Essentials of Statistics

Mario F. Triola

Chapter 8

Hypothesis Testing - all with Video Answers

Educators

Section 1

Basics of Hypothesis Testing

Problem 1

Vitamin $\mathrm{C}$ and Aspirin 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 $325 \mathrm{mg}$ of aspirin. When testing claims about the mean contents of the pills, which would have more serious implications: rejection of the Spring Valley vitamin $\mathrm{C}$ 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?

Kari Hasz

Kari Hasz

Numerade Educator

Problem 2

Estimates and Hypothesis Tests 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 $98.6^{\circ} \mathrm{F}$. What is the difference between estimating and hypothesis testing?

Kari Hasz

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 $174.1 \mathrm{~cm}$ "?

Kari Hasz

Numerade Educator

Problem 4

Interpreting $P$ -value 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?
Identifying $H_{0}$ and $H_{1} . \quad$ In Exercises $5-8$, do the following:
a. Express the original claim in symbolic form.
b. Identify the null and alternative hypotheses.

Kari Hasz

Numerade Educator

Problem 5

Online Data 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.

Kari Hasz

Numerade Educator

Problem 6

Cell Phone 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.

Kari Hasz

Numerade Educator

Problem 7

Pulse Rates Claim: The mean pulse rate (in beats per minute, or bpm) of adult males is equal to $69 \mathrm{bpm}$. 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.

Kari Hasz

Numerade Educator

Problem 8

Pulse Rates 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.

Kari Hasz

Numerade Educator

Problem 9

Exercise 5 "Online Data"

Kari Hasz

Numerade Educator

Problem 10

Exercise 6 "Cell Phone"

Kari Hasz

Numerade Educator

Problem 11

Exercise 7 "Pulse Rates"

Kari Hasz

Numerade Educator

Problem 12

Exercise 6 "Cell Phone"

Kari Hasz

Numerade Educator

Problem 13

Refer to the exercise identified and find the value of the test statistic.
Exercise 5 "Online Data"

Kari Hasz

Numerade Educator

Problem 14

Refer to the exercise identified and find the value of the test statistic.
Exercise 6 "Cell Phone"

Kari Hasz

Numerade Educator

Problem 15

Refer to the exercise identified and find the value of the test statistic.
Exercise 7 "Pulse Rates"

Kari Hasz

Numerade Educator

Problem 16

Refer to the exercise identified and find the value of the test statistic.
Exercise 8 "Pulse Rates"

Kari Hasz

Numerade Educator

Problem 17

The test statistic of $z=1.00$ is obtained when testing the claim that $p>0.3$.
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 $H_{0}$ ?

Kari Hasz

Numerade Educator

Problem 18

The test statistic of $z=-2.50$ is obtained when testing the claim that $p<0.75$.
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 $H_{0}$ ?

Kari Hasz

Numerade Educator

Problem 19

The test statistic of $z=2.01$ is obtained when testing the claim that $p \neq 0.345$.
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 $H_{0}$ ?

Kari Hasz

Numerade Educator

Problem 20

The test statistic of $z=-1.94$ is obtained when testing the claim that $p=3 / 8$.
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 $H_{0}$ ?

Kari Hasz

Numerade Educator

Problem 21

Exercise 17
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 $H_{0}$ ?

Kari Hasz

Numerade Educator

Problem 22

Exercise 18
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 $H_{0}$ ?

Kari Hasz

Numerade Educator

Problem 23

Exercise 19
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 $H_{0}$ ?

Kari Hasz

Numerade Educator

Problem 24

Exercise 20
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 $H_{0}$ ?

Kari Hasz

Numerade Educator

Problem 25

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 $0.3257$.

Kari Hasz

Numerade Educator

Problem 26

Original claim: Fewer than $90 \%$ of adults have a cell phone. The hypothesis test results in a $P$ -value of $0.0003$.

Kari Hasz

Numerade Educator

Problem 27

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 $0.0095$.

Kari Hasz

Numerade Educator

Problem 28

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 $0.3045$.

Kari Hasz

Numerade Educator

Problem 29

The proportion of people who write with their left hand is equal to $0.1$.

Kari Hasz

Numerade Educator

Problem 30

The proportion of people with blue eyes is equal to $0.35$.

Kari Hasz

Numerade Educator

Problem 31

The proportion of adults who use the Internet is greater than $0.87$.

Kari Hasz

Numerade Educator

Problem 32

The proportion of people who require no vision correction is less than $0.25$.

Kari Hasz

Numerade Educator

Problem 33

Interpreting Power 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.

Kari Hasz

Numerade Educator

Problem 34

Calculating Power 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

Finding Sample 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 level 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 $0.55$.

Victor Salazar

Numerade Educator