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 1/2, the probability of having a girl is 1/2, and this is not affected by how many boys or girls have previously been born. determine the probabilities of each sample proportion. sample proportion of girls probability (type integers or simplified fractions.)

Explanation:

Step1: Calculate sample proportion for each outcome

• For 'bb': The number of girls is 0. The sample proportion of girls $p=\frac{0}{2} = 0$.
• For 'bg' and 'gb': The number of girls is 1. The sample proportion of girls $p=\frac{1}{2}$.
• For 'gg': The number of girls is 2. The sample proportion of girls $p=\frac{2}{2}=1$.

Step2: Determine probabilities for each sample - proportion

Since the four outcomes 'bb', 'bg', 'gb', 'gg' are equally - likely, each has a probability of $\frac{1}{4}$.

• The probability that the sample proportion of girls is 0 (outcome 'bb') is $P(p = 0)=\frac{1}{4}$.
• The probability that the sample proportion of girls is $\frac{1}{2}$ (outcomes 'bg' and 'gb') is $P(p=\frac{1}{2})=\frac{2}{4}=\frac{1}{2}$ (because there are 2 favorable outcomes out of 4).
• The probability that the sample proportion of girls is 1 (outcome 'gg') is $P(p = 1)=\frac{1}{4}$.

Step3: Calculate the mean of the sample proportions

The mean of a discrete - probability distribution $\mu_p=\sum x\cdot P(x)$.
\[

$$\begin{align*} \mu_p&=0\times\frac{1}{4}+\frac{1}{2}\times\frac{1}{2}+1\times\frac{1}{4}\\ &=0 + \frac{1}{4}+\frac{1}{4}\\ &=\frac{1}{2} \end{align*}$$

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

Answer:

Sample proportion of girls	Probability
$\frac{1}{2}$	$\frac{1}{2}$
1	$\frac{1}{4}$

The mean of the sample proportions equals the proportion of girls in two births. The result suggests that a sample proportion is an unbiased estimator of a population proportion.