Question 4.31: At a large university, the mean age of the students is 22.3 ......

Let $X_{1},…, X_{64}$ be the ages of the 64 students in the sample. We wish to find $P(\overline{X} > 23)$. Now the population from which the sample was drawn has mean 𝜇 = 22.3 and variance $\sigma^{2} = 16$. The sample size is n = 64. It follows from the Central Limit Theorem (expression 4.33) that $\overline{X} ∼  N(22.3,\ 0.25)$. The z-score for 23 is

$\overline{X} ∼ N (μ, \frac{σ^2}{n})$          approximately          (4.33)

$z = \frac{23\ −\ 22.3}{\sqrt{0.25}} = 1.40$

From the $z$ table, the area to the right of 1.40 is 0.0808. Therefore $P(\overline{X} > 23) = 0.0808$. See Figure 4.18.