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Guilty Verdicts in Court Cases A reporter on cnn.com stated in July 2010 that \(95 \%\) of all court cases that go to trial result in a guilty verdict. To test the accuracy of this claim, we collect a random sample of 2000 court cases that went to trial and record the proportion that resulted in a guilty verdict. (a) What is/are the relevant parameter(s)? What sample statistic(s) is/are used to conduct the test? (b) State the null and alternative hypotheses. (c) We assess evidence by considering how likely our sample results are when \(H_{0}\) is true. What does that mean in this case?

Exercise and the Brain It is well established that exercise is beneficial for our bodies. Recent studies appear to indicate that exercise can also do wonders for our brains, or, at least, the brains of mice. In a randomized experiment, one group of mice was given access to a running wheel while a second group of mice was kept sedentary. According to an article describing the study, "The brains of mice and rats that were allowed to run on wheels pulsed with vigorous, newly born neurons, and those animals then breezed through mazes and other tests of rodent IQ"10 compared to the sedentary mice. Studies are examining the reasons for these beneficial effects of exercise on rodent (and perhaps human) intelligence. High levels of BMP (bone- morphogenetic protein) in the brain seem to make stem cells less active, which makes the brain slower and less nimble. Exercise seems to reduce the level of BMP in the brain. Additionally, exercise increases a brain protein called noggin, which improves the brain's ability. Indeed, large doses of noggin turned mice into "little mouse geniuses," according to Dr. Kessler, one of the lead authors of the study. While research is ongoing in determining which effects are significant, all evidence points to the fact that exercise is good for the brain. Several tests involving these studies are described. In each case, define the relevant parameters and state the null and alternative hypotheses. (a) Testing to see if there is evidence that mice allowed to exercise have lower levels of BMP in the brain on average than sedentary mice (b) Testing to see if there is evidence that mice allowed to exercise have higher levels of noggin in the brain on average than sedentary mice (c) Testing to see if there is evidence of a negative correlation between the level of BMP and the level of noggin in the brains of mice

4.20 Taste Test A taste test is conducted between two brands of diet cola, Brand \(A\) and Brand \(B\), to determine if there is evidence that more people prefer Brand A. A total of 100 people participate in the taste test. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) Give an example of possible sample results that would provide strong evidence that more people prefer Brand A. (Give your results as number choosing Brand \(\mathrm{A}\) and number choosing Brand B.) (c) Give an example of possible sample results that would provide no evidence to support the claim that more people prefer Brand A. (d) Give an example of possible sample results for which the results would be inconclusive: The sample provides some evidence that Brand \(\mathrm{A}\) is preferred but the evidence is not strong.

Describe tests we might conduct based on Data 2.3 , introduced on page \(66 .\) This dataset, stored in ICUAdmissions, contains information about a sample of patients admitted to a hospital Intensive Care Unit (ICU). For each of the research questions below, define any relevant parameters and state the appropriate null and alternative hypotheses. Is the average age of ICU patients at this hospital greater than \(50 ?\)

Indicate whether the analysis involves a statistical test. If it does involve a statistical test, state the population parameter(s) of interest and the null and alternative hypotheses. Polling 1000 people in a large community to determine if there is evidence for the claim that the percentage of people in the community living in a mobile home is greater than \(10 \%\)

Indicate whether the analysis involves a statistical test. If it does involve a statistical test, state the population parameter(s) of interest and the null and alternative hypotheses. Testing 50 people in a driving simulator to find the average reaction time to hit the brakes when an object is seen in the view ahead.

Arsenic in Chicken Data 4.5 on page 228 discusses a test to determine if the mean level of arsenic in chicken meat is above 80 ppb. If a restaurant chain finds significant evidence that the mean arsenic level is above \(80,\) the chain will stop using that supplier of chicken meat. The hypotheses are $$ \begin{array}{ll} H_{0}: & \mu=80 \\ H_{a}: & \mu>80 \end{array} $$ where \(\mu\) represents the mean arsenic level in all chicken meat from that supplier. Samples from two different suppliers are analyzed, and the resulting p-values are given: Sample from Supplier A: p-value is 0.0003 Sample from Supplier B: p-value is 0.3500 (a) Interpret each p-value in terms of the probability of the results happening by random chance. (b) Which p-value shows stronger evidence for the alternative hypothesis? What does this mean in terms of arsenic and chickens? (c) Which supplier, \(\mathrm{A}\) or \(\mathrm{B}\), should the chain get chickens from in order to avoid too high a level of arsenic?

Multiple Sclerosis and Sunlight It is believed that sunlight offers some protection against multiple sclerosis (MS) since the disease is rare near the equator and more prevalent at high latitudes. What is it about sunlight that offers this protection? To find out, researchers \({ }^{15}\) injected mice with proteins that induce a condition in mice comparable to MS in humans. The control mice got only the injection, while a second group of mice were exposed to UV light before and after the injection, and a third group of mice received vitamin D supplements before and after the injection. In the test comparing UV light to the control group, evidence was found that the mice exposed to UV suppressed the MS-like disease significantly better than the control mice. In the test comparing mice getting vitamin D supplements to the control group, the mice given the vitamin \(\mathrm{D}\) did not fare significantly better than the control group. If the p-values for the two tests are 0.472 and 0.002 , which p-value goes with which test?

Definition of a P-value Using the definition of a p-value, explain why the area in the tail of a randomization distribution is used to compute a p-value.

Classroom Games Two professors \(^{18}\) at the University of Arizona were interested in whether having students actually play a game would help them analyze theoretical properties of the game. The professors performed an experiment in which students played one of two games before coming to a class where both games were discussed. Students were randomly assigned to which of the two games they played, which we'll call Game 1 and Game \(2 .\) On a later exam, students were asked to solve problems involving both games, with Question 1 referring to Game 1 and Question 2 referring to Game 2 . When comparing the performance of the two groups on the exam question related to Game 1 , they suspected that the mean for students who had played Game 1 ( \(\mu_{1}\) ) would be higher than the mean for the other students \(\mu_{2},\) so they considered the hypotheses \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1}>\mu_{2}\) (a) The paper states: "test of difference in means results in a p-value of \(0.7619 . "\) Do you think this provides sufficient evidence to conclude that playing Game 1 helped student performance on that exam question? Explain. (b) If they were to repeat this experiment 1000 times, and there really is no effect from playing the game, roughly how many times would you expect the results to be as extreme as those observed in the actual study? (c) When testing a difference in mean performance between the two groups on exam Question 2 related to Game 2 (so now the alternative is reversed to be \(H_{a}: \mu_{1}<\mu_{2}\) where \(\mu_{1}\) and \(\mu_{2}\) represent the mean on Question 2 for the respective groups), they computed a p-value of \(0.5490 .\) Explain what it means (in the context of this problem) for both p-values to be greater than \(0.5 .\)