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.
determine the probabilities of each sample proportion.

Explanation:

Step1: Calculate sample proportion for each outcome

• For \(bb\): Number of girls \(= 0\). Sample proportion \(p=\frac{0}{2}=0\)
• For \(bg\): Number of girls \(= 1\). Sample proportion \(p = \frac{1}{2}=0.5\)
• For \(gb\): Number of girls \(= 1\). Sample proportion \(p=\frac{1}{2}=0.5\)
• For \(gg\): Number of girls \(= 2\). Sample proportion \(p=\frac{2}{2}=1\)

Step2: Determine probabilities

Since all four outcomes (\(bb\), \(bg\), \(gb\), \(gg\)) are equally likely, each has a probability of \(\frac{1}{4}\)

• Proportion \(0\): Only from \(bb\). Probability \(P(0)=\frac{1}{4}\)
• Proportion \(0.5\): From \(bg\) and \(gb\). Probability \(P(0.5)=\frac{2}{4}=\frac{1}{2}\)
• Proportion \(1\): Only from \(gg\). Probability \(P(1)=\frac{1}{4}\)

Step3: Calculate mean of sample proportions

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The proportion of girls in two - births (population proportion) is also \(0.5\) (since \(P(\text{girl})=\frac{1}{2}\) for each birth and for two births, expected number of girls \(=2\times\frac{1}{2} = 1\), proportion \(=\frac{1}{2}\))

Answer:

Sample proportion of girls	Probability
\(0.5\)	\(\frac{1}{2}\)
\(1\)	\(\frac{1}{4}\)

The mean of the sample proportions (\(0.5\)) equals the proportion of girls in two births (\(0.5\)). This result suggests that the sample proportion is an unbiased estimator of the population proportion.