The U.S. Center for Disease Control reports that in year 1900, the mean life expectancy is 47.6 years for whites and 33 years for nonwhites. (Click here for reference data ${ }^{2}$ ) Suppose a survey of randomly selected death records for white and nonwhite people born in 1900 from a certain county is conducted. of the 121 whites surveyed, the mean life span was 48.2 years with a standard deviation of 11.1 years and of the 97 nonwhites, the mean life span was 36.3 years with a standard deviation of 14.1 years. Conduct a hypothesis test at the 0.05 level of significance to determine whether there was no difference in mean life spans in the county for whites and nonwhites in year 1900.
Preliminary:
a. Is it safe to assume that
$n_{w} \leq 5 \%$ of all white people borm in 1900 and
$n_{n v} \leq 5 \%$ of all nonwhite people born in 1900 ?
Yes
No
b. Is $n_{w} \geq 30$ and $n_{n w} \geq 30$ ?
No
Yes
Test the claim:
a. Determine the null and alternative hypotheses.
\[
\begin{array}{l}
H_{0}: \mu_{w} \sqrt{v} \mu_{n w} \\
H_{a}: \mu_{w} ? v \mu_{n w}
\end{array}
\]
b. Determine the test statistic. Round to four decimal places.
\[
t=
\]
c. Find the $p$-value. Round to 4 decimals.
\[
p \text {-value }=
\]
Answer
Since the p-value is less than the significance level of 0.05, we reject the null hypothesis. There is sufficient evidence to conclude that there is a difference in mean life spans in the county for whites and nonwhites in year 1900. The final answer is \(\boxed{6.7941}\) for the test statistic and \(\boxed{0.0000}\) for the p-value.
Step 1 ：State the null and alternative hypotheses. The null hypothesis is that the mean life span of whites and nonwhites is the same, while the alternative hypothesis is that the mean life span of whites and nonwhites is not the same. In mathematical terms, this can be written as: \[H_{0}: \mu_{w} = \mu_{n w}\] \[H_{a}: \mu_{w} \neq \mu_{n w}\]
Step 2 ：Calculate the test statistic. The test statistic for a two-sample t-test is calculated as the difference between the sample means divided by the standard error of the difference. The standard error of the difference is calculated as the square root of the sum of the squares of the standard deviations divided by their respective sample sizes. The test statistic is \(t = 6.7941\)
Step 3 ：Find the p-value. The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. The p-value is \(p = 0.0000\)
Step 4 ：Since the p-value is less than the significance level of 0.05, we reject the null hypothesis. There is sufficient evidence to conclude that there is a difference in mean life spans in the county for whites and nonwhites in year 1900. The final answer is \(\boxed{6.7941}\) for the test statistic and \(\boxed{0.0000}\) for the p-value.
