Question

when two births are randomly selected, the sample space for genders is bb, bg, gb, and gg. assume that these 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.

determine the probabilities of each sample proportion.

does the mean of the sample proportions equal the proportion of girls in two births?

a. yes, both the mean of the sample proportions and the population proportion are \\( \frac { 1 } { 4 } \\)

b. yes, both the mean of the sample proportions and the population proportion are \\( \frac { 1 } { 2 } \\)

c. yes, both the mean of the sample proportions and the population proportion are \\( \frac { 1 } { 3 } \\)

d. no, the mean of the sample proportions and the population proportion are not equal

Explanation:

Step1: Calculate the mean of the sample proportions

The formula for the mean of a discrete probability distribution is $\mu=\sum xP(x)$.
For $x = 0$, $P(x)=\frac{1}{4}$; for $x = 0.5$, $P(x)=\frac{1}{2}$; for $x = 1$, $P(x)=\frac{1}{4}$.
So, $\mu=(0\times\frac{1}{4})+(0.5\times\frac{1}{2})+(1\times\frac{1}{4})$.

Step2: Simplify the expression

First, $0\times\frac{1}{4} = 0$.
Second, $0.5\times\frac{1}{2}=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$.
Third, $1\times\frac{1}{4}=\frac{1}{4}$.
Then, $\mu=0+\frac{1}{4}+\frac{1}{4}=\frac{1 + 1}{4}=\frac{1}{2}$.
The proportion of girls in two births (population proportion) is also $\frac{1}{2}$ (since for each birth, $P(\text{girl})=\frac{1}{2}$, and for two - independent births, using the formula for the expected value of a binomial distribution $np$ where $n = 2$ and $p=\frac{1}{2}$, $np=2\times\frac{1}{2} = 1$ girl on average, and the proportion is $\frac{1}{2}$).

Answer:

B. Yes, both the mean of the sample proportions and the population proportion are $\frac{1}{2}$