Question

Decreasing the sample size from $750$ to $375$ would multiply the standard deviation by

a. 2.

b.$\sqrt{2}$

c. 1/2.

d. $\sqrt{\frac{1}{2}}$

e. none of these.

Short answer

The correct option is (b)$\sqrt{2}$

Step-by-step solution

Given information

Current sample size is $750$ and it becomes $375$

Concept

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

Calculation

The current sample size is $750$ and it will be reduced to $375$ implying that n sample will be reduced to $\frac{n}{2}$

$$\begin{gathered} {\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}} \\ =\sqrt{\frac{p(1-p)}{n/2}} \\ =\sqrt{\frac{2p(1-p)}{n}} \\ =\sqrt{2}{\sigma }_{\hat{p}} \end{gathered}$$

Standard deviation is multiplied by $\sqrt{2}$ Hence, the correct option is (b).