Chapter 9: Testing a Claim

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.

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

An opinion poll asks a random sample of adults whether they favor banning ownership of handguns by private citizens. A commentator believes that more than half of all adults favor such a ban. The null and alternative hypotheses you would use to test this claim are

a).$H_0:\hat{p}=0.5;H_a:\hat{p}>0.5$

(b) $H_0:p=0.5;H_a:p>0.5$

(c) $H_0:p=0.5;H_a:p<0.5$

(d) $H_0:p=0.5;H_a:p\ne 0.5$

(e) $H_0:p>0.5;H_a:p=0.5$

Simon reads a newspaper report claiming that 12% of all adults in the United States are left-handed. He wonders if 12% of the students at his large public high school are left-handed. Simon chooses an SRS of 100 students and records whether each student is right- or left-handed.

Stating hypotheses State the appropriate null and alternative hypotheses in each of the following cases.

(a) The average height of $18$-year-old American women is $64.2$inches. You wonder whether the mean height of this year's female graduates from a large local high school (over $3000$students) differs from the national average. You measure an SRS of $48$female graduates and find that $\overline{X}=63.1$inches.

(b) Mr. Starnes believes that less than $75\%$of the students at his school completed their math homework last night. The math teachers inspect the homework assignments from a random sample of students at the school to help Mr. Starnes test his claim.

- Check conditions for carrying out a test about a population proportion or mean.

- Interpret $P$-values in context.

Are TV commercials louder than their surrounding programs? To find out, researchers collected data on $50$randomly selected commercials in a given week. With the television’s volume at a fixed setting, they measured the maximum loudness of each commercial and the maximum loudness in the first $30$seconds of regular programming that followed. Assuming conditions for inference are met, the most appropriate method for answering the question of interest is

(a) a one-proportion z test.

(b) a one-proportion z interval.

(c) a paired t test.

In planning a study of the birth weights of babies whose mothers did not see a

doctor before delivery, a researcher states the hypotheses as

$\begin{array}{c}{H}_0:\mu <1000\text{grams}\\ H_a:\mu =900\text{grams}\end{array}$