Virginia is vegetarian, and she is the mother of four young daughters. She wants to investigate whether vegetarian mothers are more likely to have daughters than sons. She asks around her friends until she has counted 50 children of vegetarian women, comprising X = 40 daughters and 10 sons. She conducts the test X ∼ Binomial(n, p)

H0 : p = 0.5 H1 : p , 0.5, and calculates the P-value is 0.00002. Noting that many of Virginia’s friends are the mothers of her daughters’ friends, say whether each of the following statements is true or false.

Briefly explain each of your answers.

a. Because of the study design, the hypothesis p = 0.5 might be badly wrong even if vegetarianism has no impact on the sex of the offspring.

b. Despite the study design, it is valid to conclude that X is almost certainly not Binomial(50, 0.5).

c. Virginia’s test is invalid because of the study design. If she did a two-sample test instead, testing whether her vegetarian friends have the same daughter-probability as her non-vegetarian friends, it would be a lot more valid.

d. Because the sample size is quite small, the P-value won’t be very reliable so the observation X = 40 could be compatible with H0.