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Statistical Inference: A Short Course by Michael J. Panik

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13.10 Applications of the F Statistic to Regression Analysis

13.10.1 Testing the Significance of the Regression Relationship Between X and Y

We previously used a t test to determine if X contributes significantly to the variation in Y. If H0: β1 = 0 is true (we posit “no linear relationship” between X and Y), then the sole source of variation in Y is the random disturbance term ε since the population regression sum of squares is zero. Now, it can be demonstrated that the statistic

(13.28) equation

Under H0: β1 = 0, Equation (13.28) becomes

(13.28.1) equation

Here the appropriate alternative hypothesis is H1: β1 ≠ 0 so that the critical region is img(we have a one-tail alternative using the upper tail of the F distribution). So if we reject H0 in favor of H1, then we can safely conclude that there exists a statistically significant linear relationship between X and Y at the 100 α% level.

Example 13.10

Using the information provided in Table 12.4, let us conduct a significance test of the linear relationship between X and Y for α = 0.05. Here we test H0: β1 = 0, against H1: β1 ≠ 0 with img. Using Equation (13.28.1) ...

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