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Elementary Statistics: Picturing the World

Ron Larson, Betsy Farber

Chapter 8

Hypothesis Testing with Two Samples - all with Video Answers

Educators

Section 1

Testing the Difference Between Means (Independent Samples, $\sigma_{1}$ and $\sigma_{2}$ Known)

03:23

Problem 1

What is the difference between two samples that are dependent and two samples that are independent? Give an example of each.

Sneha Ravi
Sneha Ravi
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04:09

Problem 2

Explain how to perform a two-sample $z$ -test for the difference between two population means using independent samples with $\sigma_{1}$ and $\sigma_{2}$ known.

Sneha Ravi
Sneha Ravi
Numerade Educator
02:26

Problem 3

Describe another way you can perform a hypothesis test for the difference between the means of two populations using independent samples with $\sigma_{1}$ and $\sigma_{2}$ known that does not use rejection regions.

Sneha Ravi
Sneha Ravi
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01:42

Problem 4

What conditions are necessary in order to use the $z$ -test to test the difference between two population means?

Sneha Ravi
Sneha Ravi
Numerade Educator
01:18

Problem 5

Classify the two samples as independent or dependent. Explain your reasoning.
Sample 1: The maximum bench press weights for 53 football players
Sample 2: The maximum bench press weights for the same 53 football players after completing a weight lifting program

Sneha Ravi
Sneha Ravi
Numerade Educator
01:13

Problem 6

Classify the two samples as independent or dependent. Explain your reasoning.
Sample 1: The IQ scores of 60 females
Sample 2: The IQ scores of 60 males

Sneha Ravi
Sneha Ravi
Numerade Educator
01:24

Problem 7

Classify the two samples as independent or dependent. Explain your reasoning.
Sample 1: The average speed of 23 powerboats using an old hull design
Sample 2: The average speed of 14 powerboats using a new hull design

Sneha Ravi
Sneha Ravi
Numerade Educator
01:16

Problem 8

Classify the two samples as independent or dependent. Explain your reasoning.
Sample 1: The commute times of 10 workers when they use their own vehicles
Sample 2: The commute times of the same 10 workers when they use public transportation

Sneha Ravi
Sneha Ravi
Numerade Educator
02:13

Problem 9

Use the TI-84 Plus display to make a decision to reject or fail to reject the null hypothesis at the level of significance. Make your decision using the standardized test statistic and using the P-value. Assume the sample sizes are equal.
$$\alpha=0.05$$
CAN'T COPY THE FIGURE

Sneha Ravi
Sneha Ravi
Numerade Educator
02:22

Problem 10

Use the TI-84 Plus display to make a decision to reject or fail to reject the null hypothesis at the level of significance. Make your decision using the standardized test statistic and using the P-value. Assume the sample sizes are equal.
$$\alpha=0.01$$
CAN'T COPY THE FIGURE

Sneha Ravi
Sneha Ravi
Numerade Educator
02:49

Problem 11

Test the claim about the difference between two population means $\mu_{1}$ and $\mu_{2}$ at the level of significance $\alpha .$ Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
Claim: $\mu_{1}=\mu_{2} ; \alpha=0.1$
Population statistics: $\sigma_{1}=3.4$ and $\sigma_{2}=1.5$ Sample statistics: $\bar{x}_{1}=16, n_{1}=29$ and $\bar{x}_{2}=14, n_{2}=28$

Sneha Ravi
Sneha Ravi
Numerade Educator
02:51

Problem 12

Test the claim about the difference between two population means $\mu_{1}$ and $\mu_{2}$ at the level of significance $\alpha .$ Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
Claim: $\mu_{1}>\mu_{2} ; \alpha=0.10$
Population statistics: $\sigma_{1}=40$ and $\sigma_{2}=15$ Sample statistics: $\bar{x}_{1}=500, n_{1}=100$ and $\bar{x}_{2}=495, n_{2}=75$

