Discuss the below:
Q: Let (X1, X2,..., Xn) be a sample, where each Xi is a random variable of normal distribution with mean μ and variance σ². Let us suppose that n = 20, and σ² = 9. An experiment has yielded the results (X1, X2,..., X20), and we have calculated that the empirical mean x‾20 = 2.09.
1) Give a confidence interval with level of confidence 90% for μ.
2) How big would the sample have to be for the interval to be half as long?
3) Let us now suppose the variance is not known. Knowing that counting from i=120Σ(xi - x‾20)² = 14.6, give a confidence interval with level of confidence 90% for the value of μ.
4) We now suppose we know mu is known and is equal to 2. Give a confidence interval with a confidence level of 90% for the value of σ².
5) Same question if we do not know the value of μ.