Question

The height of women ages​ 20-29 is normally​ distributed, with a mean of 64.8 inches. Assume sigmaequals2.5 inches. Are you more likely to randomly select 1 woman with a height less than 66.4 inches or are you more likely to select a sample of 16 women with a mean height less than 66.4 ​inches? Explain.

Answer 1

Let X be the random variable denoting the height of women ages 20-29.

Thus, X ~ N(64.8, 2.5) i.e. (X - 64.8)/2.5 ~ N(0,1).

P(X < 66.4) = P[(X - 64.8)/2.5 < (66.4 - 64.8)/2.5] = P[(X - 64.8)/2.5 < 0.64] = (0.64) = 0.7389. (Ans).

Let M be the sample mean height of 16 women.

Thus, E(M) = 64.8, s.d.(M) = 2.5/ = 0.6250.

The probability that the sample mean height of 16 women is less than 66.4 inches = P(M < 66.4) = P[(M - 64.8)/0.6250 < (66.4 - 64.8)/0.6250] = P[(M - 64.8)/0.6250 < 2.56] = (2.56) = 0.9948. (Ans).

So, the second possibility is more likely. (Ans).