Given x1 and x2 distributions that are normal or approximately normal with unknown Ï1 and Ï2, the

Given x1 and x2 distributions that are normal or approximately normal with unknown σ1 and σ2, the value of t corresponding to 1 - 2 has a distribution that is approximated by a Student€™s t distribution. We use the convention that the degrees of freedom are approximately the smaller of n1 - 1 and n2 - 1. However, a more accurate estimate for the appropriate degrees of freedom is given by Satterthwaite€™s formula:

where s1, s2, n1, and n2 are the respective sample standard deviations and sample sizes of independent random samples from the x1 and x2 distributions. This is the approximation used by most statistical software. When both n1 and n2 are 5 or larger, it is quite accurate. The degrees of freedom computed from this formula are either truncated or not rounded.
(a) In Problem 15, we tested whether the population average crime rate μ2 in the Rocky Mountain region is higher than that in New England, μ1. The data were n1 = 10, 1 ‰ˆ 3.51, s1 ‰ˆ 0.81, n2 = 12, 2 ‰ˆ 3.87, and s2 ‰ˆ 0.94. Use Satterthwaite€™s formula to compute the degrees of freedom for the Student€™s t distribution.
(b) When you did Problem 15, you followed the convention that degrees of freedom d.f. = smaller of n1 - 1 and n2 - 1. Compare this d.f. with that found by Satterthwaite€™s formula.

Transcribed Image Text:

d.f. 1 12. n, - 1