Essentials of Statistics

Mario F. Triola

Chapter 9

Inferences From Two Samples - all with Video Answers

Educators

Section 1

Two Proportions

Problem 1

In the largest clinical trial ever conducted, 401,974 children were randomly assigned to two groups. The treatment group consisted of 201,229 children given the Salk vaccine for polio, and 33 of those children developed polio. The other 200,745 children were given a placebo, and 115 of those children developed polio. If we want to use the methods of this section to test the claim that the rate of polio is less for children given the Salk vaccine, are the requirements for a hypothesis test satisfied? Explain.

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 2

For the sample data given in Exercise 1 , consider the Salk vaccine treatment group to be the first sample. Identify the values of $n_{1}, \hat{p}_{1}, \hat{q}_{1}, n_{2}, \hat{p}_{2}, \hat{q}_{2}, \bar{p}$, and $\bar{q}$. Round all values so that they have six significant digits.

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 3

Refer to the hypothesis test described in Exercise 1 .
a. Identify the null hypothesis and the alternative hypothesis.
b. If the $P$ -value for the test is reported as "less than 0.001," what should we conclude about the original claim?

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 4

Using Confidence Intervals
a. Assume that we want to use a $0.05$ significance level to test the claim that $p_{1}<p_{2}$. Which is better: A hypothesis test or a confidence interval?
b. In general, when dealing with inferences for two population proportions, which two of the following are equivalent: confidence interval method; $P$ -value method; critical value method?
c. If we want to use a $0.05$ significance level to test the claim that $p_{1}<p_{2}$, what confidence level should we use?
d. If we test the claim in part (c) using the sample data in Exercise 1, we get this confidence interval: $-0.000508<p_{1}-p_{2}<-0.000309 .$ What does this confidence interval suggest about the claim?

Sheryl Ezze

Numerade Educator

Problem 5

The New York Times published an article about a study in which Professor Denise Korniewicz and other Johns Hopkins researchers subjected laboratory gloves to stress. Among 240 vinyl gloves, $63 \%$ leaked viruses. Among 240 latex gloves, $7 \%$ leaked viruses. See the accompanying display of the Statdisk results. Using a $0.01$ significance level, test the claim that vinyl gloves have a greater virus leak rate than latex gloves.

Ivan Kochetkov

Numerade Educator

Problem 6

Carpal tunnel syndrome is a common wrist complaint resulting from a compressed nerve, and it is often the result of extended use of repetitive wrist movements, such as those associated with the use of a keyboard. In a randomized controlled trial, 73 patients were treated with surgery and 67 were found to have successful treatments. Among 83 patients treated with splints, 60 were found to have successful treatments (based on data from "Splinting vs Surgery in the Treatment of Carpal Tunnel Syndrome," by Gerritsen et al., Journal of the American Medical Association, Vol. 288, No. 10 ). Use the accompanying StatCrunch display with a $0.01$ significance level to test the claim that the success rate is better with surgery.

Sheryl Ezze

Numerade Educator

Problem 7

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
The Chapter Problem involved passenger cars in Connecticut and passenger cars in New York, but here we consider passenger cars and commercial trucks. Among 2049 Connecticut passenger cars, 239 had only rear license plates. Among 334 Connecticut trucks, 45 had only rear license plates (based on samples collected by the author). A reasonable hypothesis is that passenger car owners violate license plate laws at a higher rate than owners of commercial trucks. Use a $0.05$ significance level to test that hypothesis.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 8

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
In a study of Burger King drive-through orders, it was found that 264 orders were accurate and 54 were not accurate. For McDonald's, 329 orders were found to be accurate while 33 orders were not accurate (based on data from QSR magazine). Use a $0.05$ significance level to test the claim that Burger King and McDonald's have the same accuracy rates.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Relative to accuracy of orders, does either restaurant chain appear to be better?

Sheryl Ezze

Numerade Educator

Problem 9

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
Among 198 smokers who underwent a "sustained care" program, 51 were no longer smoking after six months. Among 199 smokers who underwent a "standard care" program, 30 were no longer smoking after six months (based on data from "Sustained Care Intervention and Postdischarge Smoking Cessation Among Hospitalized Adults," by Rigotti et al., Journal of the American Medical Association, Vol. 312, No. 7). We want to use a $0.01$ significance level to test the claim that the rate of success for smoking cessation is greater with the sustained care program.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Does the difference between the two programs have practical significance?

Sheryl Ezze

Numerade Educator

Problem 10

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
Since the Hawk-Eye instant replay system for tennis was introduced at the U.S. Open in 2006 , men challenged 2441 referee calls, with the result that 1027 of the calls were overturned. Women challenged 1273 referee calls, and 509 of the calls were overturned. We want to use a $0.05$ significance level to test the claim that men and women have equal success in challenging calls.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Based on the results, does it appear that men and women have equal success in challenging calls?

