12.3.4 - Hypothesis Testing

We can use statistical inference (i.e., hypothesis testing) to draw conclusions about how the population of \(y\) values relates to the population of \(x\) values, based on the sample of \(x\) and \(y\) values.

The equation \(Y=\beta_0+\beta_1 x\) describes this relationship in the population. Within this model there are two parameters that we use sample data to estimate: the \(y\)-intercept (\(\beta_0\) estimated by \(b_0\)) and the slope (\(\beta_1\) estimated by \(b_1\)). We can use the five step hypothesis testing procedure to test for the statistical significance of each  separately. Note, typically we are only interested in testing for the statistical significance of the slope because that tells us that \(\beta_1 \neq 0\) which means that \(x\) can be used to predict \(y\). When \(\beta_1 = 0\) then the line of best fit is a straight horizontal line and having information about \(x\) does not change the predicted value of \(y\); in other words, \(x\) does not help us to predict \(y\). If the value of the slope is anything other than 0, then the predict value of \(y\) will be different for all values of \(x\) and having \(x\) helps us to better predict \(y\).

We are usually not concerned with the statistical significance of the \(y\)-intercept unless there is some theoretical meaning to \(\beta_0 \neq 0\). Below you will see how to test the statistical significance of the slope and how to construct a confidence interval for the slope; the procedures for the \(y\)-intercept would be the same.

1. Check assumptions and write hypotheses

The assumptions of simple linear regression are linearity, independence of errors, normality of errors, and equal error variance. You should check all of these assumptions before preceding.

Research Question Is the slope in the population different from 0? Is the slope in the population positive? Is the slope in the population negative?
Null Hypothesis, \(H_{0}\) \(\beta_1 =0\) \(\beta_1= 0\) \(\beta_1= 0\)
Alternative Hypothesis, \(H_{a}\) \(\beta_1\neq 0\) \(\beta_1> 0\) \(\beta_1< 0\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
2. Calculate the test statistic

Minitab will compute the \(t\) test statistic:

\(t=\dfrac{b_1}{SE(b_1)}\) where \(SE(b_1)=\sqrt{\dfrac{\dfrac{\sum (e^2)}{n-2}}{\sum (x- \overline{x})^2}}\)

3. Determine the p-value

Minitab will compute the p-value for the non-directional hypothesis \(H_a: \beta_1 \neq 0 \)

If you are conducting a one-tailed test you will need to divide the p-value in the Minitab output by 2.

4. Make a decision

If \(p\leq \alpha\) reject the null hypothesis. If \(p>\alpha\) fail to reject the null hypothesis.

5. State a "real world" conclusion

Based on your decision in Step 4, write a conclusion in terms of the original research question.