Hypothesis Tests of the Slope Coefficient

We frequently are interested in testing whether knowledge o an independent variable X is useful in explaining the values of Y. For example, we may want to test whether a linear relationship exists between X and Y. If X and Y are not linearly related, then in the population regression line E(Y_i) = b₀ + b₁X_i, we should have b₁ = 0. If b₁ = 0, the values of X are of no use in predicting Y, and the population regression line will be a horizontal line. If we reject the hypothesis that b₁ = 0, then we are saying that the values of X are helpful in predicting Y.

The null hypothesis does not always have the form H₀: b₁ = 0, although this is by far the most frequent case.

Suppose an economist claims that, in the U.S., annual income is related to years of education and the slope of the population regression line is approximately b₁ = $2000. That is, the economist claims that an increase of one year in education tends to be associated with an increase of approximately $2000 in annual income. Suppose we want to test the null hypothesis that the slope of the population regression line is b₁ = $2000. Then we would test the null hypothesis H₀: b₁ = $2000 against the two sided alternative hypothesis H₁: b₁ ≠ $2000.

Test of hypothesis concerning the value of b₁ are based on the fact that if the basic assumptions of the simple linear regression model hold, then the random variable t = (b-hat₁ - B₁)/Sb-hat₁ follows the student t distribution with n - 2 degrees of freedom.

The two key measures are the standard error of the parameter and the critical value for the t-distribution associated with the t-test of statistical significance.

• S_b1 = sqrt [ ∑(y_i - y-hat_i)² / (n - 2) ] / sqrt [ ∑(x_i - x-bar)² ]
• The t-distribution requires that the number of degrees of freedom be known. For a linear regression with two parameters estimated (the two parameters are the slope and intercept), the number of degrees of freedom is (n - 2), where n is the number of observations. (Generally, the number of degrees of freedom equals the number of observations less the number of parameters estimated.)

The decision rule: reject H₀ in favor of H₁ if t < -t_{α/2, n-2} or if t > t_{α/2, n-2}.

Regarding the calculated t-statistic used to test whether the slope coefficient is equal to zero:

• It is equal to the t-statistic to test whether the pairwise correlation is zero: t = r sqrt (n - 2) / sqrt (1 - r²)
• It is related to the F-distributed test statistic: t² = F

Hypothesis Tests of the Intercept

Conducting hypothesis tests and calculating confidence intervals for the intercept parameter b₀ is not done as often as it is for the slope parameter b₁. The reason for this becomes clear upon reviewing the meaning of b₀. The intercept parameter b₀ is the mean of the responses at x = 0.

We can perform the hypothesis testing in a similar manner. One thing to note is the equation for the standard error of the intercept is different. However, there's no need to memorize it.

Hypothesis Tests of Slope When Independent Variable Is an Indicator Variable

A dummy (indicator) variable takes on 1 and 0 only. The number 1 and 0 have no numerical (quantitative) meaning. The two numbers are used to represent groups. In short dummy variable is categorical (qualitative).

For instance, we may have a sample (or population) that includes both female and male. Then a dummy variable can be defined as D = 1 for female and D = 0 for male. Such a dummy variable divides the sample into two subsamples (or two sub-populations): one for female and one for male.

Hypothesis testing can be performed in a similar manner when independent variable is a dummy variable.