Question

It has been suggested that residents in the rural areas of the South have a different life-span than in the North. In a random sample of 50 residents of the rural South, the mean life span was 75.9 with a sample standard deviation of 9.8. A random sample of 75 residents of the North, the mean life span was 82.4 with a sample standard deviation of 6.1. Is the sample evidence strong enough to claim that those in the South have shorter or longer lives? Use a significance level of 5%.Use a test statistic.

Answer 1

Solution: In order to determine if the residents in the rural areas of the South have a different life-span than in the North, we construct our null and alternative hypotheses as H0: mu1 = mu2 vs Ha: mu1 mu2 where mu1 and mu2 are the unknown mean life spans of the residents of the rural areas of South and North.

The test statistic used for this test is T= (x1bar-x2bar)/sqrt((s1*s1/n1)+(s2*s2/n2)) where x1bar, x2bar are the sample means, s1,s2 are the sample standard deviations and n1,n2 are the sample sizes. sqrt refers to the square root function.

We reject H0 if |T(observed)| > t(alpha/2,v) where t(alpha/2,v) is the upper alpha/2 point of a Student's t distribution with "v" degrees of freedom. Alpha is the level of significance.

v = (((s1*s1/n1)+(s2*s2/n2))^2) / (((s1*s1/n1)^2)/(n1-1))+(((s2*s2/n2)^2)/(n2-1))

Here T(observed) = -4.181008 and t(alpha/2,v) = 1.99241.Thus we see that |T(observed)| > t(alpha/2,v).

Thus we reject H0 and conclude on the basis of the given sample at a 5% level of significance that the sample evidence is strong enough to claim that those in the South have shorter or longer lives than that of the people living in rural areas of North.