.Regression Practice Answers STA 3024 Spring, 1999 D. Meeter
In a set of 40 observations, the total SS was 5,000.
1. y was regressed on x1; the SSreg was 400. How many d.f. does SSreg have? 1 What is SSE? 4,600 How many d.f. does SSE have? 38 Is b 1 significantly different from zero? F = (400/1)/(4600/38) = 3.30, with 1, 38 d.f. p>0.05. No. What is your estimate of s 2 and how many d.f. does it have? s2 = 4600/38 = 121; 38 d.f.
2. Then y was regressed on x1 and x2, with x2 listed first. The SSE was 4,500; how many d.f. does it have? 37 What is your estimate of s 2 and how many d.f. does it have? s2 = 4500/37 = 121.6, 37 d.f. What is SSreg? 500 How many d.f.? 2 The program prints x2: 300
extra due to x1: 200
How many d.f. for the 300? 1 For the 200? 1 How can it be that the SS "due to" x1 is different?
If x1 and x2 are correlated, SS(x1) and SS(x1 after x2 in model) will be different.
Test the hypothesis that b 1 = 0 by
a) a partial F test; F = (200/1)/(4500/37) = 1.64, 1, 37 d.f. Do not reject.
b) a test which assumes that x1 was added last; This is the same test.
c) a test in which the complete model is x1 and x2 and the reduced model is just x2. Same!
(These are three ways of saying the same thing. The t ratios in Minitab, if squared, give the partial F ratios, so they accomplish the same test. What is the t ratio corresponding to the above test?) t = Ö 1.64 = 1.28, but it could be -1.28, depending on the sign of the regression coefficient.
3. Then y was regressed on x1, x2, and x3, in that order. The result was
x1: 400
extra due to x2: 400
extra due to x3: 100
What is SSreg? 900 How many d.f.? 3 What is SSE? 4100 How many d.f.? 36 What is your estimate of s 2? 4100/36 = 113.9 How many d.f.? 36 Note that the SS for x1 is the same as in 1, because it was entered first in this three-predictor regression. What do you notice about the SS for x2 in this regression compared to the previous one? It got larger! This phenomenon is unusual but not impossible. Usually the SSreg for a variable gets smaller as other variables are entered into the model.
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Suppose you had data on your company for salaries (y) for men and women, along with their years of experience (x). You fit a regression to the data with men as the reference category, and get a prediction equation:
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Assuming that a) the model is adequate, and b) all coefficients are statistically significant, what is the prediction equation for men? y-hat = 40 + 3x For women? y-hat = 43 + 3.5x Interpret these equations in words. Men start out at 40 (thousand?) and increase by 3 for each year of experience. Women start out at 43 and increase by 3.5 for each year of experience. What salary would you predict for a woman with 0 year's experience? 40 For a man? 43 The same, with 10 year's experience? 70 for a man and 88 for a woman.