Testing a Claim

Are boys more likely? We hear that newborn babies are more likely to be boys than girls. Is this true? A random sample of 25,468 firstborn children included 13,173 boys. Boys do make up more than half of the sample, but of course, we don't expect a perfect 50-50 split in a random sample.

(a) To what population can the results of this study be generalized: all children or all firstborn children? Justify your answer.

(b) Do these data give convincing evidence that boys are more common than girls in the population? Carry out a significance test to help answer this question.

People of taste are supposed to prefer fresh-brewed coffee to the instant variety. On the other hand, perhaps many coffee drinkers just want their caffeine fix. A sceptic claims that only half of all coffee drinkers prefer fresh-brewed coffee. To test this claim, we ask a random sample of 50 coffee drinkers in a small city to take part in a study. Each person tastes two unmarked cups-one containing instant coffee and one containing fresh-brewed coffee-and says which he or she prefers. We find that 36 of the 50 choose fresh coffee.

(a) We presented the two cups to each coffee drinker in a random order, so that some people tasted the fresh coffee first, while others drank the instant coffee first. Why do you think we did this?

(b) Do these results give convincing evidence that coffee drinkers favour fresh-brewed over instant coffee? Carry out a significance test to help answer this question.

Bullies in middle school A University of Illinois study on aggressive behavior surveyed a random sample of 558 middle school students. When asked to describe their behavior in the last 30 days, 445 students said their behavior included physical aggression, social ridicule, teasing, name-calling, and issuing threats. This behavior was not defined as bullying in the questionnaire.  Is this evidence that more than three-quarters of the students at that middle school engage in bullying behavior? To find out, Maurice decides to perform a significance test. Unfortunately, he made a few errors along the way. Your job is to spot the mistakes and correct them.

$\begin{array}{r}{H}_0:p=0.75\\ H_a:\stackrel{-}{p}>0.797\end{array}$

where p= the true mean proportion of middle school students who engaged in bullying.

- A random sample of 558 middle school students was surveyed.

- 558(0.797)=444.73 is at least 10.

The probability that the null hypothesis is true is only 0.0138, so we reject $H_0$. This proves that more than three-quarters of the school engaged in bullying behavior.

The French naturalist Count Buffon (1707-1788) tossed a coin 4040 times. He got 2048 heads. That's a bit more than one-half. Is this evidence that Count Buffon's coin was not balanced? To find out, Luisa decides to perform a significance test. Unfortunately, she made a few errors along the way. Your job is to spot the mistakes and correct them.

$\begin{array}{r}{H}_0:\mu >0.5\\ H_a:\overline{x}=0.5\end{array}$

- Independent 4040(0.5)=2020 and 4040(1-0.5)=2020 are both at least 10 .

- Normal There are at least 40,400 coins in the world.

$t=\frac{0.5-0.507}{\sqrt{\frac{0.5\left(0.5\right)}{4040}}}=-0.89;P_{\text{-value }}=1-0.1867=0.8133$

Reject $H_0$ because the P-value is so large and conclude that the coin is fair.

A state's Division of Motor Vehicles (DMV) claims that 60 % of teens pass their driving test on the first attempt. An investigative reporter examines an SRS of the DMV records for 125 teens; 86 of them passed the test on their first try. Is this good evidence that the DMV's claim is incorrect? Carry out a test at the $\alpha =0.05$ significance level to help answer this question.

We want to be rich In a recent year, 73 % of first-year college students responding to a national survey identified "being very well-off financially" as an important personal goal. A state university finds that 132 of an SRS of 200 of its first-year students say that this goal is important. Is there good evidence that the proportion of all first-year students at this university who think being very well-off is important differs from the national value, 73 %? Carry out a test at the $\alpha =0.05$ significance level to help answer this question.

(a) Construct and interpret a 95% confidence interval for the proportion of all teens in the state who passed their driving test on the first attempt.

(b) Explain what the interval in part (a) tells you about the DMV's claim.

(a) Construct and interpret a 95 % confidence interval for the true proportion p of all first-year students at the university who would identify being well-off as an important personal goal.

(b) Explain what the interval in part (a) tells you about whether the national value holds at this university.

