# Births: Sampling Distribution of Sample Proportion 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?

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Births: Sampling Distribution of Sample Proportion 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?

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Step 1

Given
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.
The two values are randomly selected with replacement from population {b,g}, four possible outcomes are
bb, bg, gb, and gg.

Case -1  bb
Proportion of girl births = 0/2 = 0
probability of bb = 1/4 = 0.25

Case - 2 bg
Proportion of girl births = 1/2 = 0.5
probability of bg = 1/4 = 0.25

Case - 3 gb
Proportion of girl births = 1/2 = 0.5
probability of gb = 1/4 = 0.25

Case - 4 gg
Proportion of girl births = 2/2 = 0.5
probability of gg = 1/4 = 0.25

The table representing the sample distribution of sample proportion of girls from two births is shown below

Step 2

The mean of sample proportions equal the proportion of girls in two births is
= Number of girls in two births / ...

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