Chapter 8.1, Problem 30E

a.

To determine

Identify the best among the three given conclusions when $H_0$ is rejected.

Answer to Problem 30E

The statement that gives the best conclusion of the hypotheses when $H_0$ is rejected is option (i), “The classes are successful.”

Explanation of Solution

A random sample 60 students attended the series of coaching class and had the mean IQ score of 100. It is given that $\mu$ be the population mean IQ score.

The given test hypothesis is:

Null hypothesis

 $H_0:\mu =100$

That is, the classes are not successful.

Alternate hypothesis

$H_1:\mu >100$

That is, the classes are successful.

Three statements regarding the conclusion of the hypothesis for the given study are given.

Reasons for correct answer:

If $H_0$ is rejected, It means that it is evident that the classes are successful.

Hence, the statement that gives the best interpretation of the hypotheses is option (i), “The classes are successful.”

Reasons for incorrect answers:

If the null hypothesis $H_0$ is rejected, then conclude that the alternate $H_1$ is true or null hypothesis $H_0$ is false.

Here, the surveyor rejects the null hypothesis $H_0$. Hence, option (ii) is incorrect.

Moreover, the surveyor rejects the null hypothesis $H_0$. As a result, it is appropriate to reject the null hypothesis $H_0$. Hence, option (ii) is incorrect.

Now, rejecting the null hypothesis $H_0$ means that it is not true. Thus, there is no chance that the classes might be successful. Hence, option (iii) is incorrect.

Statistics Concept Introduction

Conclusion of the hypotheses test:

• If the null hypothesis $H_0$ is rejected, then conclude that the alternate $H_1$ is true or null hypothesis $H_0$ is false.
• If the null hypothesis $H_0$ is not rejected, then conclude that the null hypothesis $H_0$ might be true.

b.

To determine

Identify the best among the three given conclusions when $H_0$ is not rejected.

Answer to Problem 30E

The statement that gives the best conclusion of the hypotheses when $H_0$ is not rejected is option (iii), “The classes might not be successful”

Explanation of Solution

Three statements regarding the interpretation of the hypothesis for the given study are given.

Reasons for correct answer:

If $H_0$ is not rejected, It means that it is evident that the classes might not be successful.

Hence, the statement that gives the best interpretation of the hypotheses is option (iii), “The classes might not be successful.”

Reasons for incorrect answers:

If the null hypothesis $H_0$ is not rejected, then conclude that the alternate $H_1$ is not true or null hypothesis $H_0$ is might be true.

Here, the surveyor failed to reject the null hypothesis $H_0$. However, failure to reject $H_0$ does not necessarily mean that $H_0$ is true. It simply implies that $H_0$ might be true. Hence, option (i) is incorrect.

Now, failing to reject the null hypothesis $H_0$ means that it may be true. Thus, there is chance for the classes might not be successful. Hence, option (ii) is incorrect.

c.

To determine

Identify the type of error.

Answer to Problem 30E

The error committed is type II error.

Explanation of Solution

It is assumed that the classes are successful, but the conclusion made on the part of surveyor is that the classes might not be successful.

Type I error:

Type I error is defined as the probability of rejecting the null hypothesis, when it is actually true.

If Type I error is committed, the surveyor concludes that the classes are not successful, when in reality, the classes are successful.

Type II error:

Type II error is defined as the probability of failure to reject the null hypothesis when alternate hypothesis is actually true.

If Type II error is committed, the surveyor concludes that the classes are successful, when in reality, the classes might not be successful.

If further analysis reveals that in reality, the conclusion made is that the classes might not be successful, then the surveyor will have committed the Type I error.

The error committed is Type I error by concluding that “classes might not be successful”, when in actual “the classes are successful”.

d.

To determine

Decide whether it is possible to make Type I error when the classes are not successful.

Answer to Problem 30E

Yes, it is possible to make Type I error.

Explanation of Solution

It is assumed that the classes are not successful.

Possible decisions made with a hypothesis test:

Correct decision:

• Rejecting $H_0$ when it is false.
• Failing to reject $H_0$ when it might be true.

Wrong decision:

• Rejecting $H_0$ when it might be true leads to Type I error.
• Failing to reject $H_0$ when it is false leads Type II error.

Now, rejecting $H_0$ when it might actually be false leads to Type I error. However when it is already known that $H_0$ is correct, rejecting $H_0$ will be the incorrect decision. As a result, committing a Type I error will also be possible.

Thus, it is possible to make Type I error in this context.

e.

To determine

Decide whether it is possible to make Type II error when the classes are not successful.

Answer to Problem 30E

No, it is not possible to make Type II error.

Explanation of Solution

It is assumed that the classes are not successful.

Now, failure to reject $H_0$ when it might actually be false leads to Type II error. However when it is already known that $H_0$ is correct, rejecting $H_0$ will be the incorrect decision. As a result, committing a Type II error will also be impossible.

Thus, it is not possible to make Type II error in this context.