Given x1 and x2 distributions that are normal or approximately normal with unknown σ1 and σ2, the value of t corresponding to x1 − x2 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 n1 − 1 and n2 − 1. However, a more accurate estimate for the appropriate degrees of freedom is given by Satterthwaite's formula: d.f. ≈ s12 n1 + s22 n2 2 1 n1 − 1 s12 n1 2 + 1 n2 − 1 s22 n2 2 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) 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 = 19, x1 ≈ 3.51, s1 ≈ 0.82, n2 = 12, x2 ≈ 3.87, and s2 ≈ 1.09. 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. =

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.6: Exponential And Logarithmic Equations

Problem 64E

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Given x₁ and x₂ distributions that are normal or approximately normal with unknown σ₁ and σ₂, the value of t corresponding to x₁ − x₂ 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 n₁ − 1 and n₂ − 1. However, a more accurate estimate for the appropriate degrees of freedom is given by Satterthwaite's formula:

d.f. ≈

s₁²

n₁

s₂²

n₂

	2

n₁ − 1

s₁²

n₁

	2

n₂ − 1

s₂²

n₂

	2

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) 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 = 19, x₁ ≈ 3.51, s₁ ≈ 0.82, n₂ = 12, x₂ ≈ 3.87, and s₂ ≈ 1.09. 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. =

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