Question

In Exercise 8.146 on page 345,we introduced one-sided one mean-t-intervals. The following relationship holds between hypothesis test and confidence intervals for one-mean t-procedures: For a right-tailed hypothesis test at the significance level $\alpha$, the null hypothesis $H_0:\mu ={\mu }_0$will be rejected in favor of the alternative hypothesis data-custom-editor="chemistry" $H_a:\mu >{\mu }_0$if and only if ${\mu }_0$is less than or equal to the $\left(1-\alpha \right)$-level lower confidence bound for $\mu$. In each case, illustrate the preceding relationship by obtaining the appropriate lower confidence bound and comparing the result to the conclusion of the hypothesis test in the specified exercise.

Part (a): Exercise 9.114 (both parts)

Part (b) Exercise9.115

Short answer

Part (a): Both conclusions are same.

This means that the conclusion for confidence interval is same as the conclusion for hypothesis test.

Part (b): Both conclusions are same.

This means, that the conclusion for confidence interval is same as the conclusion for hypothesis test.

Step-by-step solution

Part (a) Step 1. Given information.

Consider the given question,

The null hypothesis is$H_0:\mu ={\mu }_0$.

The alternative hypothesis is$H_a:\mu >{\mu }_0$.

Part (a) Step 2. Compute the confidence interval.

On using the MINITAB procedure,

1. Choose Stat>Basic Statistics>1-Sample t.
2. In Samples in Column, enter the column of Distance.
3. In Perform hypothesis test, enter the test mean as 180.
4. Check Options, enter Confidence level as 95.
5. Choose greater than in alternative.
6. Click OK in all dialogue boxes.

Test of $\mu =180vs>180$

Hence, from the MINITAB output, the 90% lower bound is 180.42.

Here, the population mean is 180, which lies below the 95% lower bound. Then, the null hypothesis is rejected at5%level.

Part (a) Step 3. Using the Hypothesis test.

Using the Hypothesis test

From Problem 9-144E, the P-value is 0.031. The P-value is lesser than the level of significance. Therefore, the null hypothesis is rejected.

It can be concluded that the club will hit the ball more than 180 years.

Hence, both conclusions are same. It means, that the conclusion for confidence interval is same as the conclusion for hypothesis test.

On computing the 99% confidence interval,

1. Choose Stat>Basic Statistics>1-Sample t.
2. In Samples in Column, enter the column of Distance.
3. In Perform hypothesis test, enter the test mean as 180.
4. Check Options, enter Confidence level as 99.
5. Choose greater than in alternative.
6. Click OK in all dialogue boxes.

Part (a) Step 4. Take one sample T.

Hence, from the MINITAB output, the 99% lower bound is 178.91.

Here, the population mean is 180, which lies above the 99% lower bound. Then, the null hypothesis is not rejected at 1% level.

Using the Hypothesis test,

From Problem 9-144E, the P-value is 0.031. The P-value is greater than the level of significance. Therefore, the null hypothesis is not rejected.

Thus, there is no evidence to infer that the club will hit the ball more than 180 years.

Hence, both conclusions are same. It means, that the conclusion for confidence interval is same as the conclusion for hypothesis test.

Part (b) Step 1. Compute the confidence interval.

On computing the 95% confidence interval,

1. Choose Stat>Basic Statistics>1-Sample t.
2. In Samples in Column, enter the column of Angle.
3. In Perform hypothesis test, enter the test mean as 45.
4. Check Options, enter Confidence level as 95.
5. Choose greater than in alternative.
6. Click OK in all dialogue boxes.

Hence, from the MINITAB output, the 95% lower bound is 45.12.

Here, the population mean is 45, which lies above the 95% lower bound. Then, the null hypothesis is rejected at 5% level.

Using the Hypothesis test,

From Problem 9-145E, the P-value is 0.045. The P-value is lesser than the level of significance. Therefore, the null hypothesis is rejected.

Thus, the dat provide sufficient evidence to conclude that on average the angle between the body and head of an alligator during a death roll is greater than${45}^{\circ }$.

Hence, both conclusions are same. It means, that the conclusion for confidence interval is same as the conclusion for hypothesis test.