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Georgianna claims that in a small city renowned for its music school, the average child takes less than 5 years of piano lessons. We have a random sample of 20 children from the city, with a mean of 4.6 years of piano lessons and a standard deviation of 2.2 years. (a) Evaluate Georgianna's claim (or that the opposite might be true) using a hypothesis test. (b) Construct a \(95 \%\) confidence interval for the number of years students in this city take piano lessons, and interpret it in context of the data. (c) Do your results from the hypothesis test and the confidence interval agree? Explain your reasoning.

Researchers interested in lead exposure due to car exhaust sampled the blood of 52 police officers subjected to constant inhalation of automobile exhaust fumes while working traffic enforcement in a primarily urban environment. The blood samples of these officers had an average lead concentration of \(124.32 \mu \mathrm{g} / \mathrm{l}\) and a SD of \(37.74 \mu \mathrm{g} / \mathrm{l} ;\) a previous study of individuals from a nearby suburb, with no history of exposure, found an average blood level concentration of \(35 \mu \mathrm{g} / \mathrm{l} .\) (a) Write down the hypotheses that would be appropriate for testing if the police officers appear to have been exposed to a different concentration of lead. (b) Explicitly state and check all conditions necessary for inference on these data. (c) Regardless of your answers in part (b), test the hypothesis that the downtown police officers have a higher lead exposure than the group in the previous study. Interpret your results in context.

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.

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. \(^{36}\) (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?

A professor who teaches a large introductory statistics class (197 students) with eight discussion sections would like to test if student performance differs by discussion section, where each discussion section has a different teaching assistant. The summary table below shows the average final exam score for each discussion section as well as the standard deviation of scores and the number of students in each section. $$ \begin{array}{rrrrrrrrrr} \hline & \text { Sec 1 } & \text { Sec 2 } & \text { Sec 3 } & \text { Sec 4 } & \text { Sec 5 } & \text { Sec 6 } & \text { Sec 7 } & \text { Sec 8 } \\ \hline n_{i} & 33 & 19 & 10 & 29 & 33 & 10 & 32 & 31 \\ \bar{x}_{i} & 92.94 & 91.11 & 91.80 & 92.45 & 89.30 & 88.30 & 90.12 & 93.35 \\\ s_{i} & 4.21 & 5.58 & 3.43 & 5.92 & 9.32 & 7.27 & 6.93 & 4.57 \\ \hline \end{array} $$ The ANOVA output below can be used to test for differences between the average scores from the different discussion sections. $$ \begin{array}{lrrrrr} \hline & \text { Df } & \text { Sum Sq } & \text { Mean Sq } & \text { F value } & \operatorname{Pr}(>\mathrm{F}) \\ \hline \text { section } & 7 & 525.01 & 75.00 & 1.87 & 0.0767 \\ \text { Residuals } & 189 & 7584.11 & 40.13 & & \\ \hline \end{array} $$ Conduct a hypothesis test to determine if these data provide convincing evidence that the average score varies across some (or all) groups. Check conditions and describe any assumptions you must make to proceed with the test.

Determine if the following statements are true or false, and explain your reasoning for statements you identify as false. If the null hypothesis that the means of four groups are all the same is rejected using ANOVA at a \(5 \%\) significance level, then ... (a) we can then conclude that all the means are different from one another. (b) the standardized variability between groups is higher than the standardized variability within groups. (c) the pairwise analysis will identify at least one pair of means that are significantly different. (d) the appropriate \(\alpha\) to be used in pairwise comparisons is \(0.05 / 4=0.0125\) since there are four groups.