# The t Distribution in Testing the Slope

One of the important points in this chapter concerned the use of Student's t test to test the null hypothesis that the true slope in the population is 0.00 (i.e., the hypothesis that there is no linear relationship between X and Y). The applet named SlopeTest illustrates the meaning of this test with population data that support the null hypothesis.

Click on the button labeled "10 sets." This will draw 10 samples of 5 pairs of scores. You drew from a population where the true slope, and therefore the true correlation, is 0.00, so you know that the variables are not linearly related. For each slope that you obtained, the applet calculated a t statistic using the formula for t given in Section 10.6. The 10 t values are given near the top of the display. The distribution of these 10 values is given at the right, and the plot of the 5 observations for your 10th set is given at the left. Each time you click the "10 sets" button you will draw 10 new sets of observations, calculate their slopes and associated t values, and add those to the plot on the right. If you click the "100 Sets" button you will accumulate 100 t values at a time.

• Run this applet again. First generate one set at a time and note the resulting variation in t and how the regression line changes with every sample.
• Then accumulate 100 sets at a time and notice how the distribution of t smooths out. Notice that our t values only rarely exceed +3.00. (In fact, the critical value of t on 3 df is +3.18.)
• Now go to the applet below, where I have set it to draw 15 pairs of observations at a time. Repeat what you did with the first applet.