Statistics: HW 7Deadline: April 18 at 8 p.m.Problem 1. Suppose we have n data points drawn from a N(μ, 52) distribution, wherethe value of μ is unknown.(a) (1 point) Suppose we have 16 data points and that the sample mean is ˉx = 20.Construct a precise 95% confidence interval for the μ.(b) (1 point) Explain why the confidence interval for the mean when n = 100 is alwaysnarrower than the confidence interval when n = 10.(c) (1 point) What is the smallest value of n so that the 95% confidence interval for themean will have width less than 1.0? (Still with σ2 = 52)Problem 2. Adult mayflies live anywhere from 30 minutes to 1 day, depending on thespecies. Data for one species was collected by tracking 10 mayflies. The recorded lifespansin hours were17.68, 13.69, 11.22, 11.05, 13.86, 14.47, 14.50, 13.47, 10.04, 13.10(a) (1 point) Compute a 95% confidence interval for the mean lifetime of this species ofmayfly.(b) (1 point) What assumptions did you make in part (a)?(c) (1 point) Compute a 95% confidence interval for the standard deviation of distributionof the lifetime of a mayfly.(d) (1 point) Based on the sample variance of this data estimate the number of datapoints you would need to make the width of the 95% confidence interval for themean less than or equal to 1 hour.(e) (1 point) Is the value of n in part (d) guaranteed to be sufficient? Explain yourreasoning.Problem 3. Finish the t-test task from the practice:(a) (4 points) Build the distribution of p-values when H0 is true(b) (8 points) Build the distribution of p-values when H0 is false. Try to decrease thesample size, while holding the mean the same. Try to increase the mean whileholding the sample size the same. Explain your results.(c) (4 points) Build the distribution of p-values when xi ～ Exp(1) and sample sizen = 5. Explain your results.Problem 4. Consider the z-test (slides 35–37):(a) (4 points) Build the distribution of p-values when H0 is true. Please consider differentdistributions.(b) (8 points) Build the distribution of p-values when H0 is false. Please consider differentdistributions. Try to decrease the sample size, while holding the mean thesame. Try to increase the mean while holding the sample size the same. Explainyour results.(c) (4 points) Build the distribution of p-values when xi ～ Cauchy and sample sizen = 1000. Explain your results.转自：http://www.7daixie.com/2019050148152891.html
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