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 11

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 306 people over the age of 55 , 68 dream in black and white, and among 298 people under the age of 25,13 dream in black and white (based on data from "Do We Dream in Color?" by Eva Murzyn, Consciousness and Cognition, Vol. 17, No. 4). We want to use a $0.01$ significance level to test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion of those under 25 .
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. An explanation given for the results is that those over the age of 55 grew up exposed to media that was mostly displayed in black and white. Can the results from parts (a) and (b) be used to verify that explanation?

Ivan Kochetkov

Numerade Educator

Problem 12

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
OxyContin (oxycodone) is a drug used to treat pain, but it is well known for its addictiveness and danger. In a clinical trial, among subjects treated with OxyContin, 52 developed nausea and 175 did not develop nausea. Among other subjects given placebos, 5 developed nausea and 40 did not develop nausea (based on data from Purdue Pharma L.P.). Use a $0.05$ significance level to test for a difference between the rates of nausea for those treated with OxyContin and those given a placebo.
a. Use a hypothesis test.
b. Use an appropriate confidence interval.
c. Does nausea appear to be an adverse reaction resulting from OxyContin?

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 13

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2823 occupants not wearing seat belts, 31 were killed. Among 7765 occupants wearing seat belts, 16 were killed (based on data from "Who Wants Airbags?" by Meyer and Finney, Chance, Vol. 18, No. 2 ). We want to use a $0.05$ significance level to test the claim that seat belts are effective in reducing fatalities.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. What does the result suggest about the effectiveness of seat belts?

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 14

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
A study investigated survival rates for in-hospital patients who suffered cardiac arrest. Among 58,593 patients who had cardiac arrest during the day, 11,604 survived and were discharged. Among 28,155 patients who suffered cardiac arrest at night, 4139 survived and were discharged (based on data from "Survival from In-Hospital Cardiac Arrest During Nights and Weekends," by Peberdy et al., Journal of the American Medical Association, Vol. 299, No. 7). We want to use a $0.01$ significance level to test the claim that the survival rates are the same for day and night.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Based on the results, does it appear that for in-hospital patients who suffer cardiac arrest, the survival rate is the same for day and night?

Sheryl Ezze

Numerade Educator

Problem 15

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
Rhinoviruses typically cause common colds. In a test of the effectiveness of echinacea, 40 of the 45 subjects treated with echinacea developed rhinovirus infections. In a placebo group, 88 of the 103 subjects developed rhinovirus infections (based on data from "An Evaluation of Echinacea Angustifolia in Experimental Rhinovirus Infections," by Turner et al., New England Journal of Medicine, Vol. 353, No. 4 ). We want to use a $0.05$ significance level to test the claim that echinacea has an effect on rhinovirus infections.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Based on the results, does echinacea appear to have any effect on the infection rate?

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 16

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
In a randomized controlled trial in Kenya, insecticide-treated bednets were tested as a way to reduce malaria. Among 343 infants using bednets, 15 developed malaria. Among 294 infants not using bednets, 27 developed malaria (based on data from "Sustainability of Reductions in Malaria Transmission and Infant Mortality in Western Kenya with Use of Insecticide-Treated Bednets," by Lindblade et al., Journal of the American Medical Association, Vol. 291, No. 21 ). We want to use a $0.01$ significance level to test the claim that the incidence of malaria is lower for infants using bednets.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Based on the results, do the bednets appear to be effective?

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 17

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
A study was conducted to investigate the association between cell phone use and hemispheric brain dominance. Among 216 subjects who prefer to use their left ear for cell phones, 166 were right-handed. Among 452 subjects who prefer to use their right ear for cell phones, 436 were right-handed (based on data from "Hemispheric Dominance and Cell Phone Use," by Seidman et al., JAMA Otolaryngology-Head \& Neck Surgery, Vol. 139, No. 5 ). We want to use a $0.01$ significance level to test the claim that the rate of right-handedness for those who prefer to use their left ear for cell phones is less than the rate of right-handedness for those who prefer to use their right ear for cell phones. (Try not to get too confused here.)
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 18

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
A trial was conducted with 75 women in China given a 100 -yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from "The Denomination Effect," by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a $0.05$ significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. If the significance level is changed to $0.01$, does the conclusion change?

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 19

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
In a study of treatments for very painful "cluster" headaches, 150 patients were treated with oxygen and 148 other patients were given a placebo consisting of ordinary air. Among the 150 patients in the oxygen treatment group, 116 were free from headaches 15 minutes after treatment. Among the 148 patients given the placebo, 29 were free from headaches 15 minutes after treatment (based on data from "High-Flow Oxygen for Treatment of Cluster Headache," by Cohen, Burns, and Goadsby, Journal of the American Medical Association, Vol. 302, No. 22). We want to use a $0.01$ significance level to test the claim that the oxygen treatment is effective.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Based on the results, is the oxygen treatment effective?

