I cant get the same answer twice.

Question

The height of women ages​ 20-29 is normally​ distributed, with a mean of

64.6

inches. Assume

σ=2.9

inches. Are you more likely to randomly select 1 woman with a height less than

66.6

inches or are you more likely to select a sample of

17

women with a mean height less than

66.6

​inches? Explain.

Answer 1

You have a normal distribution with mean 64.6 and standard deviation 2.9. Call the cumulative density function F(x). The first option (1 woman less than 66.6 inches) is answered by P = F(66.6). I get P = 0.7547945.

Option 2: probability of a random sample of 17 women having a mean height less than 66.6. For this, we need to know the standard error of the mean, which is 2.9 / sqrt(16) = 0.725. For this sample, the distribution is normal with mean 64.6 and standard deviation 0.725. So the likelihood here is P = F(66.6) = 0.9970977

So option 2 is much more likely. The simple way to think about this is that the sample mean (mean of n random samples of a distribution) is less dispersed than the parent distribution.