find the possible samples, proportion of males in each sample, mean of sampling distribution

Question

A genetics experiment involves a population of fruit flies consisting of 3 males named Barry, Carlos, and Denny and 1 female named Erin. Assume that two fruit flies are randomly selected with replacement.  A) After listing the possible samples and finding the proportion of males in each sample, use a table to describe the sampling distribution of the proportion.

Proportion of males        Probability

             0

            .5

             1

 

B) Find the mean of Sampling distribution. C) Is the mean of Sampling distribution [from (b)] equal to the population proportion of males? If so, does the mean of sampling distribution of proportions always equal the population proportion?           

This is the problem.

Answer 1

 “With replacement” means that the same fly can be selected twice.

Possible outcomes: FF, FM, MF, MM

Corresponding probabilities: ¼², ¼×¾, ¾×¼,¾²

That is: 1/16, 3/16, 3/16, 9/16

A)

P(no males)=1/16=0.0625 (probability of female selected twice when p=0.25)

P(0.5 proportion is 0.5 of 2=1 male)=3/16+3/16=3/8=0.375

P(both males)=9/16=0.5625

B)

In the sample, we have proportions 0, 0.5 and 1 corresponding to 0, 1, 2 males selected with frequency 1, 2 and 1, and probabilities 0.0625, 0.375 and 0.5625.

Mean proportion=(0+0.5+1)/3=0.5, corresponding to 1 male.

C)

Population proportion of males=3/4=0.75, which is not the same as B, so we conclude that the sampling distribution of proportions generally doesn’t equal the population proportion.