tests involving independent differences

Question

you wish to test the following claim (Ha) at a significance level of α=0.10.

      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=20 with a mean of M1=52.6 and a standard deviation of SD1=14.7 from the first population. You obtain a sample of size n2=18n2=18 with a mean of M2=41.7 and a standard deviation of SD2=9.1 from the second population.

What is the critical value for this test? For this calculation, use the conservative under-estimate for the degrees of freedom as mentioned in the textbook. (Report answer accurate to three decimal places.) 

critical value = ±

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

test statistic =

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) = (20-1)*14.7^2 + (18-1)*9.1^2 / (20 +18- 2) = 153.15

Sp = 12.38

t-calc = (mean1 - mean2)/ (Sp * sqrt(1/n1 + 1/n2)) = (52.6-41.7)/(12.38 * sqrt(1/20 + 1/18) = 2.71

since this is a two tailed test, the critical value is +/- t alpha/2 = +/- t0.05 at n1 + n2 -2 = 36 degrees of freedom = +/- 1.697 (tabled value for 30 DOF, since 36 not in table, so use lower value to be conservative).

since t-calc > 1.697, would reject null hypothesis at alpha = 0.1.