“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.