In late 2009, the Pew Internet and American Life Project asked a random sample of U.S. adults, “Do you ever . . . use Twitter or another service to share updates about yourself or to see updates about others?” According to Pew, the resulting 95% confidence interval is (0.167, 0.213).15 Can we use this interval to conclude that the actual proportion of U.S. adults who would say they Twitter differs from 0.20? Justify your answer.

Losing weight A Gallup Poll found that 59% of the people in its sample said “Yes” when asked, “Would you like to lose weight?” Gallup announced: “For

results based on the total sample of national adults, one can say with 95% confidence that the margin of (sampling) error is 􀁯3 percentage points.”16 Can we use this interval to conclude that the actual proportion of U.S. adults who would say they want to lose weight differs from 0.55? Justify your answer.

Teens and sex The Gallup Youth Survey asked a random sample of U.S. teens aged 13 to 17 whether they thought that young people should wait to have sex until marriage.17 The Minitab output below shows the results of a significance test and a 95% confidence interval based on the survey data.

(a) Define the parameter of interest.

(b) Check that the conditions for performing the significance test are met in this case.

(c) Interpret the P-value in context.

(d) Do these data give convincing evidence that the actual population proportion differs from 0.5? Justify your answer with appropriate evidence.

Reporting cheating What proportion of students

are willing to report cheating by other students? A

student project put this question to an SRS of 172

undergraduates at a large university: “You witness two

students cheating on a quiz. Do you go to the professor?”

The Minitab output below shows the results of a

significance test and a 95% confidence interval based

on the survey data.18

(a) Define the parameter of interest.

(b) Check that the conditions for performing the significance test are met in this case.

(c) Interpret the P-value in context.

(d) Do these data give convincing evidence that the actual population proportion differs from 0.15? Justify your answer with appropriate evidence.

After once again losing a football game to the archrival, a college’s alumni association conducted a survey to see if alumni were in favor of firing the

coach. An SRS of 100 alumni from the population of all living alumni was taken, and 64 of the alumni in the sample were in favor of firing the coach.

Suppose you wish to see if a majority of living alumni are in favor of firing the coach. The appropriate test statistic is

a. $z=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.64(0.36)}{100}}}}$

b.$t=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.64(0.36)}{100}}}}$

c.$z=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.5(0.5)}{100}}}}$

d.$z=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.64(0.36)}{64}}}}$

e.$z=\frac{0.5-0.64}{\sqrt{{\displaystyle \frac{0.5(0.5)}{100}}}}$

The z statistic for a test of H0 :p = 0.4 versus Ha : p > 0.4 is z = 2.43. This test is

(a) not significant at either  $\alpha =0.05$ or $\alpha =0.01$.

(b) significant at $\alpha =0.05$ but not at $\alpha =0.01$

(c) significant at $\alpha =0.01$but not at $\alpha =0.05$.

(d) significant at both $\alpha =0.05$and $\alpha =0.01$.

(e) inconclusive because we don’t know the value of ˆp .

Which of the following $95\%$ confidence intervals would lead us to reject H0 : $p = 0.30$ in favor of

Ha :p not equal to $0.30$ at the $5\%$ significance level?

Which of the following 95% confidence intervals would lead us to reject $H0 :p = 0.30$in favor of $Ha :p \ne 0.30$at the 5% significance level?

(a) (0.29, 0.38) (c) (0.27, 0.31) (e) None of these

(b) (0.19, 0.27) (d) (0.24, 0.30)

Packaging CDs (6.2, 5.3) A manufacturer of compact discs (CDs) wants to be sure that their CDs will fit inside the plastic cases they have bought for packaging. Both the CDs and the cases are circular. According to the supplier, the plastic cases vary Normally with mean diameter  $\mu =4.2 inches$ and a standard deviation $\sigma = 0.05 inches$. The CD manufacturer decides to produce CDs with mean diameter $\mu =4 inches$. Their diameters follow a Normal distribution with $\sigma =0.1 inches$.

(a) Let X = the diameter of a randomly selected CD and Y = the diameter of a randomly selected case. Describe the shape, center, and spread of the distribution of the random variable X = Y. What is the importance of this random variable to the CD manufacturer?

