Question

An SRS of size $100$is taken from Population A with proportion $0.8$of successes. An independent SRS of size $400$is taken from Population B with proportion $0.5$of successes. The sampling distribution of the difference (A − B) in sample proportions has what mean and standard deviation?

a. mean$=0.3$; standard deviation $=1.3$

b. mean$=0.3$; standard deviation $=0.40$

c. mean$=0.3$; standard deviation $=0.047$

d. mean$=0.3$; standard deviation $=0.0022$

e. mean$=0.3$; standard deviation $=0.0002$

Short answer

Option (c) mean $=0.3$; standard deviation $=0.047$

Step-by-step solution

Given information

We need to find mean and standard deviation for the (A - B).

Simplify

As given in the question :

$$\begin{gathered} \widehat{{\mathrm{p}}_1}=0.8 \\ {\mathrm{n}}_1=100 \\ \widehat{{\mathrm{p}}_2}=0.5 \\ {\mathrm{n}}_2=400 \end{gathered}$$

Mean $\mathrm{\mu }=\widehat{{\mathrm{p}}_1}-\widehat{{\mathrm{p}}_2}=0.8-0.5=0.3$

Standard deviation :

$$\begin{gathered} \sigma =\sqrt{\frac{\widehat{p_1}(1-\widehat{p_1})}{n_1}+\frac{\widehat{p_2}(1-\widehat{p_2})}{n_2}} \\ =\sqrt{\frac{0.8(1-0.8)}{100}+\frac{0.5(1-0.5)}{400}} \\ =0.047 \end{gathered}$$

Therefore, option(c) is correct.