Let β_k* denote some hypothesized value of the regression coefficient β_k. If the basic assumptions hold, then the appropriate test statistic is

t = (b_i - β_k*)/s_bi

Case 1: Suppose we wish to test the null hypothesis H₀: β_k = β_k* or H₀: β_k ≤ β_k* against the one-sided alternative hypothesis H₁: β_k > β_k* using a level of significance α. Use the decision rule

Reject H₀ in favor of H₁ if t > t_{α, v}

where the degrees of freedom of the Student t distribution is v = (T - K - 1).

Case 2: To test the null hypothesis H₀: β_k = β_k* or H₀: β_k ≥ β_k* against the one-sided alternative hypothesis H₁: β_k < β_k*, use the decision rule

Reject H₀ in favor of H₁ if t < t _{α, v}

Case 3: To test the null hypothesis H₀: β_k = β_k* against the two-sided alternative hypothesis H₁: β_k ≠ β_k*, use the decision rule

Reject H₀ in favor of H₁ if t < - t_{α/2, v} or t > t_{α/2, v}

Continue with the example in los a. It is reasonable to conjecture either that engine size has no effect on gasoline mileage or that the larger the engine, the lower the mileage. Thus, it is natural to test the null hypothesis H₀: β₁ = 0 against the one-sided alternative hypothesis H₁: β₁ < 0. Test this hypothesis using a 5% level of significance.

The data is shown below:

The t-statistic column reports the results of a t-test on the hypothesis that the population value of the regression intercept or slope coefficient equals 0.
t-statistic = (estimated regression coefficient - hypothesized population value of the regression coefficient) / standard error of the regression coefficient
= estimated regression coefficient / standard error of the regression coefficient
= column 1 / column 2.

We have α = 0.05, T = 10, and K = 2. The appropriate degrees of freedom is v = 10 - 2 - 1 = 7. The critical value of t having 7 degrees of freedom is -t0.05, 7 = -1.895.
The hypothesized value is β₁* = 0. To test H₀, we calculate the t statistic t = (b₁ - 0)/s_b1 = -4.0129 / 1.737 = -2.31.

Because this observed t statistic is less than the critical value, it falls in the rejection region. Thus, if we use α = 0.05, we reject H₀ in favor of H₁. However, if we had used the slightly different level of significance α = 0.025, then the critical value of the test would have been -t_0.025 = -2.365. In this case, the observed t statistic would have fallen in the acceptance region and we would not have rejected H₀.

In the model, the variable X₂ measures the weight of the car. We would expect heavy cars to get lower gasoline mileage than light cars, so it is natural to test H₀: β₂ = 0 against H₁: β₂ < 0. The appropriate t statistic is -4.48. This t statistic falls in the critical region if we use a 5% level of significance, so we reject the null hypothesis.

We showed that if we performed a one-tailed test using a 2.5% level of significance, we would fail to reject the null hypothesis that β₁ = 0, but if we used a 5% level of significance we would reject the null hypothesis. Now the question is: Should we keep the variable X₁ in the model or remove it and express gasoline mileage solely as a function of car weight? This is a difficult question to answer, and the decision depends partly on the opinions of the investigator. In los o when we discuss what is called the multicollinearity problem, we shall get more insight into why it is difficult to determine whether β₁ = 0.