Question 15.2: Refer again to Example 22 in Chapter 1 (see also Example 1 i......
Refer again to Example 22 in Chapter 1 (see also Example 1 in this chapter), and construct a confidence interval for F(3) with approximately 95% confidence coefficient, where F is the d.f. of the r.v.’s describing the GPA scores.
In this example, n = 34 and the number of the observations which are ≤ 3 are 18 (the following; 2.36, 2.36, 2.66, 2.68, 2.48, 2.46, 2.63, 2.44, 2.13, 2.41, 2.55, 2.80, 2.79, 2.89, 2.91, 2.75, 2.73, and 3.00). Then
F_{34}(3)={\frac{18}{34}}={\frac{9}{17}}\simeq0.529,\qquad\sqrt{{\frac{F_{34}(3)[1-F_{34}(3)]}{34}}}\simeq0.086,
and therefore the required (observed) confidence interval is:
[0.529-1.96\times0.086,\,0.529+1.96\times0.086]\simeq[0.360,0.698].
REMARK 1 It should be pointed out that the confidence interval given by (8) is of limited usefulness, because the value of (the large enough) nfor which (8) holds depends on x.