropriate to give. Assess for rectal bleeding or diarrhea.
- Check the medication, dose, volume, and expiration date.
-
Follow the seven rights of medication administration
throughout
the procedure (at least three times or according to institutional policy).
- During medication reconciliation
- When preparing the medication
- At the bedside
- Perform hand hygiene.
- Don clean gloves for the administration of rectal suppositories. Never apply with a bare hand.
- Identify the client and verify allergies and reaction. (If the institution uses barcode scanning, scan the client’s ID band and the barcode on the package using the protocol recommended by the institution.)
- Perform the third medication check at the bedside.
-
Explain the medication to the client:
- Name (brand and generic)
- Dosage
- Indication, rationale, or reason for the drug to be given
- Frequency
- Route
- Adverse effects
- Position the client on their left side with the uppermost leg flexed toward the waist. (This position is called the Sim’s position or left lateral position.)
- Provide privacy. Drape the client with a sheet.
- Remove the foil or plastic wrapping from the medication (see Figure 2.14).
- Lubricate the suppository with the water-soluble gel. Consider lubricating your gloved finger to support the client’s comfort during this process. Never use petroleum-based products for lubrication because this may affect the absorption of the medication.
- The suppository is usually shaped similarly to a bullet. Insert the rounded end into the rectum while instructing the client to take a deep breath and then exhale.
- Insert the suppository along the side of the rectal wall, at least 1 inch beyond the internal rectal sphincter.
- Instruct the client to remain on their left side for approximately 20 minutes to allow the suppository to be absorbed. If the medication is being given to stimulate defecation, it may take 20–30 minutes for that to occur. If the medication is given for other reasons, such as fever or nausea, it may take as long as an hour. Check the pharmaceutical information for specifics.
- Remove gloves and perform hand hygiene.
- Document in the medication administration record.
- Evaluate the client’s response to the drug(s) within the appropriate time frame.
Nursing Implications for Rectal Administration
The nurse should do the following for clients who are taking drugs rectally:
- Do not insert a rectal suppository into stool. Palpate the rectal wall for the presence of feces.
- Have the client defecate prior to inserting the suppository, if possible.
- Never divide suppositories.
- Loss of sphincter control may be seen in older clients. Have a bedpan handy.
- Suppositories may be administered by the client. Be clear in your teaching of the process.
- It is important to educate clients who are self-administering suppositories that these drugs are to be given rectally, not orally.