Q.9.129 - Elementary Statistics

Question

Left-Tailed Hypothesis Tests and CIs. In Exercise $8.146$ on page 345 , we introduced one-sided one-mean t-intervals. The following relationship holds between hypothesis tests and confidence intervals for one-mean t-procedures: For a left-tailed hypothesis test at the significance level α, the null hypothesis H0:μ=μ0 will be rejected in favor of the alternative hypothesis $H_o:\mu <{\mu }_0$ if and only if μ0 is greater than or equal to the (1−α)-level upper confidence bounif for μ. In each case, illustrate the preceding relationship by obtaininy the appropriate upper confidence bound and comparing the result to the conclusion of the hypothesis test in the specified exercise.
a. Exercise 9.117
b. Exercise 9.118

Step-by-Step Solution

Answer

(a) The result of the confidence interval test is the same as the result of the hypotheses test.

(b) The result of the confidence interval test is the same as the result of the hypotheses test.

Part (a) Step 1: Given information

To determine the hypothesis test's conclusion.

Step 2: Explanation

Given that, the expressions are $H_{0 }:\mu ={\mu }_0$, $H_{0 }:\mu <{\mu }_0$

Calculation:

Confidence interval: 

MINITAB can be used to calculate the 95 percent confidence interval.
MINITAB output is,
$95\%$ upper bound is $0.6581$
The population mean is 0.9, which is higher than the 95% upper bound. As a result, null hypotheses are rejected at a 5% level.
Hypothesis test:

The $P$ - value for issue $9117E$ is $0.000$. The $P$-value is smaller than the significance level.

As a result, the null hypothesis is ruled out. As a result, the findings give enough evidence to suggest that women with peripheral artery disease had an unhealthy ABI on average.

As a result, both conclusions are the same. That is, the conclusion for the confidence interval and the hypotheses test are the same.

Part (b) Step 1: Given information

To determine the hypothesis test's conclusion.

Step 2: Explanation

$0.255$Given that, $H_0:\mu ={\mu }_0$, $H_0:\mu <{\mu }_0$

Calculation:

Confidence interval,

MINITAB is used to calculate the $90\%$ confidence interval.

Output for MINITAB is,
$2.087$ is the $90$ percent upper bound.
The population mean is $2$, which is lower than the $90\%$ upper bound. As a result, at the ten percent level, the null hypothesis is not rejected.
Hypothesis test:
The P - value for issue $9118E$ is . The significance level is bigger than the P-value.
As a result, the null hypothesis is not ruled out. As a result, the data does not support the conclusion that the average gasoline tank capacity of all dirt bikes is less than $2$ litres.
As a result, both conclusions are the same. That is, the conclusion for the confidence interval and the hypotheses test are the same.