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Could owning a cat as a child be related to mental illness later in life? Toxoplasmosis is a disease transmitted primarily through contact with cat feces, and has recently been linked with schizophrenia and other mental illnesses. Also, people infected with Toxoplasmosis tend to like cats more and are 2.5 times more likely to get in a car accident, due to delayed reaction times. The CDC estimates that about \(22.5 \%\) of Americans are infected with Toxoplasmosis (most have no symptoms), and this prevalence can be as high as \(95 \%\) in other parts of the world. A study \(^{37}\) randomly selected 262 people registered with the National Alliance for the Mentally Ill (NAMI), almost all of whom had schizophrenia, and for each person selected, chose two people from families without mental illness who were the same age, sex, and socioeconomic status as the person selected from NAMI. Each participant was asked whether or not they owned a cat as a child. The results showed that 136 of the 262 people in the mentally ill group had owned a cat, while 220 of the 522 people in the not mentally ill group had owned a cat. (a) This is known as a case-control study, where cases are selected as people with a specific disease or trait, and controls are chosen to be people without the disease or trait being studied. Both cases and controls are then asked about some variable from their past being studied as a potential risk factor. This is particularly useful for studying rare diseases (such as schizophrenia), because the design ensures a sufficient sample size of people with the disease. Can casecontrol studies such as this be used to infer a causal relationship between the hypothesized risk factor (e.g., cat ownership) and the disease (e.g., schizophrenia)? Why or why not? (b) In case-control studies, controls are usually chosen to be similar to the cases. For example, in this study each control was chosen to be the same age, sex, and socioeconomic status as the corresponding case. Why choose controls who are similar to the cases? (c) For this study, calculate the relevant difference in proportions; proportion of cases (those with schizophrenia) who owned a cat as a child minus proportion of controls (no mental illness) who owned a cat as a child. (d) For testing the hypothesis that the proportion of cat owners is higher in the schizophrenic group than the control group, use technology to generate a randomization distribution and calculate the p-value. (e) Do you think this provides evidence that there is an association between owning a cat as a child and developing schizophrenia? \(^{38}\) Why or why not?

Interpreting a P-value In each case, indicate whether the statement is a proper interpretation of what a p-value measures. (a) The probability the null hypothesis \(H_{0}\) is true. (b) The probability that the alternative hypothesis \(H_{a}\) is true. (c) The probability of seeing data as extreme as the sample, when the null hypothesis \(H_{0}\) is true. (d) The probability of making a Type I error if the null hypothesis \(H_{0}\) is true. (e) The probability of making a Type II error if the alternative hypothesis \(H_{a}\) is true.

Euchre One of the authors and some statistician friends have an ongoing series of Euchre games that will stop when one of the two teams is deemed to be statistically significantly better than the other team. Euchre is a card game and each game results in a win for one team and a loss for the other. Only two teams are competing in this series, which we'll call team A and team B. (a) Define the parameter(s) of interest. (b) What are the null and alternative hypotheses if the goal is to determine if either team is statistically significantly better than the other at winning Euchre? (c) What sample statistic(s) would they need to measure as the games go on? (d) Could the winner be determined after one or two games? Why or why not? (e) Which significance level, \(5 \%\) or \(1 \%,\) will make the game last longer?

Flying Home for the Holidays, On Time In Exercise 4.115 on page \(302,\) we compared the average difference between actual and scheduled arrival times for December flights on two major airlines: Delta and United. Suppose now that we are only interested in the proportion of flights arriving more than 30 minutes after the scheduled time. Of the 1,000 Delta flights, 67 arrived more than 30 minutes late, and of the 1,000 United flights, 160 arrived more than 30 minutes late. We are testing to see if this provides evidence to conclude that the proportion of flights that are over 30 minutes late is different between flying United or Delta. (a) State the null and alternative hypothesis. (b) What statistic will be recorded for each of the simulated samples to create the randomization distribution? What is the value of that statistic for the observed sample? (c) Use StatKey or other technology to create a randomization distribution. Estimate the p-value for the observed statistic found in part (b). (d) At a significance level of \(\alpha=0.01\), what is the conclusion of the test? Interpret in context. (e) Now assume we had only collected samples of size \(75,\) but got essentially the same proportions (5/75 late flights for Delta and \(12 / 75\) late flights for United). Repeating steps (b) through (d) on these smaller samples, do you come to the same conclusion?

