Determine the decision criterion for rejecting the null hypothesis in the given hypothesis test; i.e., describe the values of the test statistic that would result in rejection of the null hypothesis. We wish to compare the means of two populations using paired observations. Suppose that $\bar { d } = 3.125 , s _ { d } = 2.911 , \text { and } n = 8$ and that you wish to test the following hypothesis at the
10 % level of significance:
$H _ { 0 } : \mu _ { d } = 0 \text { against } H _ { 1 } : \mu _ { d } 0$
What decision rule would you use?
A) Reject $H _ { 0 }$ if the test statistic is less than 1.415 .
B) Reject $H _ { 0 }$ if the test statistic is greater than -1.145 .
C) Reject $H _ { 0 }$ if the test statistic is greater than -1.145 or less than 1.415 .
D) Reject $H _ { 0 }$ if the test statistic is greater than 1.415 .

Question

Determine the decision criterion for rejecting the null hypothesis in the given hypothesis test; i.e., describe the values of the test statistic that would result in rejection of the null hypothesis. We wish to compare the means of two populations using paired observations. Suppose that $\bar { d } = 3.125 , s _ { d } = 2.911 , \text { and } n = 8$ and that you wish to test the following hypothesis at the
10 %  level of significance:
$H _ { 0 } : \mu _ { d } = 0 \text { against } H _ { 1 } : \mu _ { d } > 0$
What decision rule would you use?
A) Reject $H _ { 0 }$ if the test statistic is less than  1.415 . 
B) Reject $H _ { 0 }$ if the test statistic is greater than  -1.145 .
C) Reject $H _ { 0 }$ if the test statistic is greater than  -1.145  or less than  1.415 .
D) Reject $H _ { 0 }$ if the test statistic is greater than  1.415 .

Quizplus · Accepted Answer

Given the hypothesis test is one-tailed ($H_1: \mu_d > 0$) and at the 10% significance level, the decision rule involves rejecting $H_0$ if the test statistic exceeds a certain critical value. Since the alternative hypothesis suggests a positive direction ($\mu_d > 0$), we look for a test statistic that is significantly greater than a threshold. Choice D correctly identifies that we reject $H_0$ if the test statistic is greater than 1.415, which aligns with the upper tail critical value for a one-tailed test at the 10% significance level with $n-1$ degrees of freedom (in this case, 7 degrees of freedom).

Determine the Decision Criterion for Rejecting the Null Hypothesis in the Given