91Ó°ÊÓ

Find the p-value. An independent random sample is selected from an approximately normal population with an unknown standard deviation. Find the p-value for the given set of hypotheses and \(T\) test statistic. Also determine if the null hypothesis would be rejected at \(\alpha=0.05\). (a) \(H_{A}: \mu>\mu_{0}, n=11, T=1.91\) (c) \(H_{A}: \mu \neq \mu_{0}, n=7, T=0.83\) (b) \(H_{A}: \mu<\mu_{0}, n=17, T=-3.45\) (d) \(H_{A}: \mu>\mu_{0}, n=28, T=2.13\)

Diamonds, Part I. Prices of diamonds are determined by what is known as the 4 Cs: cut, clarity, color, and carat weight. The prices of diamonds go up as the carat weight increases, but the increase is not smooth. For example, the difference between the size of a 0.99 carat diamond and a 1 carat diamond is undetectable to the naked human eye, but the price of a 1 carat diamond tends to be much higher than the price of a 0.99 diamond. In this question we use two random samples of diamonds, 0.99 carats and 1 carat, each sample of size \(23,\) and compare the average prices of the diamonds. In order to be able to compare equivalent units, we first divide the price for each diamond by 100 times its weight in carats. That is, for a 0.99 carat diamond, we divide the price by 99. For a 1 carat diamond, we divide the price by \(100 .\) The distributions and some sample statistics are shown below. \(^{37}\) Conduct a hypothesis test to evaluate if there is a difference between the average standardized prices of 0.99 and 1 carat diamonds. Make sure to state your hypotheses \begin{tabular}{|l|} \hline \\ \hline \end{tabular} clearly, check relevant conditions, and interpret your results in context of the data.

4.36 True or false, Part I. Determine if the following statements are true or false, and explain your reasoning for statements you identify as false. (a) When comparing means of two samples where \(n_{1}=20\) and \(n_{2}=40,\) we can use the normal model for the difference in means since \(n_{2} \geq 30\). (b) As the degrees of freedom increases, the T distribution approaches normality. (c) We use a pooled standard error for calculating the standard error of the difference between means when sample sizes of groups are equal to each other.

Coffee, depression, and physical activity. Caffeine is the world's most widely used stimulant, with approximately \(80 \%\) consumed in the form of coffee. Participants in a study investigating the relationship between coffee consumption and exercise were asked to report the number of hours they spent per week on moderate (e.g., brisk walking) and vigorous (e.g., strenuous sports and jogging) exercise. Based on these data the researchers estimated the total hours of metabolic equivalent tasks (MET) per week, a value always greater than \(0 .\) The table below gives summary statistics of MET for women in this study based on the amount of coffee consumed. 3 (a) Write the hypotheses for evaluating if the average physical activity level varies among the different levels of coffee consumption. (b) Check conditions and describe any assumptions you must make to proceed with the test. (c) Below is part of the output associated with this test. Fill in the empty cells. (d) What is the conclusion of the test?

True or false, Part II. Determine if the following statements are true or false in ANOVA, and explain your reasoning for statements you identify as false. (a) As the number of groups increases, the modified significance level for pairwise tests increases as well. (b) As the total sample size increases, the degrees of freedom for the residuals increases as well. (c) The constant variance condition can be somewhat relaxed when the sample sizes are relatively consistent across groups. (d) The independence assumption can be relaxed when the total sample size is large.