(b) Compute the probability that a randomly selected CD will fit inside a randomly selected case.

(c) The production process actually runs in batches of 100 CDs. If each of these CDs is paired with a randomly chosen plastic case, find the probability that all the CDs fit in their cases.

Cash to find work? (5.2) Will cash bonuses speed the return to work of unemployed people? The Illinois Department of Employment Security designed an experiment to find out. The subjects were 10,065 people aged 20 to 54 who were filing claims for

unemployment insurance. Some were offered \(500 if they found a job within 11 weeks and held it for at least 4 months. Others could tell potential employers that the state would pay the employer \)500 for hiring them. A control group got neither kind of bonus.

(a) Describe a completely randomized design for this experiment.

(b) How will you label the subjects for random assignment? Use Table D at line 127 to choose the first 3 subjects for the first treatment.

(c) Explain the purpose of a control group in this setting.

For the job satisfaction study described in Section 9.1, the hypotheses are 

$$\begin{gathered} H_0: \mu =0 \\ {H}_a: \mu is not equal to 0 \end{gathered}$$

where $\mu$ is the mean difference in job satisfaction scores (self-paced  machine-paced) in the population of assembly-line workers at the company. Data from a random sample of $18$ workers gave $\stackrel{⏜}{x} = 17$ and $s_x= 60$

Calculate the test statistic. Show your work.

For the job satisfaction study described in Section 9.1, the hypotheses are

$$\begin{gathered} H_0:\mu =0 \\ {H}_a:\mu \ne 0 \end{gathered}$$

where $\mu$ is the mean difference in job satisfaction scores (self-paced - machine-paced) in the population of assembly-line workers at the company? Data from a random sample of $18$ workers gave $\overline{x}=17$ and $s_x=60$

Use Table B to find the P-value. What conclusion would you draw?

For the job satisfaction study described in Section 9.1, the hypotheses are

$$\begin{gathered} H_0:\mu =0 \\ {H}_a:\mu \ne 0 \end{gathered}$$

where $\mu$ is the mean difference in job satisfaction scores (self-paced -machine-paced) in the population of assembly-line workers at the company. Data from a random sample of $18$ workers gave $\overline{x} = 17$ and $s_x= 60$

Now use your calculator to find the P-value as described in the Technology Corner. Is your result consistent with the value you obtained in Question $2$?

A college professor suspects that students at his school are getting less than 8 hours of sleep a night, on average. To test his belief, the professor asks a random sample of $28$ students, “How much sleep did you get last night?” Here are the data (in hours): $9 6 8 6 8 8 6 6.5 6 7 9 4 3 4 5 6 11 6 3 6 6 10 7 8 4.5 9 7 7$

Do these data provide convincing evidence in support of the professor’s suspicion? Carry out a significance test at the $a=0.05$ level to help answer this question.

The health director of a large company is concerned about the effects of stress on the company’s middle-aged male employees. According to the National Center for Health Statistics, the mean systolic blood pressure for males 35 to 44 years of age is 128. The health director examines the medical records of a random sample of 72 male employees in this age group. The Minitab output below displays the results of a significance test and a confidence interval.

1. Do the results of the significance test allow us to conclude that the mean blood pressure for all the company’s middle-aged male employees differs from the national average? Justify your answer.

2. Interpret the 95% confidence interval in context. Explain how the confidence interval leads to the same conclusion as in Question 1.

The health director of a large company is concerned about the effects of stress on the company’s middle-aged male employees. According to the National Center for Health Statistics, the mean systolic blood pressure for males $35$ to $44$ years of age is $128$ The health director examines the medical records of a random sample of $72$ male employees in this age group. The Minitab output below displays the results of a significance test and a confidence interval. 

Do the results of the significance test allow us to conclude that the mean blood pressure for all the company’s middle-aged male employees differs from the national average? Justify your answer.

The health director of a large company is concerned about the effects of stress on the company’s middle-aged male employees. According to the National Center for Health Statistics, the mean systolic blood pressure for males $35$ to $44$ years of age is $128$ The health director examines the medical records of a random sample of $72$ male employees in this age group. The Minitab output below displays the results of a significance test and a confidence interval. 

