Using the Central Limit Theorem · Exercise 36

X ~ N(60, 9). Suppose that you form random samples of 25 from this distribution. Let $\bar{X}$ be the random variable of averages. Let ΣX be the random variable of sums. For parts c through f, sketch the graph, shade the region, label and scale the horizontal axis for $\bar{X}$, and find the probability.

(a) Sketch the distributions of X and $\bar{X}$ on the same graph.
(b) $\bar{X}$ ~ _____(_____,_____)
(c) P($\bar{x}$ < 60) = _____
(d) Find the 30^th percentile for the mean.
(e) P(56 < $\bar{x}$ < 62) = _____
(f) P(18 < $\bar{x}$ < 58) = _____
(g) Σx ~ _____(_____,_____)
(h) Find the minimum value for the upper quartile for the sum.
(i) P(1,400 < Σx < 1,550) = _____

This is an open-ended exercise: its source gives no single answer.

OpenStax, Introductory Statistics 2e · §7.3 Using the Central Limit Theorem · Exercise 36