Given x 1 and x 2 distributions that are normal or approximately normal with unknown σ 1 and σ 2 , the value of t corresponding to x 1 − x 2 has a distribution that is approximated by a Student's t...


Givenx
1 andx
2 distributions that are normal or approximately normal with unknown σ1 and σ2, the value oft corresponding to x1 − x2 has a distribution that is approximated by a Student'st distribution. We use the convention that the degrees of freedom is approximately the smaller ofn
1 − 1 andn
2 − 1. However, a more accurate estimate for the appropriate degrees of freedom is given by Satterthwaite's formula:



d.f. ≈


























s1
2
n1

+









s2
2
n2











2











1
n1 − 1

















s1
2
n1











2

+









1
n2 − 1

















s2
2
n2











2



wheres
1,s
2,n
1, andn
2 are the respective sample standard deviations and sample sizes of independent random samples from thex
1 andx
2 distributions. This is the approximation used by most statistical software. When bothn
1 andn
2 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 μ2 in the Rocky Mountain region is higher than that in New England, μ1. The data weren
1 = 19, x1 ≈ 3.51,s
1 ≈ 0.82,n
2 = 12, x2 ≈ 3.87, ands
2 ≈ 1.09. Use Satterthwaite's formula to compute the degrees of freedom for the Student'st distribution. (Round your answer to two decimal places.)

d.f. =


Jun 02, 2022
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