Sneha Ravi
Sneha Ravi
Numerade Educator
02:49

Problem 13

Test the claim about the difference between two population means $\mu_{1}$ and $\mu_{2}$ at the level of significance $\alpha .$ Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
Claim: $\mu_{1}<\mu_{2} ; \alpha=0.05$
Population statistics: $\sigma_{1}=75$ and $\sigma_{2}=105$ Sample statistics: $\bar{x}_{1}=2435, n_{1}=35$ and $\bar{x}_{2}=2432, n_{2}=90$

Sneha Ravi
Sneha Ravi
Numerade Educator
03:24

Problem 14

Test the claim about the difference between two population means $\mu_{1}$ and $\mu_{2}$ at the level of significance $\alpha .$ Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
Claim: $\mu_{1} \leq \mu_{2} ; \alpha=0.03$
Population statistics: $\sigma_{1}=136$ and $\sigma_{2}=215$
Sample statistics: $\bar{x}_{1}=5004, n_{1}=144$ and $\bar{x}_{2}=4895, n_{2}=156$

Sneha Ravi
Sneha Ravi
Numerade Educator
01:19

Problem 15

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
To compare the braking distances for two types of tires, a safety engineer conducts 35 braking tests for each type. The mean braking distance for Type $A$ is 42 feet. Assume the population standard deviation is 4.7 feet. The mean braking distance for Type $\mathrm{B}$ is 45 feet. Assume the population standard deviation is 4.3 feet. At $\alpha=0.10,$ can the engineer support the claim that the mean braking distances are different for the two types of tires?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:18

Problem 16

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
To compare the amounts spent in the first three months by clients of two meal-replacement diets, a researcher randomly selects 20 clients of each diet. The mean amount spent for Diet $A$ is $\$ 643$ Assume the population standard deviation is $\$ 89 .$ The mean amount spent for Diet B is $\$ 588 .$ Assume the population standard deviation is $\$ 75$ At $\alpha=0.01,$ can the researcher support the claim that the mean amount spent in the first three months by clients of Diet $A$ is greater than the mean amount spent in the first three months by clients of Diet B?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:26

Problem 17

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
An energy company wants to choose between two regions in a state to install energy-producing wind turbines. A researcher claims that the wind speed in Region $A$ is less than the wind speed in Region B. To test the regions, the average wind speed is calculated for 60 days in each region. The mean wind speed in Region $A$ is 14.0 miles per hour. Assume the population standard deviation is 2.9 miles per hour. The mean wind speed in Region $\mathrm{B}$ is 15.1 miles per hour. Assume the population standard deviation is 3.3 miles per hour. At $\alpha=0.05,$ can the company support the researcher's claim?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:27

Problem 18

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
You want to buy a washing machine, and a salesperson tells you that the mean repair costs for Model $A$ and Model B are equal. You research the repair costs. The mean repair cost of
24 Model A washing machines is $\$ 208$. Assume the population standard deviation is $\$ 18 .$ The mean repair cost of 26 Model B washing machines
is $\$ 221 .$ Assume the population standard deviation is $\$ 22 .$ At $\alpha=0.05,$ can you reject the salesperson's claim?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:36

Problem 19

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
The mean ACT score for 43 male high school students is
21.1. Assume the population standard deviation is $5.0 .$ The mean ACT score for 56 female high school students is $20.9 .$ Assume the population standard deviation is $4.7 .$ At $\alpha=0.01,$ can you reject the claim that male and female high school students have equal ACT scores?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:12

Problem 20

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
A guidance counselor claims that high school students in a college preparation program have higher ACT scores than those in a general program. The mean ACT score for 49 high school students who are in a college preparation program is $22.2 .$ Assume the population standard deviation is $4.8 .$ The mean ACT score for 44 high school students who are in a general program
is $20.0 .$ Assume the population standard deviation is $5.4 .$ At $\alpha=0.10,$ can you support the guidance counselor's claim?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:35

Problem 21

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
A real estate agency says that the mean home sales price in Spring, Texas, is the same as in Austin, Texas. The mean home sales price for
25 homes in Spring is $\$ 127,414 .$ Assume the population standard deviation
is $\$ 25,875 .$ The mean home sales price for 25 homes in Austin is $\$ 112,301 .$ Assume the population standard deviation is $\$ 27,110 .$ At $\alpha=0.01,$ is there enough evidence to reject the agency's claim?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:18

