Question

Week 5 Questions

9.1

30.

IQ: Scores on a certain IQ test are known to have a mean of 100. A random sample of 60 students attend a series of coaching classes before taking the test. Let μ be the population mean IQ score that would occur if every student took the coaching classes. The classes are successful if μ > 100. A test is made of the hypotheses H0: μ = 100 versus H1: μ > 100. Consider three possible conclusions: (i) The classes are successful. (ii) The classes are not successful. (iii) The classes might not be successful. a. Which of the three conclusions is best if H0 is rejected? b. Which of the three conclusions is best if H0 is not rejected? c. Assume that the classes are successful but the conclusion is reached that the classes might not be successful. Which type of error is this? d. Assume that the classes are not successful. Is it possible to make a Type I error? Explain. e. Assume that the classes are not successful. Is it possible to make a Type II error? Explain.

Answer 1

Solution:

Given: The classes are successful if μ > 100.

A test is made of the hypotheses H₀: μ = 100 versus H₁: μ > 100.

Consider three possible conclusions:

(i) The classes are successful.

(ii) The classes are not successful.

(iii) The classes might not be successful.

Part a) Which of the three conclusions is best if H₀ is rejected?

If we reject H0, then we conclude that: mean μ > 100, that means i) The classes are successful.

Part b) Which of the three conclusions is best if H₀ is not rejected?

If H0 is not rejected that is we retain the null hypothesis H₀: μ = 100, thus  (iii) The classes might not be successful.

Part c)  Assume that the classes are successful but the conclusion is reached that the classes might not be successful. Which type of error is this?

the conclusion is reached that the classes might not be successful, that means we fail to reject H0 and actually the classes are successful , that means mean μ > 100, that is: null hypothesis is False.

Type II Error occurs when we fail to reject H0, when null hypothesis is False.

Thus type II error occurred in this situation.

Part d) Assume that the classes are not successful. Is it possible to make a Type I error? Explain.

Type I Error is Reject H0, in fact H0 is true.

the classes are not successful means H0 is true. So if we reject null hypothesis H0, then it is possible to make type I error.

Part e) Assume that the classes are not successful. Is it possible to make a Type II error? Explain.

the classes are not successful means H0 is true.

If reject H0, then we make type I error.

If we fail to reject H0, then we make correct decision, since H0 is true and we do not reject it.

Thus there is no possibility of Type II error