Statistic Problem

Question

The height of women ages 20-29 is normally distributed, with a mean of 64.2 inches. Assume σ = 2.5 inches. Are you more likely to randomly select 1 woman with a height less than 65.9 inches or are you more likely to select a sample of 18 women with a mean height less than 65.9 inches? Explain

Answer 1

Let's calculate the two probabilities.

First get a Z score for a single woman with height 65.9 inches

Z = (x - mean)//standard deviation

Z = (65.9 - 64.2)/2.5

Z = 0.68

P(Z < 0.68) = 0.7517

For a sample size of 18, to calculate the probability that the mean height is less than 65.9, the formula is similar. Except now the standard error is the population standard deviation divided by the square root of the sample size.

Z = (65.9 - 64.2)/(2.5/sqrt (18))

Z = 2.88

P(Z < 2.88) = .9980

The standard error decreases as the sample size increases, therefore increasing the probability.