In statistics, we use sample quantities to infer population quantities. For example, we use the sample mean to estimate the population mean, and further use the sampling distribution of the sample mean to construct confidence intervals for the population mean.

(a) Is it true that the sample mean will become a better estimate of the population mean as the sample size grows? Use the central limit theorem to answer this question. [Hint: consider the sampling distribution of the sample mean]

(b) ) Let x1, . . . , xn denote a sample of size n(n > 100) drawn from an unknown distribution with mean ( > 0) and variance 2 . Let x denote the sample mean; that is, x = n 1 (x1 + x2 + + xn).

Show that P (x > 1.3) < 1