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, 1 2' the probability of having a girl is and this is not affected by how many 2' assume the probability of having a boy is boys or girls have previously been born. Determine the probabilities of each sample proportion. Sample proportion of girls Probability (Type integers or simplified fractions.)

Answer 1

Given that 2 births are randomly selected. Each outcome bb, bg, gb, gg are equally likely.

a) Now we have to construct the sample proportion table. sample proportion = number of girls in selection 2 For bb, sample proportion = = 0. = ON bg, sample proportion 1 = - = 2 0.5. gb, sample proportion = 1 === 2 0.5. gg, sample proportion = = 1. probability sample proportion of girls 0 1/4 0.5 1/2 1 1/4

b) Now we have to find mean of the distribution. E(X) = Xxx* P(x) = 0 * P(0) + 0.5 * P(0.5) + 1 * P(1) 0**+0.5**+1* 1 4 0.5 Proportion of girls in 2 births = = 0.5 mean of the distribution = Proportion of girls in 2 births. sample proportion is an unbiased estimator of population proportion. :: Yes, sample proportion is an unbiased estimator of population proportion.

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