Elementary Statistics

Mario F. Triola

Chapter 8

Hypothesis Testing - all with Video Answers

Educators

Section 1

Basics of Hypothesis Testing

Problem 1

Statistical Literacy and Critical Thinking
A bottle contains a label stating that it contains Spring Valley pills with $500 \mathrm{mg}$ of vitamin $\mathrm{C}$, and another bottle contains a label stating that it contains Bayer pills with 325 mg of aspirin. When testing claims about the mean contents of the pills, which would have more serious implications: rejection of the Spring Valley vitamin C claim or rejection of the Bayer aspirin claim? Is it wise to use the same significance level for hypothesis tests about the mean amount of vitamin $\mathrm{C}$ and the mean amount of aspirin?

Nick Johnson

Nick Johnson

Numerade Educator

Problem 2

Data Set 3 "Body Temperatures" in Appendix B includes sample body temperatures. We could use methods of Chapter 7 for making an estimate, or we could use those values to test the common belief that the mean body temperature is 98.6 $^{\circ}$ F. What is the difference between estimating and hypothesis testing?

Harsh Gadhiya

Numerade Educator

Problem 3

Mean Height of Men A formal hypothesis test is to be conducted using the claim that the mean height of men is equal to $174.1 \mathrm{cm} .$
a. What is the null hypothesis, and how is it denoted?
b. What is the alternative hypothesis, and how is it denoted?
c. What are the possible conclusions that can be made about the null hypothesis?
d. Is it possible to conclude that "there is sufficient evidence to support the claim that the mean height of men is equal to $174.1 \mathrm{cm}^{\prime \prime} ?$

Harsh Gadhiya

Numerade Educator

Problem 4

The Ericsson method is one of several methods claimed to increase the likelihood of a baby girl. In a clinical trial, results could be analyzed with a formal hypothesis test with the alternative hypothesis of $p>0.5,$ which corresponds to the claim that the method increases the likelihood of having a girl, so that the proportion of girls is greater than 0.5. If you have an interest in establishing the success of the method, which of the following P-values would you prefer: $0.999,0.5,0.95,0.05,0.01,0.001 ?$ Why?

Harsh Gadhiya

Numerade Educator

Problem 5

Identifying $H_{0}$ and $H_{1}$ Do the following:
a. Express the original claim in symbolic form.
b. Identify the null and alternative hypotheses.
Claim: Most adults would erase all of their personal information online if they could. A GFI Software survey of 565 randomly selected adults showed that $59 \%$ of them would erase all of their personal information online if they could.

Harsh Gadhiya

Numerade Educator

Problem 6

Identifying $H_{0}$ and $H_{1}$ Do the following:
a. Express the original claim in symbolic form.
b. Identify the null and alternative hypotheses.
Claim: Fewer than $95 \%$ of adults have a cell phone. In a Marist poll of 1128 adults, $87 \%$ said that they have a cell phone.

Harsh Gadhiya

Numerade Educator

Problem 7

Identifying $H_{0}$ and $H_{1}$ Do the following:
a. Express the original claim in symbolic form.
b. Identify the null and alternative hypotheses.
Claim: The mean pulse rate (in beats per minute, or bpm) of adult males is equal to 69 bpm. For the random sample of 153 adult males in Data Set 1 "Body Data" in Appendix B, the mean pulse rate is 69.6 bpm and the standard deviation is 11.3 bpm.

Harsh Gadhiya

Numerade Educator

Problem 8

Identifying $H_{0}$ and $H_{1}$ Do the following:
a. Express the original claim in symbolic form.
b. Identify the null and alternative hypotheses.
Claim: The standard deviation of pulse rates of adult males is more than 11 bpm. For the random sample of 153 adult males in Data Set 1 "Body Data" in Appendix B, the pulse rates have a standard deviation of 11.3 bpm.

Harsh Gadhiya

Numerade Educator

Problem 9

Conclusions Refer to the exercise identified. Make subjective estimates to decide whether results are significantly low or significantly high, then state a conclusion about the original claim. For example, if the claim is that a coin favors heads and sample results consist of 11 heads in 20 flips, conclude that there is not sufficient evidence to support the claim that the coin favors heads (because it is easy to get II heads in 20 flips by chance with a fair coin).
Exercise 5 "Online Data"

Harsh Gadhiya

Numerade Educator

Problem 10

Conclusions Refer to the exercise identified. Make subjective estimates to decide whether results are significantly low or significantly high, then state a conclusion about the original claim. For example, if the claim is that a coin favors heads and sample results consist of 11 heads in 20 flips, conclude that there is not sufficient evidence to support the claim that the coin favors heads (because it is easy to get II heads in 20 flips by chance with a fair coin).
Exercise 6 "Cell Phone"

