In addition to the true confidence level of a confidence interval, the expected

Question

In addition to the true confidence level of a confidence interval, the expected length of an interval is important too. For example, the length of a Wald confidence interval when w = 1 and n = 40 is 0.0734 - (-0.0234) = 0.0968 (see example on p. 19). The expected length is defined as sigma^n_w = 0 L(pi cap) (n w) pi^w(1 - pi)^n - w where L(pi cap) denotes the interval length using particular pi cap. Because each length is multiplied by the corresponding probability that an interval is observed, we obtain the by the expected value of the length by summing these terms. Complete the following problems on the expected length. would like a confidence interval to be as short as possible with respect to its expected length having the correct confidence level. Why? For n = 40, pi = 0.16, and alpha = 0.05, find the expected length for the Wald interval and verify it is 0.2215. For n 40 and alpha = 0.05, construct a similar plot to Figure 1.3, but now using the expected length for the y-axis on the plots. Compare the expected lengths among the intervals. To help with the comparisons, you will want to fix the y axis on each plot to be the same (use the y lim = c(0, 0.35) argument value in each plot() function call). Alternatively to having four plots in one R Graphics window, overlay the expected lengths for all four intervals on one plot. This can be done by using one plot() function for the first expected calls to the lines() function can then be used to overlay the remaining intervals' expected lengths. Using Figure 1.3 and the plot from (c), which interval is best? Explain outline the steps that would be needed to find the estimated expected length using Monte Carlo simulation.

Explanation / Answer

Here we can see that the distribution belong to binomial distribution

a.the confidence interval we here is taken as short because they are exact

d.the shortest interval is the best

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