Question

Two-Tailed Hypothesis Tests and CIs. As we mentioned on page 413, the following relationship holds between hypothesis tests and confidence intervals: For a two-tailed hypothesis test at the significance level $\alpha$, the null hypothesis $H_0:{\mu }_1={\mu }_2$will be rejected in favor of the alternative hypothesis $H_{\mathrm{a}}:{\mu }_1\ne {\mu }_2$if and only if the ( $1-\alpha$)-level confidence interval for ${\mu }_1-{\mu }_2$does not contain $0$. In each case, illustrate the preceding relationship by comparing the reults of the hypothesis test and confidence interval in the specified xercises.

a. Exercises 10.48 and 10.54.

b. Exercises 10.49 and 10.55.

Short answer

a). The null hypothesis is not rejected since the $95\%$confidence interval contains $0$. This is in line with the two-tailed hypothesis test's stated purpose.

b). The null hypothesis is rejected because the $99\%$confidence interval does not contain $0$. This is in line with the two-tailed hypothesis test's stated purpose.

Step-by-step solution

Part (a) Step 1: Given Information

Given in the question that, as a sample, data on vehicle miles travelled (VMT) is shown for two populations of Midwestern and Southern households.

Midwestern households, ${\overline{x}}_1=16.23,s_1=4.06$, and $n_1=15$.

Southern households, ${\overline{x}}_2=17.69,s_2=4.42$, and $n_2=14$.

The significance level is $5\%$.

Part (a) Step 2: Explanation

A statement is represented as - for a two-tailed hypothesis test.

The null hypothesis $H_0:{\mu }_1={\mu }_2$will be rejected in favour of the alternate hypothesis $H_a:{\mu }_1\ne {\mu }_2$at the significance level $\alpha$if and only if the $(1-\alpha )$-level confidence interval for ${\mu }_1-{\mu }_2$does not include $0$.

The test statistic for exercise $10.48$does not fall in the rejection zone of the two-tailed hypotheses test at a significance level of $5$percent. As a result, null hypotheses are not ruled out.

The $95\%$confidence interval for exercise $10.54$is $-4.69$to $1.77$.

Part (b) Step 1: Given Information

Male and female Nigerians' spleen lengths are given :

Population $1:$Male Nigerians $;{\overline{x}}_1=11.1,s_1=0.9,n_1=91$

Population $2:$Female Nigerians: ${\overline{x}}_2=10.1,s_2=0.7,n_2=109$

The significance level is $1\%$.

Part (b) Step 2: Explanation

A statement is represented as - for a two-tailed hypothesis test.

The null hypothesis $H_0:{\mu }_1={\mu }_2$will be rejected in favour of the alternate hypothesis $H_a:{\mu }_1\ne {\mu }_2$at the significance level $\alpha$if and only if the $(1-\alpha )$-level confidence interval for ${\mu }_1-{\mu }_2$does not include $0$.

The test statistic for exercise $10.49$falls in the rejection zone of the two-tailed hypotheses test at a significance level of $1\%$. As a result, null hypotheses are ruled out.

The $99\%$confidence interval for exercise $10.55$is $0.71\mathrm{cm}$to$1.29\mathrm{cm}$.