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NFL Combine The National Football League (NFL) combine is a forum in which players who want to play in the NFL demonstrate various skills. One skill that is measured is the 40-yard dash. Go to the book’s website to obtain the data file 13_1_23 using the file format of your choice for the version of the text you are using.

(a) What type of variable is 40-yard dash time?

(b) Excluding quarterback (QB), the skill positions are cornerback (CB), safety (S), running back (RB), and wide receiver (WR). Suppose we want to know whether there is a difference between the 40-yard dash times for these positions. What would be the null and alternative hypotheses?

(c) Assuming we treat the players as a random sample of all players in each position, explain why we may use a one-way ANOVA to determine if there is a significant difference between 40-yard dash times for these positions. How many means are being compared?

(d) Does the evidence suggest a difference exists for the 40-yard dash times for the skill positions (except quarterback)? Use the  level of significance.

(e) Draw boxplots of the 40-yard dash times for the skill positions (except quarterback) to support the results from part (d).

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Work through the parts of this exercise in order, and

use them to make a generalization concerning volumes

of spheres. Leave answers in terms of p.

(a) Find the volume of a sphere having radius of 1 m.

(b) Suppose the radius is doubled to 2 m. What is the

volume?

(c) When the radius was doubled, by how many times did

the volume increase? (To find out, divide the answer

for part (b) by the answer for part (a).)

(d) Suppose the radius of the sphere from part (a) is

tripled to 3 m. What is the volume?

(e) When the radius was tripled, by how many times

did the volume increase?

(f) In general, if the radius of a sphere is multiplied by

n, the volume is multiplied by .

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Putting It Together: Time to Complete a Degree A researcher wanted to determine if the mean time to complete a bachelor’s degree was different depending on the selectivity of the first institution of higher education that was attended. The following data represent a random sample of 12th-graders who earned their degree within eight years. Probability plots indicate that the data for each treatment level are normally distributed.

(a) What type of observational study was conducted? What is the response variable?

(b) Find the sample mean for each treatment level.

(c) Find the sample standard deviation for each treatment level. Using the general rule presented in this chapter, does it appear that the population variances are the same?

(d) Use the time to degree completion for students first attending highly selective institutions to construct a 95% confidence interval estimate for the population mean.

(e) How many pairwise comparisons are possible among the treatment levels?

(f) Consider the null hypothesis . If we test this hypothesis using t-tests for each pair of treatments, use your answer from part (e) to compute the probability of making a Type I error, assuming that each test uses an  level of significance.

(g) Use the one-way ANOVA procedure to determine if there is a difference in the meantime to degree completion for the different types of initial institutions. If the null hypothesis is rejected, use Tukey’s test to determine which pairwise differences are significant using a familywise error rate of .

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Lower Your Cholesterol Researcher Francisco Fuentes and his colleagues wanted to determine the most effective diet for reducing LDL cholesterol, the so-called “bad” cholesterol, among three diets: (1) a saturated-fat diet: 15% protein, 47% carbohydrates, and 38% fat (20% saturated fat, 12% monounsaturated fat, and 6% polyunsaturated fat); (2) the Mediterranean diet: 47% carbohydrates, and 38% fat (10% saturated fat, 22% monounsaturated fat, and 6% polyunsaturated fat); and (3) the US National Cholesterol Education Program or NCEP-1 Diet: 10% saturated fat, 12% monounsaturated fat, and 6% polyunsaturated fat. Participants in the study were shown to have the same levels of LDL cholesterol before the study and were randomly assigned to one of the three diets, or treatment groups. After 28 days, their LDL cholesterol levels were recorded. The data in the following table are based on this study.

(a) Why is this study a completely randomized design?

(b) What is the response variable? What is the explanatory variable that is controlled and set at three levels?

(c) The participants were randomly assigned to one of three treatment groups. What is the purpose of randomization in this study?

(d) State the null and alternative hypotheses.

(e) Verify that the requirements to use the one-way ANOVA procedure are satisfied. Normal probability plots indicate that the sample data come from normal populations.

(f) Are the mean LDL cholesterol levels different at the  level of significance?

(g) Draw boxplots of the LDL cholesterol levels for the three groups to support the analytic results obtained in part (f).

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