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Equip male and female students with a small device that secretly records sound for a random 30 seconds during each \(12.5\) minute period over two days. Count the words each subject speaks during each recording period, and from this, estimate how many words per day each subject speaks. The published report includes a table summarizing six such studies. \({ }^{12}\) Here are two of the six: $$ \begin{array}{ccccc} \hline & \text { Sample Size } & & \text { Estimated Average Number (SD) of Words Spoken per Day } \\ \text { Study } & \text { Women Men } & \text { Women } & \text { Men } \\ \hline 1 & 56 & 56 & 16,177(7520) & 16,569(9108) \\ \hline 2 & 27 & 20 & 16,496(7914) & 12,867(8343) \\ \hline \end{array} $$ Readers are supposed to understand that, for example, the 56 women in the first study had \(\mathrm{x}^{-} \bar{x}=16,177\) and \(s=7520\). It is commonly thought that women talk more than men. Does either of the two samples support this idea? For each study: (a) state hypotheses in terms of the population means for men \(\left(\mu_{M}\right)\) and women \(\left(\mu_{F}\right)\). (b) find the two-sample \(t\) statistic. (c) what degrees of freedom does Option 2 use to get a conservative \(P\)-value? (d) compare your value of \(t\) with the critical values in Table C. What can you say about the \(P\)-value of the test? (e) what do you conclude from the results of these two studies?

What is the effect of concussions on the brain? Researchers measured the brain sizes (hippocampal volume in microliters) of 25 collegiate football players with a history of clinician-diagnosed concussion and 25 collegiate football players without a history of concussion. Here are the summary statistics: 18 $$ \begin{array}{lccc} \hline \text { Group } & \text { Group Size } & \text { Mean } & \text { Standard Deviation } \\ \hline \text { Concussion } & 25 & 5784 & 609.3 \\ \hline \text { Nonconcussion } & 25 & 6489 & 815.4 \\ \hline \end{array} $$ (a) Is there evidence of a difference in mean brain size between football players with a history of concussion and those without concussions? (b) The researchers in this study stated that participants were "consecutive cases of healthy National Collegiate Athletic Association Football Bowl Subdivision Division I football athletes with \((n=25)\) or without \((n=25)\) a history of clinician-diagnosed concussion ... between June 2011 and August 2013" at a U.S. psychiatric research institute specializing in neuroimaging among collegiate football players. What effect does this information have on your conclusions in part (a)?