Question

The professor of an introductory statistics course has found something interesting: there may be a relationship between scores on his first midterm and the number of years the test-takers have spent at the university. For the 63 students taking the course, the professor found that the least squares regression equation relating the two variables number of years spent by the student at the university (denoted by x) and score on the first midterm (denoted by y) is 9 - 83.52 - 2.50x. The standard error of the slope of the least squares regression line is approximately 0.96. Test for a significant linear relationship between the two variables by doing a hypothesis test regarding the population slope B (Assume that the variable y follows a normal distribution for each value of x and that the other regression assumptions are satisfied.) Use the 0.05 level of significance, and perform a two- tailed test. Then fill in the table below.

Answer 1

We are testing here whether there is a significance linear relationship between the two variables, therefore the null and the alternative hypothesis here are given as:

$H_0: \beta_1 = 0$

$H_1: \beta_1 \neq 0$

As we are doing a test for slope here, therefore this is a t test here for the slope coefficient.

The test statistic here is computed as:

$t^* = \frac{\beta_1}{SE_{\beta_1}} = \frac{-2.5}{0.96} = 2.604$

Therefore 2.604 is the test statistic value here.

For n - 2 = 61 degrees of freedom, we get the p-value from t distribution tables here as:
p = 2P( t₆₁ > 2.604) = 2*0.0058 = 0.0116

Therefore 0.012 is the required p-value here.

As the p-value here is 0.012 < 0.05 which is the level of significance here, therefore the test is significant here and we can reject the null hypothesis here. Therefore we have sufficient evidence here that there is a significant linea relationship between the number of years spent at the university and the score on the first midterm.