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Regression Adjusted R-Square

7.9 A three-variable regression gave the following results {see attachment}.
(a) What is the sample size?
(b) What is the value of the RSS?
(c) What are the d.f of the ESS and RSS?
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*Please see attachment for complete list of questions (including 7.12 and 7.15)

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7.9.
because of different definitions of "variable" in different text books, sometime the constant is treated as variable, and sometimes not. In this case we use the general form that constant is also a "variable" in the "three variable regression"

(a) from the ANOVA table, we know that sample size = df(TSS) + 1 =15
(b) RSS = TSS - ESS = 66042 - 65965= 77
(c) df (ESS) = # of variables -1 = k-1 = 2
df (RSS) = df(TSS) - df (ESS) = 14-2 = 12
(d) R squared = ESS /TSS = 66042 / 65965= 0.9988
R2 adjusted = 1- (1-R2)(n-1)/(n-k) = 1- (1-0.9988)*(15-1)/(15-3) = 0.9986

(e) to test the coefficients of X2 and X3 are simultaneously zero, we should use a F test.
Because F test employs the statistic (F) to test the hypotheses about the mean of the distributions from which a sample or a set of samples have been drawn. In this case,
Ho: X2 = X3 = 0
Ha: at least one variable is not zero.
Since we set two restrictions on the variables, we cannot use the student t test. But F test can be applied in the following form:
F = MSS(E) / MSS(R) = (ESS / df(E)) / (RSS / df(R)) = (65965 / 2) / (77 / 12) = 32982.5 / 6.42 = 5140.13
Since the 95% critical F(2,12) value is 4, less than the calculated F statistic, ...

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