Question

Suppose that we want to see if the probability of being left-handed is different for men than it is for women. And suppose that 475 students in this class each test this hypothesis at the 2% significance level using their own personalized class data set.

a.) If the probability of being left-handed is in fact the same for men as it is for women, how many students (on average) would reject the null hypothesis, and falsely conclude that the probability of being left-handed is different for men than for women?
(Assume that each student does the analysis correctly.)
(A) 466
(B) 10
(C) 451
(D) 0
(E) Impossible to tell because the probability of Type II error is unknown.
(F) 470
(G) 5
(H) 24
(I) Impossible to tell because the probability of Type I error is unknown.
b.) If the probability of being left-handed is in fact different for men than for women, how many students (on average) would fail to reject the null hypothesis?
(Assume that each student does the analysis correctly.)
(A) 466
(B) Impossible to tell because the probability of Type II error is unknown.
(C) 0
(D) 470
(E) 24
(F) 5
(G) 10
(H) 451
(I) Impossible to tell because the probability of Type I error is unknown.

Answer 1

On average, 10 students would falsely conclude that the probability of being left-handed is different for men and women. On average, 451 students would fail to reject the null hypothesis if the probabilities are different.

Step-by-step explanation:

To determine the number of students who would falsely conclude that the probability of being left-handed is different for men and women, we need to understand the concept of Type I error. Type I error occurs when the null hypothesis is rejected even though it is true. In this case, if the probability of being left-handed is indeed the same for men and women, the significance level of 2% means that there is a 2% chance of committing a Type I error. Therefore, on average, 10 students would falsely conclude that the probability of being left-handed is different for men and women.

If the probability of being left-handed is different for men and women, the students would be performing a hypothesis test correctly. In this case, failing to reject the null hypothesis would mean that they accept the null hypothesis that the probabilities are the same. Therefore, on average, 451 students would fail to reject the null hypothesis.

Answer 2

a. On average, approximately 10 students would falsely conclude that the probability of being left-handed is different for men and women.

The answer is (G) 10.

b. The answer is (B) Impossible to tell because the probability of Type II error is unknown.

How to determine the number of students

a) If the probability of being left-handed is the same for men and women, and each student correctly performs the analysis, the number of students who falsely conclude that the probability is different for men and women corresponds to the significance level of the test.

In this case, the significance level is 2%.

The significance level represents the probability of making a Type I error, which is rejecting the null hypothesis when it is true.

Therefore, out of 475 students, on average, we would expect 2% of them to falsely conclude that the probability of being left-handed is different for men and women.

Calculating 2% of 475:

0.02 * 475 = 9.5

Since we can't have a fractional number of students, we round to the nearest whole number.

Therefore, on average, approximately 10 students would falsely conclude that the probability of being left-handed is different for men and women.

The answer is (G) 10.

b) If the probability of being left-handed is different for men and women, and each student correctly performs the analysis, the number of students who fail to reject the null hypothesis depends on the power of the test, which is related to the Type II error rate.

However, the information provided does not specify the power of the test or the probability of Type II error.

Therefore, we cannot determine the number of students who would fail to reject the null hypothesis.

The answer is (B) Impossible to tell because the probability of Type II error is unknown.