Question 4.31: At a large university, the mean age of the students is 22.3 ......
At a large university, the mean age of the students is 22.3 years, and the standard deviation is 4 years. A random sample of 64 students is drawn. What is the probability that the average age of these students is greater than 23 years?
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.
