Finding the linear correlation coefficient from a Minitab display

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find the linear correlation coefficient from a Minitab display.  Here's our problem statement: The Minitab output shown below was obtained by using paired data consisting of weights in pounds of 26 cars and their highway fuel consumption amounts in miles per gallon.  Along with the paired sample data, Minitab was also given a car weight of 5,000 pounds to be used for predicting the highway fuel consumption amount.  Use the information provided in the display to determine the value of the linear correlation coefficient.  Be careful to correctly identify the sign of the correlation coefficient.  Given that there are 26 pairs of data, is there sufficient evidence to support a claim of linear correlation between the weights of cars and their highway fuel consumption amounts?

Part 1

OK, the first part of this problem asks for the linear correlation coefficient.  And to find that, we're going to take a look at the Minitab display here.  So here's the Minitab display.  And notice there's quite a bit of stuff here.  But everything we're really going to need is at the top of the display here.  The linear correlation coefficient is normally listed in software output with the variable R.  Well, R is not shown here, but we do have R squared.  So we can take this value for R squared, and if we take the square root of R squared, that leaves us with R.  So all I have to do is take the square root of this value here.  So 0.636.  Notice I'm converting from the percent to a decimal.  Take the square root, and there's my R value.

But we know that values for the linear correlation coefficient can be positive or negative.  So which is it here?  If I take a positive number and square it, I get a positive number.  If I take a negative number and square it, I also get a positive number.  So how do we know whether this is positive or negative?  Well, look at the model that they're giving us, the regression equation.  If you look at the value for the coefficient in front of your independent variable here, notice it's negative.  That means this line, when you graph it, is going to have a negative slope.  And aligned with a negative slope is a negative linear correlation coefficient.  So the value for R we're looking for is going to be negative.  Now I know that I need put in my negative sign and then put in the number we calculated rounded to three decimal places.  Excellent!

Part 2

And now the last part of the problem asks, "Is there sufficient evidence to support a claim of linear correlation?"  Well, let's go back to our Minitab display, and we're going to look at this table here with our predictor values and the actual testing that was run on each of them.  Notice here, we want the one for the model itself, which is going to be the independent variable.  There's a P-value here on the end.  So this value here is the one we want to look at.

And that's a number that's either zero or a number that's so low it's practically zero.  And so therefore we're just going to say zero. And when you have a P-value of zero, any reasonable significance level you would test against, you're going to be inside the region of rejection.  And when you're inside the reason of rejection, you reject the null hypothesis.  And every time you reject the null hypothesis, there's always sufficient evidence.  So is there sufficient evidence the answer's going to be yes.  Good job!

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