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5. A weight training and exercise program has been developed to increase the speed of a pitcher's fastball.Nine players take part in a test of the program.Each player has his pitches timed before the program is begun and after the program is over.Their paired averages are (80, 85), (73, 77), (80, 86), (83, 90), (93, 95), (90, 89), (79, 79), (91, 90), (95, 98) where the first number of a pair is the average speed before the program and the second number is the average speed after the program.At a confidence level of 95% can we conclude that theprogram increases the speed of a pitcher's fastball? 6.Given a random variablewith distributionfind the sample sizerequired for a 95% confidence interval for the mean with an error of at most 7.Given data for the final examination scores for two classes in differential equations and (a)Test the hypothesisagainst the alternativeat the significance level assuming andareandwhere (b)Test the hypothesis against the alternativeat the significance level. 8.It is suspected that a particular die used in a game of chance was not fair.It was thought that the 1-side was heavy and the 6-side was light.Data was collected concerning the number of observed 1-s when the die was rolled 5000 times.Let equal the probability of rolling a one with that die. We will test the hypothesisagainst the alternative hypothesis. (a)Give the test statistic and critical region that has an significance level. (b)If is the observed number of ones in the 5000 rolls, calculate the value of the test statistic and state your conclusion about the die. (c)Find the p-value for this test. 9. (Extra Credit)Explain clearly what a p-value is.

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