Question

If a random sample of size 81 is drawn from this

population, what is the probability that the mean

of this sample is larger than 53.5 inches?

Answer 1

Let $X=$ the sample mean: $X\sim N(\mu, \sigma^2/n).$

Given $\mu=54, \sigma=2.5, n=81.$

$$P(X>53.5)=1-P(Z\le\dfrac{53.5-54}{2.5/\sqrt{81}})$$

$$=1-P(Z\le-1.8)\approx0.96407$$

$96.407\%$