Chapter 8.1, Problem 29E

a.

To determine

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

Answer to Problem 29E

The statement that gives the best conclusion of the hypotheses when $H_0$ is rejected is option (ii), “The scale is not in calibration.”

Explanation of Solution

The researcher interested to check the calibration of the scale by using standard weight of 10 grams 100 times. It is given that $\mu$ is the population mean reading on the scale.

The given test hypotheses are:

Null hypothesis

 $H_0:\mu =10$

That is, the scale is in calibration.

Alternate hypothesis

$H_1:\mu \ne 10$

That is, the scale is out of calibration.

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

Reasons for correct answer:

If the null hypothesis $H_0$ is rejected, It means that it is evident that the scale is out of calibration.

Hence, the statement that gives the best interpretation of the hypotheses when $H_0$ is rejected is option (ii), “The scale is not in calibration.”

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 not true.

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

Now, rejecting the null hypothesis $H_0$ means that it is not true. Thus, there is no chance that the scale might be in calibration. Hence, option (ii) 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 conclusion when $H_0$ is not rejected.

Answer to Problem 29E

The statement that gives the best interpretation of the hypotheses when $H_0$ is not rejected is option (iii), “The scale might be in calibration.”

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 scale might be in calibration.

Hence, the statement that gives the best interpretation of the hypotheses when $H_0$ is rejected is option (iii), “The scale might be in calibration.”

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 is may be true. Thus, there is chance for the scale to be in calibration. Hence, option (ii) is incorrect.

c.

To determine

Identify the type of error.

Answer to Problem 29E

The error committed is Type I error.

Explanation of Solution

It is assumed that the scale is in calibration, but the conclusion made on the part of surveyor is that the scale is not in calibration.

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 scale is not in calibration, when in reality, the scale might be in calibration.

Type II error:

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

If Type II error is committed, the surveyor concludes that the scale is in calibration, when in reality, the scale is not in calibration.

If further analysis reveals that in reality, the conclusion made is that the scale is not in calibration, then the surveyor will have committed the Type I error.

The error committed is Type I error by concluding evidence as “the scale is not in calibration”, but in actual “the scale is in calibration”.

d.

To determine

Decide whether it is possible to make Type I error when the scale is not in calibration.

Answer to Problem 29E

No, it is not possible to make Type I error when the scale is not in calibration.

Explanation of Solution

It is assumed that the scale is not in calibration.

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.

From the given information, it is known that the scale is not in calibration.

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 not correct, rejecting $H_0$ will not be the incorrect decision. As a result, committing a Type I error will also be impossible.

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

e.

To determine

Decide whether it is possible to make Type II error when the scale is not in calibration.

Answer to Problem 29E

Yes, it is possible to make Type II error when the scale is not in calibration.

Explanation of Solution

It is given that the scale is not in calibration.

Failure to reject $H_0$ when it is actually false leads to Type II error. Hence, if $H_0$ is not rejected even when it is false, the Type II error would be committed.

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