91Ó°ÊÓ

In Exercises 7 to 10, explain what's wrong with the stated hypotheses. Then give correct hypotheses. A change is made that should improve student satisfaction with the parking situation at a local high school. Right now, \(37 \%\) of students approve of the parking that's provided. The null hypothesis \(H_{0}: p>0.37\) is tested against the alternative \(H_{a}: p=0.37\)

For the study of Jordanian children in Exercise 2 , the sample mean hemoglobin level was \(11.3 \mathrm{~g} / \mathrm{dl}\) and the sample standard deviation was \(1.6 \mathrm{~g} / \mathrm{dl} .\) A significance test yields a \(P\) -value of 0.0016 . (a) Explain what it would mean for the null hypothesis to be true in this setting. (b) Interpret the \(P\) -value in context.

A student performs a test of \(H_{0}: p=0.75\) versus \(H_{a}: p>0.75\) and gets a \(P\) -value of \(0.99 .\) The student writes: "Because the \(P\) -value is greater than \(0.75,\) we reject \(H_{0} .\) The data prove that \(H_{a}\) is true." Explain what is wrong with this conclusion.

Experiments on learning in animals sometimes measure how long it takes mice to find their way through a maze. The mean time is 18 seconds for one particular maze. A researcher thinks that a loud noise will cause the mice to complete the maze faster. She measures how long each of 10 mice takes with a noise as stimulus. The appropriate hypotheses for the significance test are (a) \(H_{0}: \mu=18 ; H_{a}: \mu \neq 18\). (b) \(H_{0}: \mu=18 ; H_{a}: \mu>18\). (c) \(H_{0}: \mu<18 ; H_{a}: \mu=18\) (d) \(H_{0}: \mu=18 ; H_{a}: \mu<18\). (e) \(H_{0}: \bar{x}=18 ; H_{a}: \bar{x}<18\).

In the sample, \(\hat{p}=158 / 300=0.527 .\) The resulting \(P\) -value is 0.18 . What is the correct interpretation of this \(P\) -value? (a) Only \(18 \%\) of the city residents support the tax increase. (b) There is an \(18 \%\) chance that the majority of residents supports the tax increase. (c) Assuming that \(50 \%\) of residents support the tax increase, there is an \(18 \%\) probability that the sample proportion would be 0.527 or higher by chance alone. (d) Assuming that more than \(50 \%\) of residents support the tax increase, there is an \(18 \%\) probability that the sample proportion would be 0.527 or higher by chance alone. (e) Assuming that \(50 \%\) of residents support the tax increase, there is an \(18 \%\) chance that the null hypothesis is true by chance alone.

Of the 24,611 degrees in mathematics given by U.S. colleges and universities in a recent year, \(70 \%\) were bachelor's degrees, \(24 \%\) were master's degrees, and the rest were doctorates. Moreover, women earned \(43 \%\) of the bachelor's degrees, \(41 \%\) of the master's degrees, and \(29 \%\) of the doctorates. (a) How many of the mathematics degrees given in this year were earned by women? Justify your answer. (b) Are the events "degree earned by a woman" and "degree was a bachelor's degree" independent? Justify your answer using appropriate probabilities. (c) If you choose 2 of the 24,61 l mathematics degrees at random, what is the probability that at least 1 of the 2 degrees was earned by a woman? Show your work.

Walking to school A recent report claimed that \(13 \%\) of students typically walk to school. \({ }^{10}\) DeAnna thinks that the proportion is higher than 0.13 at her large elementary school, so she surveys a random sample of 100 students to find out.

Are boys more likely? We hear that newborn babies are more likely to be boys than girls. Is this true? \(\mathrm{A}\) random sample of 25,468 firstborn children included 13,173 boys. (a) Do these data give convincing evidence that firstborn children are more likely to be boys than girls? (b) To what population can the results of this study be generalized: all children or all firstborn children? Justify your answer.

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. $$ \begin{array}{l} H_{0}: \mu>0.5 \\ H_{a}: \bar{x}=0.5 \end{array} $$ \(\bullet\quad\) \(10 \%: 4040(0.5)=2020\) and \(4040(1-0.5)=2020\) are both at least 10 . \(\bullet\quad\) Large Counts: There are at least 40,400 coins in the world. \(t=\frac{0.5-0.507}{\sqrt{\frac{0.5(0.5)}{4040}}}=-0.89 ; P\) -value \(=1-0.1867=0.8133\) Reject \(H_{0}\) because the \(P\) -value is so large and conclude that the coin is fair.

A drug manufacturer claims that fewer than \(10 \%\) of patients who take its new drug for treating Alzheimer's disease will experience nausea. To test this claim, a significance test is carried out of $$ \begin{array}{l} H_{0}: p=0.10 \\ H_{a}: p<0.10 \end{array} $$ You learn that the power of this test at the \(5 \%\) significance level against the alternative \(p=0.08\) is 0.29 . (a) Explain in simple language what "power \(=0.29 "\) means in this setting. (b) You could get higher power against the same alternative with the same \(\alpha\) by changing the number of measurements you make. Should you make more measurements or fewer to increase power? Explain. (c) If you decide to use \(\alpha=0.01\) in place of \(\alpha=0.05\), with no other changes in the test, will the power increase or decrease? Justify your answer. (d) If you shift your interest to the alternative \(p=0.07\) with no other changes, will the power increase or decrease? Justify your answer.