LOST AND CONFUSED

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? 

Determine the probabilities of each sample proportion

Sample proportion of girls probability
0          0                1            ?
0.5       0.25            0.5         ?
0.25     0.75            0.75       ?

Answer 1

Each of the 4 outcomes is equally likely, so each outcome has probability 1/4.

Since we have 2 births, we get the sampling proportion by dividing by 2.

Then, P(0) = P(0 girls) = P(bb) = 1/4 = .25

P(.5) = P(1 girl) = P(bg or gb) = 2/4 = 1/2 = .5

P(1) = P(2 girls) = P(gg) = 1/4 = .25

 

Then, the mean of the sample proportions is the sum of x P(x) = 0 * 1/4 + .5 * .5 + 1 * .25 = 0 + .25 + .25 = .5

 

The population proportion is .5 (this is what generates equally likely outcomes).

 

Thus, the sample proportion of .5 equals the population of .5, so it is unbiased.