Interpret the $95\%$ confidence interval in context. Explain how the confidence interval leads to the same conclusion as in Question $1$

Attitudes The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures students’ attitudes toward school and study habits. Scores range from $0 to 200$. The mean score for U.S. college students is about $115$. A teacher suspects that

older students have better attitudes toward school. She gives the SSHA to an SRS of $45$ of the over $1000$ students at her college who are at least $30$ years of age. Check the conditions for carrying out a significance test of the teacher’s suspicion.

Hemoglobin is a protein in red blood cells that carries oxygen from the lungs to body tissues. People with fewer than 12 grams of hemoglobin per deciliter of blood (g/dl) are anemic. A public health official in Jordan suspects that Jordanian children are at risk of anemia. He measures a random sample of 50 children. Check the conditions for carrying out a significance test of the official’s suspicion.

Anemia Hemoglobin is a protein in red blood cells that carries oxygen from the lungs to body tissues. People with fewer than $12$ grams of hemoglobin per deciliter of blood $(g/dl)$ are anemic. A public health official in Jordan suspects that Jordanian children are at risk of anemia. He measures a random sample of $50$ children. Check the conditions for carrying out a significance test of the official’s suspicion.

A retailer entered into an exclusive agreement with a supplier who guaranteed to provide all products at competitive prices. The retailer eventually began to purchase supplies from other vendors who offered better prices. The original supplier filed a lawsuit claiming violation of the agreement. In defense, the retailer had an audit performed on a random sample of 25 invoices. For each audited invoice, all purchases made from other suppliers were examined and compared with those offered by the original supplier. The percent of purchases on each invoice for which an alternative supplier offered a lower price than the original supplier was recorded.26 For example, a data value of 38 means that the price would be lower with a different supplier for 38% of the items on the invoice. A histogram and some computer output for these data are shown below. Explain why we should not carry out a one-sample t test in this setting.

Paying high prices? A retailer entered into an exclusive agreement with a supplier who guaranteed to provide all products at competitive prices. The retailer eventually began to purchase supplies from other vendors who offered better prices. The original supplier filed a lawsuit claiming violation of the agreement. In defense, the retailer had an audit performed on a random sample of $25$ invoices. For each audited invoice, all purchases made from other suppliers were examined and compared with those offered by the original supplier. The percent of purchases on each invoice for which an alternative supplier offered a lower price than the original supplier was recorded.$26$ For example, a data value of $38$ means that the price would be lower with a different supplier for $38\%$ of the items on the invoice. A histogram and some computer output

for these data are shown below. Explain why we should not carry out a one-sample t-test in this setting.

The composition of the earth’s atmosphere may have changed over time. To try to discover the nature of the atmosphere long ago, we can examine the gas in bubbles inside ancient amber. Amber is tree resin that has hardened and been trapped in rocks. The gas in bubbles within amber should be a sample of the atmosphere at the time the amber was formed. Measurements on 9 specimens of amber from the late Cretaceous era (75 to 95 million years ago) give these percent of nitrogen:

63.4  65.0  64.4  63.3  54.8  64.5  60.8  49.1  51.0

Explain why we should not carry out a one-sample t test in this setting.

Ancient air The composition of the earth’s atmosphere may have changed over time. To try to discover the nature of the atmosphere long ago, we can examine the gas in bubbles inside ancient amber. Amber is tree resin that has hardened and been trapped in rocks. The gas in bubbles within amber should be a sample of the atmosphere at the time the amber was formed. Measurements on $9$ specimens of amber from the late Cretaceous era ($75 to 95$ million years ago) give these percents of nitrogen:

$63.4 65.0 64.4 63.3 54.8 64.5 60.8 49.1 51.0$ Explain why we should not carry out a one-sample t test in this setting.

In the study of older students’ attitudes from Exercise 63, the sample mean SSHA score was 125.7 and the sample standard deviation was 29.8.

(a) Calculate the test statistic.

(b) Find the P-value using Table B. Then obtain a more precise P-value from your calculator.

Attitudes In the study of older students’ attitudes from Exercise $63$, the sample mean SSHA score was $125.7$ and the sample standard deviation was $29.8$.

(a) Calculate the test statistic.

(b) Find the P-value using Table $B$. Then obtain a more precise P-value from your calculator.

