31. A simple random sample of size 100 is selected from a population with p = .40. a. What is the expected value of p? b. What is the standard error of p? c. Show the sampling distribution of p. d. What does the sampling distribution of p show? 32. A population proportion is .40. A simple random sample of size 200 will be taken and the sample proportion p will be used to estimate the population proportion. a. What is the probability that the sample proportion will be within 1.03 of the popula- tion proportion? b. What is the probability that the sample proportion will be within 1.05 of the popula- tion proportion? 33. Assume that the population proportion is .55. Compute the standard error of the propor- tion, 0, for sample sizes of 100, 200, 500, and 1000. What can you say about the size of the standard error of the proportion as the sample size is increased? 34. The population proportion is .30. What is the probability that a sample proportion will be within +.04 of the population proportion for each of the following sample sizes? a. n = 100 b. n = 200 C. n= 500 d. n = 1000 e. What is the advantage of a larger sample size