When two births are randomly selected, the sample space for genders is bb, bg, g

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. 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 1/3. B. Yes, both the mean of the sample proportions and the population proportion are 1/4. C. Yes, both the mean of the sample proportions and the population proportion are 1/2. D. No, the mean of the sample proportions and the population proportion are not equal. Does the result suggest that a sample proportion is an unblased estimator of a population proportion? A. No, because the sample proportions and the population proportion are not the same. B. Yes, because the sample proportions and the population proportion are the same. C. No, because the sample proportions and the population proportion are the same. D. Yes, because the sample proportions and the population proportion are not the same.

Explanation / Answer

Sample proportion of girls Probability

0 (0.5 * 0.5) = 0.25 [ As the only possible choice is bb]

1/2 (0.5*0.5) + (0.5*0.5) = 0.5 [As the possible choices are bg and gb]

1 (0.5 * 0.5) = 0.25 [ As the only possible choice is gg]

The mean of the sample proportion of girls in two births = (0 * 0.25) + (0.5 * 0.5) + (1 * 0.25) = 0.25 + 0.25 = 0.5.

So the correct choice is: Option(C): Yes, both the sample proportion and the population proportion are 1/2. (Ans).

The result suggest sample proportion is an unbiased estimator of population proportion.So the correct choice is:

Option(B): Yes, because the sample proportions and population proportion are the same. (Ans).

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