Problem 1. Suppose a N (up) random variable accurately describes the distribution of annual al- lowances in USD for children in a population of interest. Let this variable be X m N (pa, 0). Suppose the population standard deviation is known to be a = 259 USD and we wish to estimate the population mean ,u in USD. From a simple random sample of size n = 144 drawn from the population, we calculate the sample mean to be :7: = 1117.8. a) Derive a 96% condence interval for it based on the distribution of it. (6pts) b) Suppose 200 other researchers replicate the process in (a) each with their own independent simple random sample; that is, all of them use the same sample size of n = 144 for each sample to derive 96% condence intervals, but their samples are all different and independent of each other. What is the expected number of these intervals that are bad condence intervals; i.e., they do not, in fact, contain the true population mean? Note: this is determining the number of bad condence intervals due to chance, not an error on any researcher's part. This relates to the distribution of the sample mean. (4pm)