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 one half ​, the probability of having a girl is one half ​, and this is not affected by how many boys or girls have previously been born.

Answer 1

solution:

probability of havin a boy = probability of having a girl = 1/2 = 0.5

also it is given that the four outcomes are equally likely

the sample proportion is the number of girls devided by the number of children in the sample:

the following is the table that describes the sample proportion of girls in the two births

sample	proportion of girls	probability
bb	0.0	= 1/2*1/2 = 1/4
bg	= 1/2 = 0.5	= 1/2*1/2 = 1/4
gb	= 1/2 = 0.5	= 1/2*1/2 = 1/4
gg	= 2/2 = 1	= 1/2*1/2 = 1/4

now the below table describe the sampling distribution for the number of girls in two births

proportion of girs(X)	P(X)
0	= 1/4 = 0.25
0.5	= 2/4 = 0.5
1	= 1/4 = 0.25

the population proportion is 0.5, because the probability of having a boy = girl = 0.5,so we assume the number of boys and girls in the population are equal.

mean number of the sampling distribution of the sample proportion is

E(X) = (0*0.25) + (0.5*0.5) + (1*0.25) =0.5

thus we note the population proportion and mean of the sampling distribution of the sample proportion are equal.