Question 1 Let Y₁ ~ Normal (μ, o²), i = 1, 2, ..., n. independent and identically distributed. Suppose is unknown, and the standard deviation of each observation o = 2

is known. The sample size n = 100. To estimate the population mean μ, we consider the sample mean Y = Y₁+Y₂++, which is the estimator for μ. n Hint for Q1: https://online.stat.psu.edu/stat500/lesson/4 https://online.stat.psu.edu/stat500/lesson/5 https://online.stat.psu.edu/stat500/lesson/6 71 1. The distribution of the sample mean Ỹ = ₁+++ is (A) with expectation (B) and standard deviation (C) Fill out (A), (B), and (C) Remark The distribution of an estimator is called a "Sampling Distribution" • The standard deviation of an estimator is called a "Standard Error" 2. Consider a random variable U where U = (sampling distributions) (confidence intervals) (hypothesis testing for one mean) Ý-μ SE(Y) Show that the distribution of U is N(0, 1). - Y – μ o/√√n Hint: use the fact that for a normal random variable X~ N(o,), the standardized normal random variable follows N(0, 1). This is an important fact that you should remember! 3. Based on 3, find a 95% confidence interval for (the derivation is not required). In other words, specify the point etimate and margin of error./nU = Show that the distribution of U is N(0, 1). Ỹ -" SE(Y) Hint: use the fact that for a normal random variable X random variable follows N (0, 1). This is an important fact that you should remember! U = 3. Based on 3, find a 95% confidence interval for (the derivation is not required). In other words, specify the point etimate and margin of error. 4. Now suppose that is unknown. It is a well-known fact that the distribution of standardized mean based on the estimated standard error Y-" o/√√n Ý - μ SE(Y) TS N(o,), the standardized normal is the t distribution with df = n - 1. Based on this fact, find a 95% confidence interval for μ (the derivation is not required). In other words, specify the point etimate and margin of error. Y - 3 S/√n 5. Suppose we want to test whether Ho: μ = 3 and H₁ : μ‡ 3. Recall that n = 100. We consider a test statistic = Ý - μ S/√n Y - 3 S/10 where Y is the sample mean and S² is the sample variance (S = √S² is the sample standard deviation). 5-1. Find the observed test statistic for the data whose sample mean is = 2.7 and the sample standard deviation s = 2.1. 5-2. Find the p-value. Hint: You can use the pt function in R to compute probabilities such as P(T > c) for any number c when T ~ t(df).