Question

A Three students independently collect different samples of size 20 to test a null hypothesis that the average number of eggs in a fairy wren's nest is 2. In reality, the null hypothesis is true, but the students do not know this. Each student sets the level of significance of his/her hypothesis test at 0.05.

What is the probability that at least one student will commit a Type I error?

a. 0.86

b. 0.05

c. 0.95

d. 0.14

B In a two-tailed hypothesis test of means, if the null mean is false, and the distance between the true mean and the null mean increases,

a. The power of the test decreases

b. The probability of making a Type 2 error decreases

c. The probability that a sample mean will fall in the retention region increases

d. The probability of making a Type I error decreases

e. (a) and (c)

C Three students independently collect different samples of size 20 to test a null hypothesis that the average number of eggs in a fairy wren's nest is 2. In reality, the null hypothesis is true, but the students do not know this. Each student sets the level of significance of his/her hypothesis test at 0.05.

What is the probability that at least one student will commit a Type I error?

a. 0.86

b. 0.05

c. 0.95

d. 0.14

Answer 1

A.

The probability that at least one student will commit a Type I error=

= 1 - Probability that no student will commit a Type I error

Probability of a student committing a Type I error = level of significance = 0.05

Probability of a student not committing a Type I error = 1 - 0.05 = 0.95

Probability that no student out of the 3 will commit a Type I error = 0.95³ = 0.86

The probability that at least one student will commit a Type I error=

= 1 - Probability that no student will commit a Type I error = 1 -0.86 = 0.14

Correct option : d. 0.14

B.

Probability of Type II error = Prob( Accepting null hypothesis / Alternative hypothesis is true)

We are given that Null hypothesis is false which means Alternative hypothesis is true.

Now, as the distance between the true mean and the null mean increases , we tend to reject null hypothesis more often. Hence, according to definition of Type II error , this will decrease the probability.

Hence, probability of Type II error decreases.

Correct option :  b. The probability of making a Type 2 error decreases

C. (This part is same as part A)

The probability that at least one student will commit a Type I error=

= 1 - Probability that no student will commit a Type I error

Probability of a student committing a Type I error = level of significance = 0.05

Probability of a student not committing a Type I error = 1 - 0.05 = 0.95

Probability that no student out of the 3 will commit a Type I error = 0.95³ = 0.86

The probability that at least one student will commit a Type I error=

= 1 - Probability that no student will commit a Type I error = 1 -0.86 = 0.14

Correct option : d. 0.14