Chapter 4: Problem 7
State the null and alternative hypotheses for the statistical test described. Testing to see if there is evidence that a mean is less than 50 .
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Give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.7\) vs \(H_{a}: p<0.7\) Sample data: \(\hat{p}=125 / 200=0.625\) with \(n=200\)
Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) (a) 95\% confidence interval for \(p: \quad 0.53\) to 0.57 (b) \(95 \%\) confidence interval for \(p: \quad 0.41\) to 0.52 (c) 99\% confidence interval for \(p: \quad 0.35\) to 0.55
Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: \rho=0\) vs \(H_{a}: \rho \neq 0\). In addition, in each case for which the results are significant, give the sign of the correlation. (a) \(95 \%\) confidence interval for \(\rho: 0.07\) to 0.15 . (b) \(90 \%\) confidence interval for \(\rho:-0.39\) to -0.78 . (c) \(99 \%\) confidence interval for \(\rho:-0.06\) to 0.03 .
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?
The consumption of caffeine to benefit alertness is a common activity practiced by \(90 \%\) of adults in North America. Often caffeine is used in order to replace the need for sleep. One study \(^{24}\) compares students' ability to recall memorized information after either the consumption of caffeine or a brief sleep. A random sample of 35 adults (between the ages of 18 and 39 ) were randomly divided into three groups and verbally given a list of 24 words to memorize. During a break, one of the groups takes a nap for an hour and a half, another group is kept awake and then given a caffeine pill an hour prior to testing, and the third group is given a placebo. The response variable of interest is the number of words participants are able to recall following the break. The summary statistics for the three groups are in Table 4.9. We are interested in testing whether there is evidence of a difference in average recall ability between any two of the treatments. Thus we have three possible tests between different pairs of groups: Sleep vs Caffeine, Sleep vs Placebo, and Caffeine vs Placebo. (a) In the test comparing the sleep group to the caffeine group, the p-value is \(0.003 .\) What is the conclusion of the test? In the sample, which group had better recall ability? According to the test results, do you think sleep is really better than caffeine for recall ability? (b) In the test comparing the sleep group to the placebo group, the p-value is 0.06 . What is the conclusion of the test using a \(5 \%\) significance level? If we use a \(10 \%\) significance level? How strong is the evidence of a difference in mean recall ability between these two treatments? (c) In the test comparing the caffeine group to the placebo group, the p-value is 0.22 . What is the conclusion of the test? In the sample, which group had better recall ability? According to the test results, would we be justified in concluding that caffeine impairs recall ability? (d) According to this study, what should you do before an exam that asks you to recall information?
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