Question: please answer the following problem. 1. (10 pt) Let Y, Y,,... Yo denote a random sample of size 10 taken from a normal distribution with

please answer the following problem.

1. (10 pt) Let Y, Y,,... Yo denote a random sample of size 10 taken from a normal distribution with mean 2 and variance o. o is unknown. a. (3) Show that U1= Y1 - 2 follows the standard normal distribution using the method of moment-generating-function. b. (1) According to part (a), what is the distribution of U 2 Y1 - 2 2 c. (2) Knowing that U = 10 Yi - 2 has a chi-square distribution with 10 degrees of freedom, show i=1 that U is a pivotal quantity for o. d. (4) Use U, given in part (c), as a pivotal quantity to find a one-sided 90% confidence interval with an upper bound for o

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