Hypothesis Test of the 
Coefficient of Regression 
Using the t-Test

 

Example: A sample of 10 families revealed the following figures for family size and the amount spent on food per week.  

Size 3 6 5 6 6 3 4 4 5 3
$ 99 104 151 129 142 111 74 91 119 91


a. Compute the coefficient of correlation.
b. Determine the coefficient of determination.
c. Can we conclude that there is a positive association between the amount spent on food and the family size?  Use a level of significance of a = 5%.

Solution: “Can we ... a positive association ... food and the family size" means the same as “the coefficient of correlation is greater than zero.”  Therefore, the null and alternate hypotheses are H0: r £ 0 and H1: r > 0, respectively.  Follow the steps below to solve the problem using the TI-83.  [NOTE: If the p-value < a, reject the null hypothesis; otherwise, do not reject the null hypothesis.]

 

Press STAT, EDIT, 1:Edit, and enter the data in L1 (independent variable) and L2 (dependent variable). 

 

Press STAT, TESTS, and press the down arrow to E:LinRegTTest. 
Press ENTER.

 

Select the alternate hypothesis > 0.

Note the line RegEQ:Y1. This will help produce the graph later.

Use the down arrow to get to calculate and press ENTER.

Results:

Since the p-value is 0.0365 < 0.05, reject the null hypothesis.

Conclude that there is a positive relationship between the amount spent on food and family size.

The coefficient of correlation is 0.5892, and the coefficient of determination is 0.3471.

Thus, about 34.7% of the variation in food expense is accounted for by variation in family size.

 
To graph the scattered diagram with the linear regression line, press ZOOM 9.