Anemia For the study of Jordanian children in Exercise $64$, the sample mean hemoglobin level was $11.3$ mg/dl and the sample standard deviation was $1.6$ mg/dl.

(a) Calculate the test statistic.

(b) Find the P-value using Table $B$. Then obtain a

more precise P-value from your calculator.

Suppose you carry out a significance test of $H_0:\mu =5$ versus $H_0:\mu >5$ based on a sample of size n =20 and obtain t =  1.81.

(a) Find the P-value for this test using (i) Table B and (ii) your calculator. What conclusion would you draw at the 5% significance level? At the 1% significance level?

(b) Redo part (a) using an alternative hypothesis of $H_0:\mu \ne 5$

One-sided test Suppose you carry out a significance test of $H_0:\mu =5$ versus $H_a:\mu >5$ based on a sample of size $n = 20$ and obtain $t = 1.81$.

(a) Find the P-value for this test using (i) Table $B$ and (ii) your calculator. What conclusion would you draw at the $5\%$ significance level? At the $1\%$ significance level?

(b) Redo part (a) using an alternative hypothesis of $H_a:\mu \ne 5$.

The one-sample t statistic from a sample of n =25 observations for the two-sided test of

$\begin{array}{r}{H}_0:\mu =64\\ H_d:\mu \ne 64\end{array}$

has the value t =- 1.12.

(a) Find the P-value for this test using (i) Table B and (ii) your calculator. What conclusion would you draw at the 5% significance level? At the 1% significance level?

(b) Redo part (a) using an alternative hypothesis of $H_a:\mu <64$

Two-sided test The one-sample t statistic from a sample of $n = 25$ observations for the two-sided test of $H_0 : \mu = 64; H_a : \mu \ne 64$has the value $t =-1.12$.

(a) Find the P-value for this test using (i) Table $B$ and (ii) your calculator. What conclusion would you draw at the $5\%$ significance level? At the $1\%$ significance level?

(b) Redo part (a) using an alternative hypothesis of $H_a : \mu < 64$.

Cola makers test new recipes for loss of sweetness during storage. Trained tasters rate the sweetness before and after storage From experience, the population distribution of sweetness losses will be close to Normal. Here are the sweetness losses (sweetness before storage minus sweetness after storage) found by tasters from a random sample of 10 batches of a new cola recipe:

2.0  0.4  0.7   2.0   0.4   2.2   1.3    1.2   1.1   2.3

Are these data good evidence that the cola lost sweetness? Carry out a test to help you answer this question.

Sweetening colas Cola makers test new recipes for loss of sweetness during storage. Trained tasters rate the sweetness before and after storage. From experience, the population distribution of sweetness losses will be close to Normal. Here are the sweetness losses (sweetness before storage minus sweetness after storage) found by tasters from a random sample of $10$ batches of a new cola recipe:

$2.0 0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3$

Are these data good evidence that the cola lost sweetness? Carry out a test to help you answer this question.

Heat through the glass How well materials conduct heat matters when designing houses, for example. Conductivity is measured in terms of watts of heat power transmitted per square meter of surface per degree Celsius of temperature difference on the two sides of the material. In these units, glass has conductivity about $1$. The National Institute of Standards and Technology provides exact data on properties of materials. Here are measurements of the heat conductivity of $11$ randomly selected pieces of a particular type of glass:

$1.11 1.07 1.11 1.07 1.12 1.08 1.08 1.18 1.18 1.18 1.12$

Is there convincing evidence that the conductivity of this type of glass is greater than $1$? Carry out a test to help you answer this question.

The recommended daily allowance (RDA) of calcium for women between the ages of 18 and 24 years is 1200 milligrams (mg). Researchers who were involved in a large-scale study of women’s bone health suspected that

their participants had significantly lower calcium intakes than the RDA. To test this suspicion, the researchers measured the daily calcium intake of a random sample of 36 women from the study who fell in the desired age range. The Minitab output below displays descriptive statistics for these data, along with the results of a significance test.

(a) Determine whether there are any outliers. Show your work.

(b) Interpret the P-value in context.

