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Use the t-distribution and the sample results to complete the test of the hypotheses. Use a \(5 \%\) significance level. Assume the results come from a random sample, and if the sample size is small, assume the underlying distribution is relatively normal. Test \(H_{0}: \mu=500\) vs \(H_{a}: \mu \neq 500\) using the sample results \(\bar{x}=432, s=118,\) with \(n=75\).

It is generally recommended that adults sleep at least 8 hours each night. One of the authors recently asked some of her students (undergraduate and graduate students at Harvard) how many hours each had slept the previous night, curious as to whether her students are getting enough sleep. The data are displayed in Figure \(6.18 .\) The 12 students sampled averaged $$ \begin{aligned} &\text { 40 Belguermi, A., "Pigeons discriminate between human feeders," }\\\ &\text { Animal Cognition, } 2011 ; 14: 909-914 . \end{aligned} $$ 6.2 hours of sleep with a standard deviation of 1.70 hours. Assuming this sample is representative of all her students, and assuming students need at least 8 hours of sleep a night, does this provide evidence that, on average, her students are not getting enough sleep?

In the mid-1990s a Nabisco marketing campaign claimed that there were at least 1000 chips in every bag of Chips Ahoy! cookies. A group of Air Force cadets collected a sample of 42 bags of Chips Ahoy! cookies, bought from locations all across the country, to verify this claim. \({ }^{41}\) The cookies were dissolved in water and the number of chips (any piece of chocolate) in each bag were hand counted by the cadets. The average number of chips per bag was \(1261.6,\) with standard deviation 117.6 chips. (a) Why were the cookies bought from locations all over the country? (b) Test whether the average number of chips per bag is greater than 1000 . Show all details. (c) Does part (b) confirm Nabisco's claim that every bag has at least 1000 chips? Why or why not?

If random samples of the given sizes are drawn from populations with the given proportions: (a) Find the mean and standard error of the distribution of differences in sample proportions, \(\hat{p}_{A}-\hat{p}_{B}\) (b) If the sample sizes are large enough for the Central Limit Theorem to apply, draw a curve showing the shape of the sampling distribution. Include at least three values on the horizontal axis. Samples of size 80 from population \(A\) with proportion 0.40 and samples of size 60 from population \(B\) with proportion 0.10

Has Support for Capital Punishment Changed over Time? The General Social Survey (GSS) has been collecting demographic, behavioral, and attitudinal information since 1972 to monitor changes within the US and to compare the US to other nations. \({ }^{46}\) Support for capital punishment (the death penalty) in the US is shown in 1974 and in 2006 in the two-way table in Table \(6.6 .\) Find a \(95 \%\) confidence interval for the change in the proportion supporting capital punishment between 1974 and 2006. Is it plausible that the proportion supporting capital punishment has not changed? $$ \begin{array}{rrcl} \hline \text { Year } & \text { Favor } & \text { Oppose } & \text { Total } \\\ \hline 1974 & 937 & 473 & 1410 \\ 2006 & 1945 & 870 & 2815 \\ \hline \end{array} $$

(a) Find the relevant sample proportions in each group and the pooled proportion. (b) Complete the hypothesis test using the normal distribution and show all details. Test whether there is a difference between two groups in the proportion who voted, if 45 out of a random sample of 70 in Group 1 voted and 56 out of a random sample of 100 in Group 2 voted.

(a) Find the relevant sample proportions in each group and the pooled proportion. (b) Complete the hypothesis test using the normal distribution and show all details. Table 6.10 gives flight arrival numbers from a random sample of flights for two airlines. Test whether there is a difference between the two airlines in the percent of flights that arrive late.

Refer 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 invasive breast cancer? $$ \begin{array}{lcc} \hline \text { Condition } & \text { HRT Group } & \text { Placebo Group } \\ \hline \text { Cardiovascular Disease } & 164 & 122 \\ \text { Invasive Breast Cancer } & 166 & 124 \\ \text { Cancer (all) } & 502 & 458 \\ \text { Fractures } & 650 & 788 \\ \hline \end{array} $$

Refer 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 cancer of any kind? $$ \begin{array}{lcc} \hline \text { Condition } & \text { HRT Group } & \text { Placebo Group } \\ \hline \text { Cardiovascular Disease } & 164 & 122 \\ \text { Invasive Breast Cancer } & 166 & 124 \\ \text { Cancer (all) } & 502 & 458 \\ \text { Fractures } & 650 & 788 \\ \hline \end{array} $$

In the dataset ICUAdmissions, the variable Service indicates whether the ICU (Intensive Care Unit) patient had surgery (1) or other medical treatment (0) and the variable Sex gives the gender of the patient (0 for males and 1 for females.) Use technology to test at a \(5 \%\) level whether there is a difference between males and females in the proportion of ICU patients who have surgery.