Q.6 - The Practice of Statistics for AP

Question

 A researcher initially plans to take an SRS of size n from a population that has a mean of 80 and a standard deviation of $20$. If he were to double his sample size (to $2n$), the standard deviation of the sampling distribution of the sample mean would be multiplied by

(a) $2$.

(b)$\frac{1}{\sqrt{2}}$ . 

(c) $2$.

(d) $\frac{1}{2}$.

(e) $\frac{1}{\sqrt{2n}}$

Step-by-Step Solution

Answer

The correct answer is (b)

Step 1 Given Information

From the given data, we have,

$$\begin{gathered} {\mu }_x=\mu \\ =80 \\ {\sigma }_{-x}=\frac{\sigma }{\sqrt{n}} \\ =20 \end{gathered}$$

Step 2 Explanation

Given the sample size is doubled. So $n$ becomes $2 n$.

New Standard deviation of the sample distribution is,

$$\begin{gathered} {\sigma }_x=\frac{\sigma }{\sqrt{2n}} \\ =\frac{1}{\sqrt{2}}\frac{\sigma }{\sqrt{n}} \end{gathered}$$

Hence the answer is (b)