(c) Do these data give convincing evidence to support the researchers’ suspicion? Carry out a test to help your answer

Healthy bones The recommended daily allowance (RDA) of calcium for women between the ages of $18$ and $24$ years is $1200$ milligrams (mg). Researchers who were involved in a large-scale study of women’s bone health suspected that their participants had significantly lower calcium intakes than the RDA. To test this suspicion, the researchers measured the daily calcium intake of a random sample of $36$ women from the study who fell in the desired age range. The Minitab output below displays descriptive statistics for these data, along with the results of a significance test.

(a) Determine whether there are any outliers. Show your work.

(b) Interpret the P-value in context.

(c) Do these data give convincing evidence to support the researchers’ suspicion? Carry out a test pg $571$ to help you answer this question.

Growing tomatoes An agricultural field trial compares the yield of two varieties of tomatoes for commercial use. Researchers randomly selected $10$Variety A and $10$ Variety B tomato plants. Then the researchers divide in half each $10$ small plots of land in different locations. For each plot, a coin toss determines which half of the plot gets a Variety A plant; a Variety B plant goes in the other half. After harvest, they compare the yield in pounds for the plants at each location. The $10$differences $(Variety A - Variety B)$ give $x = 0.34$and$s_x = 0.83$. A graph of the differences looks roughly symmetric and single-peaked with no outliers. Is there convincing evidence that Variety A has a higher mean yield? Perform a significance test using $\alpha$$=0.05$ to answer the question.

An investor with a stock portfolio worth several hundred thousand dollars sued his broker due to the low returns he got from the portfolio at a time when the stock market did well overall. The investor’s lawyer wants to compare the broker’s performance against the market as a whole. He collects data on the broker’s returns for a

random sample of 36 weeks. Over the 10-year period that the broker has managed portfolios, stocks in the Standard & Poor’s 500 index gained an average of 0.95% per month. The Minitab output below displays descriptive statistics for these data, along with the results of a significance test.

(a) Determine whether there are any outliers. Show your work.

(b) Interpret the P-value in context.

(c) Do these data give convincing evidence to support the lawyer’s case? Carry out a test to help you answer this question.

Taking stock An investor with a stock portfolio worth several hundred thousand dollars sued his broker due to the low returns he got from the portfolio at a time when the stock market did well overall. The investor’s lawyer wants to compare the broker’s performance against the market as a whole. He collects data on the broker’s returns for a random sample of$36$ weeks. Over the $10$-year period that the broker has managed portfolios, stocks in the Standard & Poor’s $500$ index gained an average of $0.95\%$ per month. The Minitab output below displays descriptive statistics for these data, along

with the results of a significance test.

(a) Determine whether there are any outliers. Show your work.

(b) Interpret the P-value in context.

(c) Do these data give convincing evidence to support the lawyer’s case? Carry out a test to help you answer this question.

An agricultural field trial compares the yield of two varieties of tomatoes for commercial use. Researchers randomly selected 10 Variety A and 10 Variety B tomato plants. Then the researchers divide half of 10 small plots of land in different locations. For each plot, a coin toss determines which half of the plot gets a Variety A plant; a Variety B plant goes in the other half. After harvest, they compare the yield in pounds for the plants at each location. The 10 differences (Variety A Variety B) give $\stackrel{-}{x}$=0.34 and sx = 0.83. A graph of the differences looks roughly symmetric and single-peaked with no outliers. Is there convincing evidence that Variety A has a higher mean yield? Perform a significance test using $\alpha =$0.05 to answer the question.

A student group claims that first-year students at a university study 2.5 hours per night during the school week. A skeptic suspects that they study less than that on average. He takes a random sample of 30 first-year students and finds that $\stackrel{-}{x}$ =137 minutes and $s_x$=45 minutes. A graph of the data shows no outliers but some skewness. Carry out an appropriate significance test at the 5% significance level. What conclusion do you draw?

Study more! A student group claims that first-year students at a university study $2.5$ hours per night during the school week. A skeptic suspects that they study less than that on average. He takes a random sample of $30$ first-year students and finds that $\overline{x} = 137$minutes and $s_x = 45$minutes. A graph of the data shows no outliers but some skewness. Carry out an appropriate significance test at the $5\%$ significance level. What conclusion do you draw?