Problem 22

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
Refer to Exercise $21 .$ Two more samples are taken, one from Spring and one from Austin. For 50 homes in Spring, $\bar{x}_{1}=\$ 124,329 .$ For 50 homes in Austin, $\bar{x}_{2}=\$ 110,483 .$ Use $\alpha=0.01 .$ Do the new samples lead to a different conclusion?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:10

Problem 23

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
A sociologist claims that children ages $6-17$ spent more time watching television in 1981 than children ages $6-17$ do today. A study was conducted in 1981 to find the time that children ages $6-17$ spent watching television on weekdays. The results (in hours per weekday) are shown below. Assume the population standard deviation is 0.6 hour.
$$\begin{array}{llllllllll}
2.0 & 2.5 & 2.1 & 2.3 & 2.1 & 1.6 & 2.6 & 2.1 & 2.1 & 2.4 \\
2.1 & 2.1 & 1.5 & 1.7 & 2.1 & 2.3 & 2.5 & 3.3 & 2.2 & 2.9 \\
1.5 & 1.9 & 2.4 & 2.2 & 1.2 & 3.0 & 1.0 & 2.1 & 1.9 & 2.2
\end{array}$$
Recently, a similar study was conducted. The results are shown below. Assume the population standard deviation is 0.5 hour.
$$\begin{array}{llllllllll}
2.9 & 1.8 & 0.9 & 1.6 & 2.0 & 1.7 & 2.5 & 1.1 & 1.6 & 2.0 \\
1.4 & 1.7 & 1.7 & 1.9 & 1.6 & 1.7 & 1.2 & 2.0 & 2.6 & 1.6 \\
1.5 & 2.5 & 1.6 & 2.1 & 1.7 & 1.8 & 1.1 & 1.4 & 1.2 & 2.3
\end{array}$$
At $\alpha=0.05,$ can you support the sociologist's claim?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:07

Problem 24

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
A sociologist claims that children ages $12-14$ spent less time sleeping in 1981 than children ages $12-14$ do today. A study was conducted in 1981 to find the time that children ages $12-14$ spent sleeping on weekdays. The results (in hours per weekday) are shown below. Assume the population standard deviation is 0.5 hour.
$$\begin{array}{cccccccccc}
7.3 & 7.5 & 7.7 & 7.8 & 6.9 & 7.9 & 8.3 & 7.9 & 8.0 & 8.3 \\
7.4 & 8.5 & 7.9 & 6.8 & 8.2 & 7.1 & 7.9 & 7.6 & 8.0 & 8.0
\end{array}$$
Recently, a similar study was conducted. The results are shown below. Assume the population standard deviation is 0.4 hour.
$$\begin{array}{llllllllll}
9.2 & 9.1 & 10.0 & 9.3 & 9.6 & 8.0 & 9.5 & 8.2 & 9.0 & 8.6 \\
9.2 & 9.2 & 8.9 & 9.1 & 8.4 & 9.0 & 8.8 & 9.1 & 8.6 & 9.0
\end{array}$$
At $\alpha=0.01,$ can you support the sociologist's claim?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
00:35

Problem 25

Explain why the null hypothesis $H_{0}: \mu_{1}=\mu_{2}$ is equivalent to the null hypothesis $H_{0}: \mu_{1}-\mu_{2}=0$.

Hossam Mohamed
Hossam Mohamed
Numerade Educator
00:28

Problem 26

Explain why the null hypothesis $H_{0}: \mu_{1} \geq \mu_{2}$ is equivalent to the null hypothesis $H_{0}: \mu_{1}-\mu_{2} \geq 0$.

