When two births are randomly selected, the sample space for genders is bb, b, 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
boys or girls have previously been born
2, the probability of having a girl is
2, and this is not affected by how many boys or girls have previously been born.
Determine the probabilities of each sample proportion.

Step 1 ：Given a sample space of four outcomes: bb, bg, gb, gg. These represent the possible combinations of genders for two births.

Step 2 ：The sample proportions of girls can be 0 (for bb), 0.5 (for bg and gb), and 1 (for gg).

Step 3 ：Since the outcomes are equally likely, the probabilities for these sample proportions can be calculated as the number of outcomes that result in that proportion divided by the total number of outcomes. Therefore, the probabilities of each sample proportion are all equal to 0.25.

Step 4 ：Next, calculate the mean of the sample proportions and compare it to the proportion of girls in two births. The mean of the sample proportions can be calculated as the sum of the product of each proportion and its corresponding probability. The proportion of girls in two births is the sum of the proportions of girls in each outcome divided by the total number of outcomes.

Step 5 ：The mean of the sample proportions is equal to the proportion of girls in two births, which is 0.5. This suggests that the sample proportion is an unbiased estimator of the population proportion, as it accurately represents the proportion of girls in the population.

Step 6 ：Finally, determine if the result suggests that a sample proportion is an unbiased estimator of a population proportion. An estimator is considered unbiased if its expected value is equal to the parameter it is estimating. In this case, the parameter is the proportion of girls in the population, and the estimator is the sample proportion. Since the mean of the sample proportions is equal to the proportion of girls in the population, this suggests that the sample proportion is an unbiased estimator of the population proportion.

Step 7 ：Final Answer: The probabilities of each sample proportion are all equal to \(\boxed{0.25}\). The mean of the sample proportions and the proportion of girls in two births are both equal to \(\boxed{0.5}\). This suggests that the sample proportion is an unbiased estimator of the population proportion.