# 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 H_{0}: β_{1} = 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)

Under H_{0}: β_{1} = 0, Equation (13.28) becomes

(13.28.1)

Here the appropriate alternative hypothesis is H_{1}: β_{1} ≠ 0 so that the critical region is (we have a one-tail alternative using the upper tail of the F distribution). So if we reject H_{0} in favor of H_{1}, 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 H_{0}: β_{1} = 0, against H_{1}: β_{1} ≠ 0 with . Using Equation (13.28.1) ...