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A very important special case of the hypotheses in Eq. (2.23) is

(2.30)

These hypotheses relate to the significance of regression. Failing to reject H₀: β₁ = 0 implies that there is no linear relationship between x and y. This situation is illustrated in Figure 2.2. Note that this may imply either that x is of little value in explaining the variation in y and that the best estimator of y for any x is (Figure 2.2a) or that the true relationship between x and y is not linear (Figure 2.2b). Therefore, failing to reject H₀: β₁ = 0 is equivalent to saying that there is no linear relationship between y and x.

Alternatively, if H₀: β₁ = 0 is rejected, this implies that x is of value in explaining the variability in y. This is illustrated in Figure 2.3. However, rejecting H₀: β₁ = 0 could mean either that the straight-line model is adequate (Figure 2.3a) or that even though there is a linear effect of x, better results could be obtained with the addition of higher order polynomial terms in x (Figure 2.3b).

Figure 2.2 Situations where the hypothesis H₀: β₁ = 0 is not rejected.

Figure 2.3 Situations where the hypothesis H₀: β₁ = 0 is rejected.

The test procedure for H₀: β₁ = 0 may be developed from two approaches. The first approach simply makes use of the t statistic in Eq. (2.27) with β₁₀ = 0, or

The null hypothesis of significance of regression would be rejected if |t₀| > t_α/2,n−2.