Question

More SkittlesWhat 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.

Short answer

Sample size is$120$

Step-by-step solution

Given Information

It is given that sample size$n=30$

Calculation

The standard deviation of sampling distribution for sample proportion is

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

We need to reduce it to half of the value.

$\frac{{\sigma }_{\hat{p}}}{2}=\frac{1}{2}\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{1}{2}·\frac{p(1-p)}{n}}=\sqrt{\frac{p(1-p)}{4n}}$

When it is reduced to half, $n$is replaced by $4n$.

It means sample size need to be multiplied by $4$.

Hence, $4n=4\left(30\right)=120$

Required sample size is$120$