test with independent differences

Question

You wish to test the following claim (Ha) at a significance level of α=0.001.

      Ho:μ1=μ2 

      Ha:μ1>μ2

You believe both populations are normally distributed, but you do not know the standard deviations for either. However, you also have no reason to believe the variances of the two populations are not equal. You obtain a sample of size n1=13 with a mean of M1=83.6 and a standard deviation of SD1=19.4 from the first population. You obtain a sample of n2=19 with a mean of M2=72.9 and a standard deviation of SD2=12.6 from the second population.

What is the test statistic for this sample? (Report answer accurate to three decimal places.) 

test statistic = 

What is the p-value for this sample? (Report answer accurate to four decimal places.) 

p-value =

Answer 1

Since we assume the variances of the two populations are the same, we can use the pooled variance in the computation of the t-statistic:

pooled variance (Sp^2) = (n1-1)s1^2 + (n2-1)*s2^2 / (n1 + n2 - 2) = (13-1)*19.4^2 + (19-1)*12.6^2 / (13 +19- 2) = 245.8

Sp = 15.67

t-calc = (mean1 - mean2)/ (Sp * sqrt(1/n1 + 1/n2)) = (83.6-72.9)/(15.67 * sqrt(1/13 + 1/19) = 1.897

since this is a right tailed test, the p-value = P(t > 1.897) at n1 + n2 - 2 = 30 degrees of freedom = 0.0337

so would not reject null hypothesis at alpha = 0.001.