Question

A research study investigated differences between male and female students. based on the study results, we can assume the population mean and standard deviation for the gpa of male students are µ = 3.5 and σ = 0.5. suppose a random sample of 100 male students is selected and the gpa for each student is calculated. what is the probability that the random sample of 100 male students has a mean gpa greater than 3.42?

Answer 1

The probability is 0.9452.

We first compute the z-score for this value:
$z=\frac{X-\mu}{\frac{\sigma}{n}}=\frac{3.42-3.5}{\frac{0.5}{\sqrt{100}}} \\ \\=\frac{-0.08}{\frac{0.5}{10}}=-0.08\div \frac{0.5}{10}=-0.08\times \frac{10}{0.5}=\frac{-0.8}{0.5}=-1.6$

Using a z-table (http://www.z-table.com) we see that the area to the left of this score, less than, is 0.0548.  We want the area greater than this, so we subtract from 1:
1-0.0548 = 0.9452