Problem 1. Suppose each X; for i = 1, 2,... is independently drawn from the same normal distribution with variance o = 4. We do not know the mean and wish to find a confidence interval it lies in. Remark: Recall that (-) refers to the cdf of a standard normal distribution, which can be looked up on a z-table. Furthermore, recall that we defined za the critical value which satisfies: Pr(Z > za) = a Here, Z is a standard normal random variable. Rearranging, we see: Pr(Za) 1 Pr(Z z) = 1 (za) So, to find the critical value za, we want to find the value of za from the z-table such that: (za)=1-a In other words, you calculate 1-a and find the entry on the z-table that equals 1-a. Wherever 1-a is, that is the value of za. So, for example, 20.005 = 2.57. Please verify you know how to find the value of Za given a. (a) Say we get n = 10 data points, and the sample average is X = 15. (Recall, we calculate X by adding up all the data points and dividing by n = 10.) What is a 99% confidence interval for ? (b) Suppose now that we have n = 100 data points, and the sample average is X = 15. (Now, this is the result of averaging 100 instead of 10 data points.) What is a 99% confidence interval for ?