Sheryl Ezze

Numerade Educator

Problem 20

20. Does Aspirin Prevent Heart Disease? In a trial designed to test the effectiveness of aspirin in preventing heart disease, 11,037 male physicians were treated with aspirin and 11,034 male physicians were given placebos. Among the subjects in the aspirin treatment group, 139 experienced myocardial infarctions (heart attacks). Among the subjects given placebos, 239 experienced myocardial infarctions (based on data from "Final Report on the Aspirin Component of the Ongoing Physicians' Health Study," New England Journal of Medicine, Vol. 321 :
$129-135)$. Use a $0.05$ significance level to test the claim that aspirin has no effect on myocardial infarctions.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Based on the results, does aspirin appear to be effective?In a trial designed to test the effectiveness of aspirin in preventing heart disease, 11,037 male physicians were treated with aspirin and 11,034 male physicians were given placebos. Among the subjects in the aspirin treatment group, 139 experienced myocardial infarctions (heart attacks). Among the subjects given placebos, 239 experienced myocardial infarctions (based on data from "Final Report on the Aspirin Component of the Ongoing Physicians' Health Study," New England Journal of Medicine, Vol. 321 :
$129-135)$. Use a $0.05$ significance level to test the claim that aspirin has no effect on myocardial infarctions.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Based on the results, does aspirin appear to be effective?

Sheryl Ezze

Numerade Educator

Problem 21

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
In a random sample of males, it was found that 23 write with their left hands and 217 do not. In a random sample of females, it was found that 65 write with their left hands and 455 do not (based on data from "The Left-Handed: Their Sinister History," by Elaine Fowler Costas, Education Resources Information Center, Paper 399519). We want to use a $0.01$ significance level to test the claim that the rate of left-handedness among males is less than that among females.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Based on the results, is the rate of left-handedness among males less than the rate of lefthandedness among females?

Sheryl Ezze

Numerade Educator

Problem 22

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
A study investigated rates of fatalities among patients with serious traumatic injuries. Among 61,909 patients transported by helicopter, 7813 died. Among 161,566 patients transported by ground services, 17,775 died (based on data from "Association Between Helicopter vs Ground Emergency Medical Services and Survival for Adults With Major Trauma," by Galvagno et al., Journal of the American Medical Association, Vol. 307, No. 15). Use a $0.01$ significance level to test the claim that the rate of fatalities is higher for patients transported by helicopter.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Considering the test results and the actual sample rates, is one mode of transportation better than the other? Are there other important factors to consider?

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 23

The sample size needed to estimate the difference between two population proportions to within a margin of error $E$ with a confidence level of $1-\alpha$ can be found by using the following expression:
$$
E=z_{\alpha / 2} \sqrt{\frac{p_{1} q_{1}}{n_{1}}+\frac{p_{2} q_{2}}{n_{2}}}
$$
Replace $n_{1}$ and $n_{2}$ by $n$ in the preceding formula (assuming that both samples have the same size) and replace each of $p_{1}, q_{1}, p_{2}$, and $q_{2}$ by $0.5$ (because their values are not known). Solving for $n$ results in this expression:
$$
n=\frac{z_{\alpha / 2}^{2}}{2 E^{2}}
$$
Use this expression to find the size of each sample if you want to estimate the difference between the proportions of men and women who own smartphones. Assume that you want $95 \%$ confidence that your error is no more than $0.03$.

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 24

In one segment of the TV series MythBusters, an experiment was conducted to test the common belief that people are more likely to yawn when they see others yawning. In one group, 34 subjects were exposed to yawning, and 10 of them yawned. In another group, 16 subjects were not exposed to yawning, and 4 of them yawned. We want to test the belief that people are more likely to yawn when they are exposed to yawning.
a. Why can't we test the claim using the methods of this section?
b. If we ignore the requirements and use the methods of this section, what is the $P$ -value? How does it compare to the $P$ -value of $0.5128$ that would be obtained by using Fisher's exact test?
c. Comment on the conclusion of the Mythbusters segment that yawning is contagious.

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 25

In the article "On Judging the Significance of Differences by Examining the Overlap Between Confidence Intervals"' by Schenker and Gentleman (American Statistician, Vol. 55, No. 3), the authors consider sample data in this statement:
"Independent simple random samples, each of size 200 , have been drawn, and 112 people in the first sample have the attribute, whereas 88 people in the second sample have the attribute."
a. Use the methods of this section to construct a $95 \%$ confidence interval estimate of the difference $p_{1}-p_{2}$. What does the result suggest about the equality of $p_{1}$ and $p_{2}$ ?
b. Use the methods of Section $7-1$ to construct individual $95 \%$ confidence interval estimates for each of the two population proportions. After comparing the overlap between the two confidence intervals, what do you conclude about the equality of $p_{1}$ and $p_{2}$ ?
c. Use a $0.05$ significance level to test the claim that the two population proportions are equal. What do you conclude?
d. Based on the preceding results, what should you conclude about the equality of $p_{1}$ and $p_{2}$ ? Which of the three preceding methods is least effective in testing for the equality of $p_{1}$ and $p_{2}$ ?

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 26

Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 10 having a common attribute. The second sample consists of 2000 people with 1404 of them having the same common attribute. Compare the results from a hypothesis test of $p_{1}=p_{2}$ (with a $0.05$ significance level) and a $95 \%$ confidence interval estimate of $p_{1}-p_{2}$.

Sheryl Ezze

Numerade Educator