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Give as much information as you can about the \(P\) -value for an upper-tailed \(F\) test in each of the following situations. a. \(\mathrm{df}_{1}=4, \mathrm{df}_{2}=15, F=5.37\) b. \(\mathrm{df}_{1}=4, \mathrm{df}_{2}=15, F=1.90\) c. \(\mathrm{df}_{1}=4, \mathrm{df}_{2}=15, F=4.89\) d. \(\mathrm{df}_{1}=3, \mathrm{df}_{2}=20, F=14.48\) e. \(\mathrm{df}_{1}=3, \mathrm{df}_{2}=20, F=2.69\) f. \(\mathrm{df}_{1}=4, \mathrm{df}_{2}=50, F=3.24\)

Employees of a certain state university system can choose from among four different health plans. Each plan differs somewhat from the others in terms of hospitalization coverage. Four random samples of recently hospitalized individuals were selected, each sample consisting of people covered by a different health plan. The length of the hospital stay (number of days) was determined for each individual selected. a. What hypotheses would you test to decide whether the mean lengths of stay are not the same for all four health plans? b. If each sample consisted of eight individuals and the value of the ANOVA \(F\) statistic was \(F=4.37\), what conclusion would be appropriate for a test with \(\alpha=0.01 ?\) c. Answer the question posed in Part (b) if the \(F\) value given there resulted from sample sizes \(n_{1}=9, n_{2}=8, n_{3}=7\), and \(n_{4}=8\).

The authors of the paper "Age and Violent Content Labels Make Video Games Forbidden Fruits for Youth鈥� (Pediatrics [2009]: 870-876) carried out an experiment to determine if restrictive labels on video games actually increased the attractiveness of the game for young game players. Participants read a description of a new video game and were asked how much they wanted to play the game. The description also included an age rating. Some participants read the description with an age restrictive label of \(7+\), indicating that the game was not appropriate for children under the age of 7 . Others read the same description, but with an age restrictive label of \(12+, 16+,\) or \(18+\). The following data for 12- to 13-year-old boys are consistent with summary statistics given in the paper. (The sample sizes in the actual experiment were larger.) For purposes of this exercise, you can assume that the boys were assigned at random to one of the four age label treatments \((7+, 12+, 16+,\) and \(18+) .\) Data shown are the boys' ratings of how much they wanted to play the game on a scale of 1 to 10 . Do the data provide convincing evidence that the mean rating associated with the game description by 12 - to 13-year-old boys is not the same for all four restrictive rating labels? Test the appropriate hypotheses using a significance level of 0.05

Give as much information as you can about the \(P\) -value of the single-factor ANOVA \(F\) test in each of the following situations. a. \(k=5, n_{1}=n_{2}=n_{3}=n_{4}=n_{5}=4, F=5.37\) b. \(k=5, n_{1}=n_{2}=n_{3}=5, n_{4}=n_{5}=4, F=2.83\) c. \(k=3, n_{1}=4, n_{2}=5, n_{3}=6, F=5.02\) d. \(k=3, n_{1}=n_{2}=4, n_{3}=6, F=15.90\) e. \(k=4, n_{1}=n_{2}=15, n_{3}=12, n_{4}=10, F=1.75\)

Leaf surface area is an important variable in plant gas-exchange rates. Dry matter per unit surface area (mg/ \(\mathrm{cm}^{3}\) ) was measured for trees raised under three different growing conditions. Let \(\mu_{1}, \mu_{2},\) and \(\mu_{3}\) represent the mean dry matter per unit surface area for the growing conditions \(1,2,\) and \(3,\) respectively. Suppose that the given \(95 \% \mathrm{~T}-\mathrm{K}\) confidence intervals are: \(\begin{array}{lccc}\text { Difference } & \mu_{1}-\mu_{2} & \mu_{1}-\mu_{3} & \mu_{2}-\mu_{3} \\ \text { Interval } & (-3.11,-1.11) & (-4.06,-2.06) & (-1.95,0.05)\end{array}\) Which of the following four statements do you think describes the relationship between \(\mu_{1}, \mu_{2},\) and \(\mu_{3} ?\) Explain your choice. a. \(\mu_{1}=\mu_{2},\) and \(\mu_{3}\) differs from \(\mu_{1}\) and \(\mu_{2}\). b. \(\mu_{1}=\mu_{3},\) and \(\mu_{2}\) differs from \(\mu_{1}\) and \(\mu_{3}\) c. \(\mu_{2}=\mu_{3},\) and \(\mu_{1}\) differs from \(\mu_{2}\) and \(\mu_{3}\) d. All three \(\mu\) 's are different from one another.

The paper "Trends in Blood Lead Levels and Blood Lead Testing among U.S. Children Aged 1 to 5 Years" (Pediatrics [2009]: e376-e385) gave data on blood lead levels (in \(\mathrm{mg} / \mathrm{dL}\) ) for samples of children living in homes that had been classified either at low, medium, or high risk of lead exposure, based on when the home was constructed. After using a multiple comparison procedure, the authors reported the following: 1\. The difference in mean blood lead level between low-risk housing and medium-risk housing was significant. 2\. The difference in mean blood lead level between low-risk housing and high-risk housing was significant. 3\. The difference in mean blood lead level between mediumrisk housing and high-risk housing was significant. Which of the following sets of T-K intervals (Set \(1,2,\) or 3\()\) is consistent with the authors' conclusions? Explain your choice. \(\mu_{L}=\) mean blood lead level for children living in low-risk housing \(\mu_{M}=\) mean blood lead level for children living in mediumrisk housing \(\mu_{H}=\) mean blood lead level for children living in high-risk housing