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.
0 $\frac{1}{4}$
0.5 $\frac{1}{2}$
1 $\frac{1}{4}$
(type integers or simplified fractions.)
does the mean of the sample proportions equal the proportion of girls in two births?
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 $\frac{1}{2}$.

Explanation:

Step1: Recall the formula for the mean of a discrete - probability distribution

The mean $\mu$ of a discrete - probability distribution is given by $\mu=\sum xP(x)$, where $x$ is the value of the random variable and $P(x)$ is the probability of that value.

Step2: Identify the values of $x$ and $P(x)$

We have three possible values for the proportion of girls in two births: $x_1 = 0$ (corresponding to the outcome $bb$), $x_2=0.5$ (corresponding to the outcomes $bg$ and $gb$), and $x_3 = 1$ (corresponding to the outcome $gg$). The probabilities are $P(x_1)=\frac{1}{4}$, $P(x_2)=\frac{2}{4}=\frac{1}{2}$, and $P(x_3)=\frac{1}{4}$.

Step3: Calculate the mean of the sample proportions

\[

$$\begin{align*} \mu&=\sum xP(x)\\ &=0\times\frac{1}{4}+0.5\times\frac{1}{2}+1\times\frac{1}{4}\\ &=0 + \frac{1}{4}+\frac{1}{4}\\ &=\frac{1}{2} \end{align*}$$

\]
The proportion of girls in two births in the population is also $\frac{1}{2}$ since the probability of having a girl in a single birth is $\frac{1}{2}$.

Answer:

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