Births: Sampling Distribution of Sample Proportion When two births are randomly selected, the sample space for genders is bb, bg, gb, and gg (where b = boy and g = girl). Assume that those four outcomes are equally likely. Construct a table that describes the sampling distribution of the sample proportion of girls from two births. Does the mean of the sample proportions equal the proportion of girls in two births? Does the result suggest that a sample proportion is an unbiased estimator of a population proportion?

The sample space for the gender of two births is provided. The four outcomes of the sample space are equally likely.

All possible samples of size 2 given in the sample space are:

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

$\hat{p}=\frac{\text{Number}\text{of}\text{girls}}{2}$

The following table shows all possible samples of size equal to 2 and the corresponding sample proportions:

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

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

The population proportion of girls in two births is computed below:

$$\begin{gathered} p=\frac{\text{Number}\text{of}\text{Girls}}{\text{Number}\text{of}\text{Births}} \\ =\frac{1}{2} \\ =0.5 \end{gathered}$$

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

The mean of the sample proportions is computed below:

$$\begin{gathered} Meanof\hat{p}=\frac{{\hat{p}}_1+{\hat{p}}_2+{\hat{p}}_3+{\hat{p}}_4}{4} \\ =\frac{0+0.5+0.5+1}{4} \\ =0.5 \end{gathered}$$

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).

An unbiased estimator is a sample statistic whose sampling distribution has a mean value equal to the population parameter.

The mean value of the sampling distribution of the sample proportion (0.5) is equal to the population proportion (0.5).

Thus, the sample proportioncan be considered as an unbiased estimator of the population proportion.