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.