Question 5.59

For Exercises 5.40 and 5.41, see page 269; and for 5.42 and 5.43, see page 271.

Question 5.59

5.59 How much less spread?

Refer to Example 5.24 in which we showed the sampling distributions of $\hat{p}$ for $n = 100$ and $n = 2500$ with $p = 0.60$ in both cases.

In terms of a multiple, how much larger is the standard deviation of the sampling distribution for $n = 100$ versus $n = 2500$ when $p = 0.60$ ?
Show that the multiple found in part (a) does not depend on the value of $p$ .

5.59

(a) 5 times as large. (b) $\begin{array}{l} \sqrt{\frac{p (1 - p)}{100}} / \sqrt{\frac{p (1 - p)}{2500}} \\ = \sqrt{\frac{p (1 - p)}{\sqrt{100}}} / \sqrt{\frac{p (1 - p)}{\sqrt{2500}}} \\ = \frac{1}{\sqrt{100}} / \frac{1}{\sqrt{2500}} = \frac{\sqrt{2500}}{\sqrt{100}} = \frac{50}{10} = 5 \end{array}$