Testing for Significant Linear Correlation; Regression Lines; Coefficients of Determination and Correlation

Brief Instructions

Press STAT and ENTER. Enter the values of the first variable under L1 and the values of the second under L2. Make sure the pairings between numbers in the first and second variable remain the same as they were in the original data. Press STAT, choose TESTS, then LinRegTTest and press ENTER. The X-list is L1, the Y-List is L2 and the Freq is 1. Choose 0 next to β & ρ: and leave the space next to RegEQ: blank. Choose Calculate and press Enter.

The test statistic is next to t = and the P-value is next to p =. If the null hypothesis is rejected, the evidence is strong enough to say there is significant linear correlation. If the null hypothesis is kept, the evidence is not strong enough to say there is significant linear correlation.

The y-intercept of the regression is a and the slope is b. Write the equation of the regression line in the form of a linear equation without using a or b.

The coefficient of determination is r 2 and the correlation coefficient is r.

Detailed Instructions

The TI 83/84 calculator is set up so that when you test for significant linear correlation, the equation of the regression line and the coefficients of determination and correlation are presented as by-products. To test for significant linear correlation, you need to be working with two variables such that each value of the first variable is paired with one value of the second variable. For this demonstration, we will use the following data:

 1st Variable 5 6 8 9 12 15 2nd Variable 12 8 11 5 8 4

Press STAT and ENTER. Enter the numbers for the first variable under L1. Enter the numbers for the second variable under L2. When you are finished, the data entry screen should look like the following:

 L1 L2 5 12 6 8 8 11 9 5 12 8 14 4 ------ ------

Notice that the first numbers are still paired with the same second numbers. If the pairing is changed, your results will most likely be wrong.

Now press STAT. Move the cursor to CALC and select LinRegTTest using the up-arrow or down-arrow buttons. Pressing Enter will bring up the following display: Make sure L1 is next to XList:, L2 is next to YList: and 1 is next to Freq:. If your first variable is in some other list besides L1, then you would enter that list next to XList. Similarly, if your second variable is in some other list besides L2, you would enter that list next to YList. Choose ≠0 next to β & ρ: and leave the space next to RegEQ: blank.

After highlighting Calculate and pressing ENTER, you should have the following output on your calculator. You will need to use the down-arrow and up-arrow buttons to see the entire output.

Since a = 13.85 and b = -0.65, the equation of the regression line is y = 13.85 - 0.65x. Notice that the equation of the regression line does not contain a or b.

The coefficient of determination, r 2 = 0.507, says that about 50.7% of the variation in the data is determined by the regression line.

The correlation coefficient, r = -0.715, indicates a moderate negative correlation.