Harsh Gadhiya

Numerade Educator

Problem 11

Conclusions Refer to the exercise identified. Make subjective estimates to decide whether results are significantly low or significantly high, then state a conclusion about the original claim. For example, if the claim is that a coin favors heads and sample results consist of 11 heads in 20 flips, conclude that there is not sufficient evidence to support the claim that the coin favors heads (because it is easy to get II heads in 20 flips by chance with a fair coin).
Exercise 7 "Pulse Rates"

Harsh Gadhiya

Numerade Educator

Problem 12

Conclusions Refer to the exercise identified. Make subjective estimates to decide whether results are significantly low or significantly high, then state a conclusion about the original claim. For example, if the claim is that a coin favors heads and sample results consist of 11 heads in 20 flips, conclude that there is not sufficient evidence to support the claim that the coin favors heads (because it is easy to get II heads in 20 flips by chance with a fair coin).
Exercise 8 "Pulse Rates"

Harsh Gadhiya

Numerade Educator

Problem 13

Refer to the exercise identified and find the value of the test statistic. (Refer to Table 8 -2 on page 362 to select the correct expression for evaluating the test statistic.)
Exercise 5 "Online Data"

Harsh Gadhiya

Numerade Educator

Problem 14

Refer to the exercise identified and find the value of the test statistic. (Refer to Table 8 -2 on page 362 to select the correct expression for evaluating the test statistic.)
Exercise 6 "Cell Phone"

Harsh Gadhiya

Numerade Educator

Problem 15

Refer to the exercise identified and find the value of the test statistic. (Refer to Table 8 -2 on page 362 to select the correct expression for evaluating the test statistic.)
Exercise 7 "Pulse Rates"

Harsh Gadhiya

Numerade Educator

Problem 16

Refer to the exercise identified and find the value of the test statistic. (Refer to Table 8 -2 on page 362 to select the correct expression for evaluating the test statistic.)
Exercise 8 "Pulse Rates"

Harsh Gadhiya

Numerade Educator

Problem 17

Do the following:
a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.
b. Find the $P$ -value. (See Figure $8-3$ on page $364 .$ )
c. Using a significance level of $\alpha=0.05,$ should we reject $H_{0}$ or should we fail to reject $H_{0} ?$
The test statistic of $z=1.00$ is obtained when testing the claim that $p>0.3.$

Harsh Gadhiya

Numerade Educator

Problem 18

Do the following:
a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.
b. Find the $P$ -value. (See Figure $8-3$ on page $364 .$ )
c. Using a significance level of $\alpha=0.05,$ should we reject $H_{0}$ or should we fail to reject $H_{0} ?$
The test statistic of $z=-2.50$ is obtained when testing the claim that $p<0.75.$

Harsh Gadhiya

Numerade Educator

Problem 19

Do the following:
a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.
b. Find the $P$ -value. (See Figure $8-3$ on page $364 .$ )
c. Using a significance level of $\alpha=0.05,$ should we reject $H_{0}$ or should we fail to reject $H_{0} ?$
The test statistic of $z=2.01$ is obtained when testing the claim that $p \neq 0.345.$

Harsh Gadhiya

Numerade Educator

Problem 20

Do the following:
a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.
b. Find the $P$ -value. (See Figure $8-3$ on page $364 .$ )
c. Using a significance level of $\alpha=0.05,$ should we reject $H_{0}$ or should we fail to reject $H_{0} ?$
The test statistic of $z=-1.94$ is obtained when testing the claim that $p=3 / 8.$

Harsh Gadhiya

Numerade Educator

Problem 21

Refer to the information in the given exercise and do the following.
a. Find the critical value(s).
b. Using a significance level of $\alpha=0.05,$ should we reject $H_{0}$ or should we fail to reject $H_{0} ?$
Exercise 17

Harsh Gadhiya

Numerade Educator

Problem 22

Refer to the information in the given exercise and do the following.
a. Find the critical value(s).
b. Using a significance level of $\alpha=0.05,$ should we reject $H_{0}$ or should we fail to reject $H_{0} ?$
Exercise 18

Harsh Gadhiya

Numerade Educator

Problem 23

Refer to the information in the given exercise and do the following.
a. Find the critical value(s).
b. Using a significance level of $\alpha=0.05,$ should we reject $H_{0}$ or should we fail to reject $H_{0} ?$
Exercise 19

Harsh Gadhiya

Numerade Educator

Problem 24

Refer to the information in the given exercise and do the following.
a. Find the critical value(s).
b. Using a significance level of $\alpha=0.05,$ should we reject $H_{0}$ or should we fail to reject $H_{0} ?$
Exercise 20

Harsh Gadhiya

Numerade Educator

Problem 25

Use a significance level of $\alpha=0.05$ and use the given information for the following:
a. State a conclusion about the null hypothesis. (Reject $H_{0}$ or fail to reject $H_{0 .}$ )
b. Without using technical terms or symbols, state a final conclusion that addresses the original claim.
Original claim: More than $58 \%$ of adults would erase all of their personal information online if they could. The hypothesis test results in a $P$ -value of 0.3257.

