Question

Suppose independent random samples that are taken to test the difference between the means of two populations (n1 = 66 and n2 =46). The variances of the populations are unknown but are assumed to be unequal. The sample standard deviations are s1=82 and s2=68. The appropriate distribution to use is the:

A) t distribution with df = 110

B ) t distribution with df = 107

C) t distribution with df = 106

D) F distribution with numerator df = 65 and denominator df = 45

Answer 1

Correct answer is,

B) t distribution with df = 107

Explanation :

Here in this scenario we have two independent sample which is assumed to be population Variances are unknown and also we assuming that the population Variance are unequal.

So in this scenario to test weather the mean of two population are same we need to construct appropriate hypothesis and for to test that Hypothesis we have to use independent sample t test i.e two sample t test because here the population standard deviations is unknown.

And Since the population Variance are unequal so the degrees of freedom of t distribution will be calculated using following formula ( n = 66, n2 = 46 )

Where, s1 = 82 & s2 = 68,

Substitute the value in above formula, then we get df = 106.67 which is rounded to 107.

So option B is correct t distribution with df = 107 is appropriate test for this scenario.

If we considered that the population Variance are equal the. The degrees of freedom is calculated using the formula which is not equal to 107.

Other options are not appropriate for here.

Thank you.