Question

A researcher initially plans to take an SRS of size 160 from a certain population and calculate the sample mean ${\mathrm{x}}^{-}\overline{x}$. Later, the researcher decides to increase the sample size so that the standard deviation of the sampling distribution of $x^{-}\overline{x}$will be half as big as when using a sample size of 160 . What sample size should the researcher use?

a. 40

b. 80

c. 320

d. 640

e. There is not enough information to determine the sample size.

Short answer

(d). The sample size should the researcher is $640$

Step-by-step solution

Given Information

Initially, a researcher intends to collect an SRS of size 160 from a specific demographic.

Explanation for correct option

The initial sample size was 160 people.

${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$

to reduce the standard deviation by half

$\frac{{\sigma }_{\overline{x}}}{2}=\frac{1}{2}\times \frac{\sigma }{\sqrt{n}}=\frac{\sigma }{2\sqrt{n}}=\frac{\sigma }{\sqrt{2^{2}n}}=\frac{\sigma }{\sqrt{4n}}$

In the equation of the standard deviation, the sample size n is replaced with 4 n, implying that the sample quadruples in size. n was the initial sample size. The sample with a standard deviation half as large

$4n=4\left(160\right)=640$

Hence, the correct option is (d)

Explanation for incorrect option

a. 40 is not the answer.

b. 80 is not the answer.

c. 320 is not the answer.

e. There is not enough information to determine the sample size is not the answer.