Q. 37 - The Practice of Statistics for AP

Question

Do you drink cereal milk? What sample size would be required to reduce the standard deviation of the sampling distribution to one-half the value you found in Exercise 35(b)? Justify your answer.

Step-by-Step Solution

Answer

Sample size would be required to reduce the standard deviation of the sampling distribution to one-half the value you found in Exercise 35(b)  is $4048.$

Step 1: Concept Introduction

The sampling distribution's mean difference indicates how much the sample statistic changes from sample to sample.

By a factor of $\mathrm{n}$, it is smaller than the population standard deviation.

Individual observations are more variable than averages.

Step 2: Explanation

Given:

$n=1012$

The standard deviation formula is:

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

In order to reduce the standard deviation to half of its existing value, then: 

$$\begin{gathered} \frac{{\sigma }_{\widehat{p}}}{2}=\frac{1}{2}\times \sqrt{\frac{p(1-p)}{n}} \\ =\sqrt{\frac{p(1-p)}{4n}} \end{gathered}$$

Let quadrupling the sample size:

$$\begin{gathered} 4n=4\left(1012\right) \\ n=4048 \end{gathered}$$

Hence, the required sample size is $4048.$