Question

9. 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 ½ the probability of having a girl is ½, and this is not affected by how many boys or girls have previously been born.

Determine the probabilities of each sample proportion.

Sample proportion of girls	Probability
(0.25, 0, 0.5)
(0.75, 0.5, 0.25)
(1, 0.5, 0.75)

​(Type integers or simplified​ fractions.)

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 ¼.

B.No, the mean of the sample proportions and the population proportion are not equal.

C.Yes, both the mean of the sample proportions and the population proportion are ½.

​D.Yes, both the mean of the sample proportions and the population proportion are 1/3.

Does the result suggest that a sample proportion is an unbiased estimator of a population​ proportion?

A.No, because the sample proportions and the population proportion are the same.

B.Yes, because the sample proportions and the population proportion are not the same.

C.No, because the sample proportions and the population proportion are not the same.

D.Yes, because the sample proportions and the population proportion are the same.

Answer 1

the sample space for genders is​ bb, bg,​ gb, and gg.

Let X be the event that no.of girls in two births

at X = 0 out comes BB

P(X=0) = 0.25

at X= 1 out comes BG,GB

P(X=1) = 0.50

at X=2 outcomes GG

P(X=2) = 0.25

so the sampling distribution of sample proportion is p̂ = x/n

P(p̂) = 0.25 -------p̂ = 0/2 = 0

P(p̂) = 0.50 -------p̂ = 1/2 = 0.5

P(p̂) = 0.25 -------p̂ = 2/2 = 1

Mean of sample proportion = Σ p̂P(p̂) = 0.50

P( girls in two births) = 1-P(no girls in two births) = 1-0.25 = 0.75

Q : Does the mean of the sample proportions equal the proportion of girls in two​ births?

ANS: No, the mean of the sample proportions and the population proportion are not equal.

Q : Does the result suggest that a sample proportion is an unbiased estimator of a population​ proportion?

Ans : Yes, because the sample proportions and the population proportion are the same.