Question

Given X, and x, distributions that are normal or approximately normal with unknown o, and on, the value of t corresponding to X, - X, has a distribution that is approximated by a Student's t distribution. We use the convention that the degrees of freedom is approximately the smaller of n - 1 and n, - 1. However, a more accurate estimate for the appropriate degrees of freedom is given by Satterthwaite's formula: 2 2 xn2 522 +$22) d.f. z _ d.f. Z- where S,,S2, ng, and n, are the respective sample standard deviations and sample sizes of independent random samples from the X, and x, distributions. This is the approximation used by most statistical software. When both n, and n, are 5 or larger, it is quite accurate. The degrees of freedom computed from this formula are either truncated or not rounded. (a) We tested whether the population average crime rate u, in the Rocky Mountain region is higher than that in New England, Uy. The data were n, = 19, X, ~ 3.51, S, ~0.87, ng = 14, X, a 3.87, and s, ~ 0.95. Use Satterthwaite's formula to compute the degrees of freedom for the Student's t distribution. (Round your answer to two decimal places.) d.f. = (b) Previously, you followed the convention that degrees of freedom d.f. = smaller of n - 1 and n, - 1. Compare this value of d.f. with that found with Satterthwaite's formula. O The previous method leads to a d.f. that is always greater than or equal to that computed by Satterthwaite's formula. O The previous method leads to a dif. that is always less than or equal to that computed by Satterthwaite's formula. O The previous method leads to a d.f. that is always equal to that computed by Satterthwaite's formula. O The previous method leads to a dif. that could be less than or greater than that computed by Satterthwaite's formula.

Answer 1

SOLUTION 1: Degrees of freedom by conventional formula

Df= Smaller of n1-1 and n2-1

= 19-1 and 14-1

minimum(18,13)

df= 13

df by satterwhite's formula

By putting all values in the formula we have

df= 26.67

The previous method lead to df that always less than or equal to the df computed by satterthwait's formula.

As per the guidelines I have done the first question please re post the rest. Thank You.