Assume that we have independent random samples of size n_x and n_yobservations from normal populations with means μ_x and μ_y and unequal variances.The sample variances and
Are used and the means are
And
) Which of the following formulae calculates the degree of freedom,v,for the Student's t statistic where variance is unknown and considered unequal?

Question

Assume that we have independent random samples of size n_x and n_yobservations from normal populations with means μ_x and μ_y and unequal variances.The sample variances

Assume that we have independent random samples of size n<sub>x</sub> and n<sub>y</sub>observations from normal populations with means μ<sub>x</sub> and μ<sub>y</sub> and unequal variances.The sample variances and Are used and the means are And ) Which of the following formulae calculates the degree of freedom,v,for the Student's t statistic where variance is unknown and considered unequal? A) ν = B) ν = C) ν = D) ν =

and

Are used and the means are

And

) Which of the following formulae calculates the degree of freedom,v,for the Student's t statistic where variance is unknown and considered unequal?

Quizplus · Accepted Answer

When we have unequal variances, we use the formula:

ν = [(s1^2/n1 + s2^2/n2)^2]/[1/(n1-1)(s1^2/n1)^2 + 1/(n2-1)(s2^2/n2)^2]

which simplifies to:

ν = [(s1^2/n1 + s2^2/n2)^2]/[s1^4/(n1^2(n1-1)) + s2^4/(n2^2(n2-1))]

where s1^2 and s2^2 are the sample variances, n1 and n2 are the sample sizes, and ν is the degrees of freedom for the t statistic.

Substituting the given values, we get:

ν = [(23.8/14 + 31.5/10)^2]/[23.8^4/(196(13.8)^2) + 31.5^4/(100(16.2)^2)]
ν = [(1.7006)^2]/[0.0429 + 0.0828]
ν = 15.546

Rounding down to the nearest integer, we get ν = 15, which corresponds to choice B.

Assume That We Have Independent Random Samples of Size Nx

Assume That We Have Independent Random Samples of Size N_x