Question

Births: Sampling Distribution of Sample Proportion For three births, assume that the genders are equally likely. Construct a table that describes the sampling distribution of the sample proportion of girls from three births. Does the mean of the sample proportions equal the proportion of girls in three births? (Hint: See Exercise 15 for two births.)

Short answer

The following table describes the sampling distribution of the sample proportions of girls in three births.

Sample proportion	Probability
0	$\frac{1}{8}$
0.33	$\frac{3}{8}$
0.67	$\frac{3}{8}$
1	$\frac{1}{8}$

The population proportion of girls (0.5) is equal to the mean of the sample proportions of girls (0.5).

Step-by-step solution

Given information

The genders of three births are considered equally likely.

Sampling distribution of sample proportions

All possible samples for the genders of the births of size equal to 3 are given as follows:

bbb

bbg

bgb

gbb

bgg

gbg

ggb

ggg

The sample proportion of girls for each sample has the following formula:

$\hat{p}=\frac{\text{Number}\text{of}\text{girls}}{3}$

The following table shows all possible samples of size equal to 3 and the corresponding sample proportions:

Sample	Sample Proportion of Girls
bbb	$$\begin{gathered} {\hat{p}}_1=\frac{0}{3} \\ =0 \end{gathered}$$
bbg	$$\begin{gathered} {\hat{p}}_2=\frac{1}{3} \\ =0.33 \end{gathered}$$
bgb	$$\begin{gathered} {\hat{p}}_3=\frac{1}{3} \\ =0.33 \end{gathered}$$
gbb	$$\begin{gathered} {\hat{p}}_4=\frac{1}{3} \\ =0.33 \end{gathered}$$
bgg	$$\begin{gathered} {\hat{p}}_5=\frac{2}{3} \\ =0.67 \end{gathered}$$$$\begin{gathered} {\hat{p}}_5=\frac{2}{3} \\ =0.67 \end{gathered}$$
gbg	$$\begin{gathered} {\hat{p}}_6=\frac{2}{3} \\ =0.67 \end{gathered}$$
ggb	$$\begin{gathered} {\hat{p}}_7=\frac{2}{3} \\ =0.67 \end{gathered}$$
ggg	$$\begin{gathered} {\hat{p}}_8=\frac{3}{3} \\ =1 \end{gathered}$$

Combining the values of proportions that are the same, the following probability values are obtained:

Sample proportion	probability
0	$\frac{1}{8}$
0.33	$\frac{3}{8}$
0.67	$\frac{3}{8}$
1	$\frac{1}{8}$

Population proportion and mean of the sample proportions

The population can be described as {b,g}.

The population proportion of girls is computed below:

$$\begin{gathered} p=\frac{1}{2} \\ =0.5 \end{gathered}$$

Thus, the population proportion of girls in three births is equal to 0.5.

The mean of the sample proportions is computed below:

$$\begin{gathered} \text{Mean}\text{of}\hat{p}=\frac{{\hat{p}}_1+{\hat{p}}_2+.....+{\hat{p}}_8}{8} \\ =\frac{0+0.33+........+1}{8} \\ =0.5 \end{gathered}$$

Thus, the mean of the sampling distribution of the sample proportion is equal to 0.5.

Here, the population proportion (0.5) is equal to the mean of the sample proportions (0.5).