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(\bar{X}>23) .Now the population from which the sample was drawn has mean μ = 22.3 and variance σ² = 16. The sample size is n = 64. It follows from the Central Limit Theorem (expression 4.33) that \bar{X}\sim N(22.3,0.25). The z-score for 23 is

\bar{X}\sim N\left(\mu, \frac{\sigma^2}{n}\right) \quad \text{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(\bar{X}>23) = 0.0808. See Figure 4.18.