For each situation described, indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level (such as \(\alpha=0.01\) ). Testing a new drug with potentially dangerous side effects to see if it is significantly better than the drug currently in use. If it is found to be more effective, it will be prescribed to millions of people.

For each situation described, indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level (such as \(\alpha=0.01\) ). Using a sample of 10 games each to see if your average score at Wii bowling is significantly more than your friend's average score.

For each situation described, indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level (such as \(\alpha=0.01\) ). Testing to see if a well-known company is lying in its advertising. If there is evidence that the company is lying, the Federal Trade Commission will file a lawsuit against them.

Influencing Voters Exercise 4.39 on page 272 describes a possible study to see if there is evidence that a recorded phone call is more effective than a mailed flyer in getting voters to support a certain candidate. The study assumes a significance level of \(\alpha=0.05\) (a) What is the conclusion in the context of thisstudy if the p-value for the test is \(0.027 ?\) (b) In the conclusion in part (a), which type of error are we possibly making: Type I or Type II? Describe what that type of error means in this situation. (c) What is the conclusion if the p-value for the test is \(0.18 ?\)

Do iPads Help Kindergartners Learn: A Subtest The Auburn, Maine, school district conducted an early literacy experiment in the fall of 2011 . In September, half of the kindergarten classes were randomly assigned iPads (the intervention group) while the other half of the classes got them in December (the control group.) Kids were tested in September and December and the study measures the average difference in score gains between the control and intervention group. \(^{41}\) The experimenters tested whether the mean score for the intervention group was higher on the HRSIW subtest (Hearing and Recording Sounds in Words) than the mean score for the control group. (a) State the null and alternative hypotheses of the test and define any relevant parameters. (b) The p-value for the test is 0.02 . State the conclusion of the test in context. Are the results statistically significant at the \(5 \%\) level? (c) The effect size was about two points, which means the mean score for the intervention group was approximately two points higher than the mean score for the control group on this subtest. A school board member argues, "While these results might be statistically significant, they may not be practically significant." What does she mean by this in this context?

Mating Choice and Offspring Fitness Does the ability to choose a mate improve offspring fitness in fruit flies? Researchers have studied this by taking female fruit flies and randomly dividing them into two groups; one group is put into a cage with a large number of males and able to freely choose who to mate with, while flies in the other group are each put into individual vials, each with only one male, giving no choice in who to mate with. Females are then put into egg laying chambers, and a certain number of larvae collected. Do the larvae from the mate choice group exhibit higher survival rates? A study \(^{44}\) published in Nature found that mate choice does increase offspring fitness in fruit flies (with p-value \(<0.02\) ), yet this result went against conventional wisdom in genetics and was quite controversial. Researchers attempted to replicate this result with a series of related experiments, \({ }^{45}\) with data provided in MateChoice. (a) In the first replication experiment, using the same species of fruit fly as the original Nature study, 6067 of the 10000 larvae from the mate choice group survived and 5976 of the 10000 larvae from the no mate choice group survived. Calculate the p-value. (b) Using a significance level of \(\alpha=0.05\) and \(\mathrm{p}\) -value from (a), state the conclusion in context. (c) Actually, the 10,000 larvae in each group came from a series of 50 different runs of the experiment, with 200 larvae in each group for each run. The researchers believe that conditions dif- fer from run to run, and thus it makes sense to treat each \(\mathrm{run}\) as a case (rather than each fly). In this analysis, we are looking at paired data, and the response variable would be the difference in the number of larvae surviving between the choice group and the no choice group, for each of the 50 runs. The counts (Choice and NoChoice and difference (Choice \(-\) NoChoice) in number of surviving larva are stored in MateChoice. Using the single variable of differences, calculate the p-value for testing whether the average difference is greater than \(0 .\) (Hint: this is a single quantitative variable, so the corresponding test would be for a single mean.) (d) Using a significance level of \(\alpha=0.05\) and the p-value from (c), state the conclusion in context. (e) The experiment being tested in parts (a)-(d) was designed to mimic the experiment from the original study, yet the original study yielded significant results while this study did not. If mate choice really does improve offspring fitness in fruit flies, did the follow-up study being analyzed in parts (a)-(d) make a Type I, Type II, or no error? (f) If mate choice really does not improve offspring fitness in fruit flies, did the original Nature study make a Type I, Type II, or no error?