(2) In order to meet specifications, widgets must have a given time coefficient of at least 20. A sample of 100 widgets is drawn from the day's production and tested and 21 of them had time coefficients less than 20. (a) Find a (i) 95% confidence interval and (ii) 98% confidence interval, both, for the pro- portion of widgets manufactured that day which fail to meet specifications. (b) For each of the two levels of confidence computed in part (a), find the sample size needed so that the corresponding confidence intervals specify the proportion to within 1 0.03. (c) If a 96% confidence interval is computed each day for 120 days, what is the probability that 115 of the computed confidence intervals cover the true proportion? (d) If a 96% confidence interval is computed each day for n days, to find the probability that at least 115 of the computed confidence intervals cover the true proportion, for what n > 150, can we safely utilize the normal approximation to the binomial? Justify your answer. (e) If a 96% confidence interval is computed each day for 500 days, what is the probability that at least 490 of the computed confidence intervals cover the true proportion