Question

A randomized controlled experiment has 50 participants, of whom 20 are women. A simple random sample of 25 participants are assigned to the treatment group and the remainder to the control group.

a) Say whether the following statement is true or false. If it is true, provide a math expression for the chance. If it is false, provide math expressions for the two chances.

?(the treatment group has 16 women) = ?(the control group has 16 women)P(the treatment group has 16 women) = P(the control group has 16 women)

b) Say whether the following statement is true or false, and justify your answer.

The event "the treatment group has 16 women" is independent of the event "the control group has 16 women".

c) Write a math expression for the chance that there are at least six women in both groups.

[Hint: This can be done by just thinking about the women in one of the groups. Any other way will prove quite a bit harder.]

Answer 1

The solution is given below.

Part a. We show that there is a bijection between the number of situations in which there are 16 women in the control group and 16 in the treatment group. This gives that the two probabilities are equal. Then we calculate the two probabilities by elementary methods.

Part b. We show that the two events are not independent by showing that their intersection has zero probability but neither of them has zero probability.

Part c. The explanation for calculating the favourable outcomes is given in the solution. This gives the probability.

Hope this helps.