Question

when two births are randomly selected, the sample space for genders is bb, bg, gb, and gg. 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? for the entire population, assume the probability of having a boy is $\frac{1}{2}$, the probability of having a girl is $\frac{1}{2}$, and this is not affected by how many boys or girls have previously been born.

a. no, the mean of the sample proportions and the population proportion are not equal
b. yes, both the mean of the sample proportions and the population proportion are $\frac{1}{3}$
c. yes, both the mean of the sample proportions and the population proportion are $\frac{1}{4}$
d. yes, both the mean of the sample proportions and the population proportion are $\frac{1}{2}$

does the result suggest that a sample proportion is an unbiased estimator of a population proportion?
a. yes, because the sample proportions and the population proportion are not the same.
b. no, because the sample proportions and the population proportion are the same.
c. yes, because the sample proportions and the population proportion are the same.
d. no, because the sample proportions and the population proportion are not the same.

Explanation:

Step1: Define sample - proportion for each outcome

For outcome bb (2 boys, 0 girls), the sample proportion of girls $p_1 = 0$.
For outcome bg (1 boy, 1 girl), the sample proportion of girls $p_2=\frac{1}{2}$.
For outcome gb (1 boy, 1 girl), the sample proportion of girls $p_3=\frac{1}{2}$.
For outcome gg (0 boys, 2 girls), the sample proportion of girls $p_4 = 1$.

Step2: Calculate the mean of the sample - proportions

The mean of the sample - proportions $\mu_{\hat{p}}$ is given by $\mu_{\hat{p}}=\frac{p_1 + p_2 + p_3 + p_4}{4}$.
Substitute the values: $\mu_{\hat{p}}=\frac{0+\frac{1}{2}+\frac{1}{2}+1}{4}=\frac{2}{4}=\frac{1}{2}$.
The population proportion of girls in a single - birth is $\frac{1}{2}$, and for two births, the expected proportion of girls is also $\frac{1}{2}$.

Step3: Determine if it is an unbiased estimator

An estimator is unbiased if the mean of the sample statistics (in this case, sample proportions) is equal to the population parameter. Since the mean of the sample proportions $\frac{1}{2}$ is equal to the population proportion $\frac{1}{2}$, the sample proportion is an unbiased estimator of the population proportion.

Answer:

For the first part: D. Yes, both the mean of the sample proportions and the population proportion are $\frac{1}{2}$
For the second part: C. Yes, because the sample proportions and the population proportion are the same.