Because the coefficient of determination Upper R squaredR2 always increases when

Question

Because the coefficient of determination Upper R squaredR2 always increases when a new independent variable is added to the model, it is tempting to include many variables in a model to force Upper R squaredR2 to be near 1. However, doing so reduces the degrees of freedom available for estimating sigma squared2, which adversely affects our ability to make reliable inferences. Suppose you want to use 1919 economic indicators to predict next year's gross domestic product (GDP). You fit the model y equals beta 0 plus beta 1 x 1 plus times times times plus beta 18 x 18 plus beta 19 x 19 plus epsilony=0+1x1+•••+18x18+19x19+ where yequals=GDP and x 1 comma x 2 comma ... comma x 19x1, x2, ..., x19 are the economic indicators. Only 2222 years of data (nequals=2222) are used to fit the model, and you obtain Upper R squaredR2equals=0.910.91. Test to see whether this impressive-looking Upper R squaredR2 is large enough for you to infer that the model is usefullong dash—that is, that at least one term in the model is important for predicting GDP. Use alphaequals=0.050.05. a Identify the null and alternate hypotheses. find the test statistic for this hypothesis test. Determine the p-value for this hypothesis test. state the conclusion for this hypothesis test.

Explanation / Answer

Ho: b1 = b2 = ..b19 =0

Ha: not ( b1 = b2 = ..b19 =0)

F = (R^2/ k )/((1-R^2)/(n-k-1)

here n = 22

k = 18 R^2 = 0.91

hence

F = (0.91/18)/(0.09/(22-18-1))

= 1.6851

p-value = 0.372672

since p-vlaue > 0.05

we fail to reject the null hypothesis

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