Q. 38 - The Practice of Statistics for AP

Question

Do you go to church? What sample size would be required to reduce the standard deviation of the sampling distribution to one-third the value you found in Exercise 36 (b)? Justify your answer.

Step-by-Step Solution

Answer

The sample size of$16,065$would be required to reduce the standard deviation of the sampling distribution to one-third the value.

Step 1: Given Information

Sample size $=1785$

Proportion of condition $=10\%$

Step 2: Explanation

We have,

$n=1785$

The standard deviation formula is

${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}$

Assuming the standard deviation should become one third of the current standard deviation:

$$\begin{gathered} \frac{\sigma \widehat{{}_p}}{3}=\frac{1}{3}\times \sqrt{\frac{p(1-p)}{n}} \\ =\sqrt{\frac{p(1-p)}{9n}} \end{gathered}$$

Multiplying the sample size by 9 will yield the following result:

$$\begin{gathered} 9n=9\left(1785\right) \\ n=16065 \end{gathered}$$

Hence, a new sample size of $16065$ is needed