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Chapter 6: Problem 92
A sample with \(n=12, \bar{x}=7.6,\) and \(s=1.6\)
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Rrefer to a study on hormone replacement therapy. Until 2002 , hormone replacement therapy (HRT), taking hormones to replace those the body no longer makes after menopause, was commonly prescribed to post-menopausal women. However, in 2002 the results of a large clinical trial \(^{56}\) were published, causing most doctors to stop prescribing it and most women to stop using it, impacting the health of millions of women around the world. In the experiment, 8506 women were randomized to take HRT and 8102 were randomized to take a placebo. Table 6.16 shows the observed counts for several conditions over the five years of the study. (Note: The planned duration was 8.5 years. If Exercises 6.205 through 6.208 are done correctly, you will notice that several of the p-values are just below \(0.05 .\) The study was terminated as soon as HRT was shown to significantly increase risk (using a significance level of \(\alpha=0.05)\), because at that point it was unethical to continue forcing women to take HRT). Does HRT influence the chance of a woman getting cardiovascular disease? $$ \begin{array}{lcc} \hline \text { Condition } & \text { HRT Group } & \text { Placebo Group } \\ \hline \text { Cardiovascular Disease } & 164 & 122 \\ \text { Invasive Breast Cancer } & 166 & 124 \\ \text { Cancer (all) } & 502 & 458 \\ \text { Fractures } & 650 & 788 \\ \hline \end{array} $$
THC vs Prochloroperazine An article in the New York Times on January 17,1980 reported on the results of an experiment that compared an existing treatment drug (prochloroperazine) with using THC (the active ingredient in marijuana) for combating nausea in patients undergoing chemotherapy for cancer. Patients being treated in a cancer clinic were divided at random into two groups which were then assigned to one of the two drugs (so they did a randomized, double- blind, comparative experiment). Table 6.15 shows how many patients in each group found the treatment to be effective or not effective. (a) Use these results to test whether the proportion of patients helped by THC is significantly higher (no pun intended) than the proportion helped by prochloroperazine. Use a \(1 \%\) significance level since we would require very strong evidence to switch to THC in this case. (b) Why is it important that these data come from a well-designed experiment? $$ \begin{array}{lccc} \hline \text { Treatment } & \text { Sample Size } & \text { Effective } & \text { Not Effective } \\ \hline \text { THC } & 79 & 36 & 43 \\ \text { Prochloroperazine } & 78 & 16 & 62 \\ \hline \end{array} $$
Situations comparing two proportions are described. In each case, determine whether the situation involves comparing proportions for two groups or comparing two proportions from the same group. State whether the methods of this section apply to the difference in proportions. (a) In a taste test, compare the proportion of tasters who prefer one brand of cola to the proportion who prefer the other brand. (b) Compare the proportion of males who voted in the last election to the proportion of females who voted in the last election. (c) Compare the graduation rate (proportion to graduate) of students on an athletic scholarship to the graduation rate of students who are not on an athletic scholarship. (d) Compare the proportion of voters who vote in favor of a school budget to the proportion who vote against the budget.
Use a t-distribution. Assume the samples are random samples from distributions that are reasonably normally distributed, and that a t-statistic will be used for inference about the difference in sample means. State the degrees of freedom used. Find the endpoints of the t-distribution with \(2.5 \%\) beyond them in each tail if the samples have sizes \(n_{1}=15\) and \(n_{2}=25\)
How big is the home field advantage in the National Football League (NFL)? In Exercise 6.240 on page 419 , we examine a difference in means between home and away teams using two separate samples of 80 games from each group. However, many factors impact individual games, such as weather conditions and the scoring of the opponent. It makes more sense to investigate this question using a matched pairs design, using scores for home and away teams matched for the same game. The data in NFLScores2011 include the points scored by the home and away team in 256 regular season games in \(2011 .\) We will treat these games as a sample of all NFL games. Estimate average home field scoring advantage and find a \(90 \%\) confidence interval for the mean difference.
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