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:16

Problem 27

Sometimes a researcher is interested in testing a difference in means other than zero. For instance, you may want to determine whether the difference between the mean annual salaries for a job differ by more than a certain amount between two states. In Exercises 27 and $28,$ you will test the difference between two means using a null hypothesis of $H_{0}: \mu_{1}-\mu_{2}=k$
$H_{0}: \mu_{1}-\mu_{2} \geq k,$ or $H_{0}: \mu_{1}-\mu_{2} \leq k .$ The standardized test statistic is still
$$
z=\frac{\left(\bar{x}_{1}-\bar{x}_{2}\right)-\left(\mu_{1}-\mu_{2}\right)}{\sigma_{\bar{x}_{1}-\bar{x}_{2}}} \quad \text { where } \quad \sigma_{\bar{x}_{1}-\bar{x}_{2}}=\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}
$$
Microbiologist Salaries Is the difference between the mean annual salaries of microbiologists in Maryland and California more than $\$ 10,000 ?$ To decide, you select a random sample of microbiologists from each state. The results of each survey are shown in the figure. Assume the population standard deviations are $\sigma_{1}=\$ 8795$ and $\sigma_{2}=\$ 9250 .$ At $\alpha=0.05,$ what should you conclude?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:29

Problem 28

Sometimes a researcher is interested in testing a difference in means other than zero. For instance, you may want to determine whether the difference between the mean annual salaries for a job differ by more than a certain amount between two states. In Exercises 27 and $28,$ you will test the difference between two means using a null hypothesis of $H_{0}: \mu_{1}-\mu_{2}=k$
$H_{0}: \mu_{1}-\mu_{2} \geq k,$ or $H_{0}: \mu_{1}-\mu_{2} \leq k .$ The standardized test statistic is still
$$
z=\frac{\left(\bar{x}_{1}-\bar{x}_{2}\right)-\left(\mu_{1}-\mu_{2}\right)}{\sigma_{\bar{x}_{1}-\bar{x}_{2}}} \quad \text { where } \quad \sigma_{\bar{x}_{1}-\bar{x}_{2}}=\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}
$$
Registered Nurse Salaries Is the difference between the mean annual salaries of registered nurses in New Jersey and Delaware equal to $\$ 10,000 ?$ To decide, you select a random sample of registered nurses from each state. The results of each survey are shown in the figure. Assume the population standard deviations are $\sigma_{1}=\$ 8345$ and $\sigma_{2}=\$ 7620 .$ At $\alpha=0.01,$ what should you conclude?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:02

Problem 29

You can construct a confidence interval for the difference between two population means $\mu_{1}-\mu_{2}, a s$ shown below, when both population standard deviations are known, and either both populations are normally distributed or both $n_{1} \geq 30$ and $n_{2} \geq 30 .$ Also, the samples must be randomly selected and independent.
$$
\left(\bar{x}_{1}-\bar{x}_{2}\right)-z_{c} \sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}<\mu_{1}-\mu_{2}<\left(\bar{x}_{1}-\bar{x}_{2}\right)+z_{c} \sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}
$$
Construct the indicated confidence interval for $\mu_{1}-\mu_{2}$.
Construct a $95 \%$ confidence interval for the difference between the mean annual salaries of microbiologists in Maryland and California using the data from Exercise 27.

Hossam Mohamed
Hossam Mohamed
Numerade Educator
00:56

Problem 30

You can construct a confidence interval for the difference between two population means $\mu_{1}-\mu_{2}, a s$ shown below, when both population standard deviations are known, and either both populations are normally distributed or both $n_{1} \geq 30$ and $n_{2} \geq 30 .$ Also, the samples must be randomly selected and independent.
$$
\left(\bar{x}_{1}-\bar{x}_{2}\right)-z_{c} \sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}<\mu_{1}-\mu_{2}<\left(\bar{x}_{1}-\bar{x}_{2}\right)+z_{c} \sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}
$$
Construct the indicated confidence interval for $\mu_{1}-\mu_{2}$.
Construct a $99 \%$ confidence interval for the difference between the mean annual salaries of registered nurses in New Jersey and Delaware using the data from Exercise $28 .$

Hossam Mohamed
Hossam Mohamed
Numerade Educator