Final answer:
In simple linear regression, the degrees of freedom for testing the significance of the slope is n - 2, where n is the number of data points. If the correlation coefficient is greater than the critical value from a table, or if the p-value is less than the significance level, the slope is considered significant.
Step-by-step explanation:
To test the significance of the slope in a simple linear regression equation, we use degrees of freedom (df). In the context of a regression analysis, the degrees of freedom for the significance test of the slope is calculated as df = n - 2, where n is the number of data points. This is because two parameters (the slope and the y-intercept) are estimated from the data.
For example, if a study with n = 10 data points computes a correlation coefficient, r = 0.801, the degrees of freedom would be df = 10 - 2 = 8. If the observed value of r is greater than the critical value obtained from a statistical table at a given significance level, here 0.632 for α = 0.05, we conclude that the correlation coefficient is significant, thus the slope is significant and the line can be used to make predictions.
The p-value associated with the test can also be used to determine significance. If the p-value is less than the chosen significance level, such as 0.026 < 0.05, we reject the null hypothesis which states there is no significant linear relationship, thus concluding that there is indeed a significant linear relationship between the variables in question.