Harsh Gadhiya

Numerade Educator

Problem 26

Use a significance level of $\alpha=0.05$ and use the given information for the following:
a. State a conclusion about the null hypothesis. (Reject $H_{0}$ or fail to reject $H_{0 .}$ )
b. Without using technical terms or symbols, state a final conclusion that addresses the original claim.
Original claim: Fewer than $90 \%$ of adults have a cell phone. The hypothesis test results in a $P$ -value of 0.0003.

Harsh Gadhiya

Numerade Educator

Problem 27

Use a significance level of $\alpha=0.05$ and use the given information for the following:
a. State a conclusion about the null hypothesis. (Reject $H_{0}$ or fail to reject $H_{0 .}$ )
b. Without using technical terms or symbols, state a final conclusion that addresses the original claim.
Original claim: The mean pulse rate (in beats per minute) of adult males is 72 bpm. The hypothesis test results in a $P$ -value of 0.0095.

Harsh Gadhiya

Numerade Educator

Problem 28

Use a significance level of $\alpha=0.05$ and use the given information for the following:
a. State a conclusion about the null hypothesis. (Reject $H_{0}$ or fail to reject $H_{0 .}$ )
b. Without using technical terms or symbols, state a final conclusion that addresses the original claim.
Original claim: The standard deviation of pulse rates of adult males is more than 11 bpm. The hypothesis test results in a $P$ -value of 0.3045.

Harsh Gadhiya

Numerade Educator

Problem 29

Type I and Type II Errors provide statements that identify the type I error and the type II error that correspond to the given claim. (Although conclusions are usually expressed in verbal form, the answers here can be expressed with statements that include symbolic expressions such as $p=0.1 .$ )
The proportion of people who write with their left hand is equal to 0.1.

Harsh Gadhiya

Numerade Educator

Problem 30

Type I and Type II Errors provide statements that identify the type I error and the type II error that correspond to the given claim. (Although conclusions are usually expressed in verbal form, the answers here can be expressed with statements that include symbolic expressions such as $p=0.1 .$ )
The proportion of people with blue eyes is equal to 0.35.

Harsh Gadhiya

Numerade Educator

Problem 31

Type I and Type II Errors provide statements that identify the type I error and the type II error that correspond to the given claim. (Although conclusions are usually expressed in verbal form, the answers here can be expressed with statements that include symbolic expressions such as $p=0.1 .$ )
The proportion of adults who use the Internet is greater than 0.87.

Harsh Gadhiya

Numerade Educator

Problem 32

Type I and Type II Errors provide statements that identify the type I error and the type II error that correspond to the given claim. (Although conclusions are usually expressed in verbal form, the answers here can be expressed with statements that include symbolic expressions such as $p=0.1 .$ )
The proportion of people who require no vision correction is less than 0.25.

Harsh Gadhiya

Numerade Educator

Problem 33

Chantix (varenicline) tablets are used as an aid to help people stop smoking. In a clinical trial, 129 subjects were treated with Chantix twice a day for 12 weeks, and 16 subjects experienced abdominal pain (based on data from Pfizer, Inc.). If someone claims that more than $8 \%$ of Chantix users experience abdominal pain, that claim is supported with a hypothesis test conducted with a 0.05 significance level. Using 0.18 as an alternative value of $p,$ the power of the test is $0.96 .$ Interpret this value of the power of the test.

Harsh Gadhiya

Numerade Educator

Problem 34

Consider a hypothesis test of the claim that the Ericsson method of gender selection is effective in increasing the likelihood of having a baby girl, so that the claim is $p>0.5 .$ Assume that a significance level of $\alpha=0.05$ is used, and the sample is a simple random sample of size $n=64.$
a. Assuming that the true population proportion is $0.65,$ find the power of the test, which is the probability of rejecting the null hypothesis when it is false. (Hint: With a 0.05 significance level, the critical value is $z=1.645,$ so any test statistic in the right tail of the accompanying top graph is in the rejection region where the claim is supported. Find the sample proportion $\hat{p}$ in the top graph, and use it to find the power shown in the bottom graph.)
b. Explain why the green-shaded region of the bottom graph represents the power of the test.

Lisa Stryjewski

Lisa Stryjewski

Numerade Educator

Problem 35

Size to Achieve Power Researchers plan to conduct a test of a gender selection method. They plan to use the alternative hypothesis of $H_{1}: p>0.5$ and a significance Ievel of $\alpha=0.05 .$ Find the sample size required to achieve at least $80 \%$ power in detecting an increase in $p$ from 0.50 to $0.55 .$ (This is a very difficult exercise. Hint: Sce Exercise $34 .$ )

Victor Salazar

Numerade Educator