Identifying a critical R-value

Intro

Howdy!  I’m Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to identify critical R values.  Here's our problem statement: For a data set of brain volumes and IQ scores of six males, the linear correlation coefficient is R = 0.568.  Use the table available below to find the critical values of R based on a comparison of the linear correlation coefficient R and the critical values.  What do you conclude about a linear correlation?

Part 1

OK, the first part asks us to identify the critical values.  You notice that there are two critical values.  If I click here on this table, I'm only seeing one of the critical values.  OK, there are actually two critical values; one is positive and one is negative.  Only the positive one is listed here.

So the next question we have to ask then is to identify which critical value of R we want?  First, we need to look at the number of pairs of data that we have.  So in our data set it says that we have six males.  So therefore, I know I'm going to be looking at 6 here on the chart, and then come over to the right, and this value right here — 0.818 — is the critical value R for a data set with six pairs of data.

This then becomes a threshold value that we compare with the R value obtained from the actual data.  If the R value from the actual data is greater than this critical value, then we're in the clear, and we can say that we have we have enough evidence to conclude there's a linear correlation.  However, if the value we get for our from the data is less than this threshold value, then we don't have that evidence; we can't conclude that there is a linear correlation.  So it's pretty easy if you're just comparing absolute values of everything.  So let's go ahead and make that comparison here.  The R-value we get from our data is 0.568.  That is less than 0.811, so therefore we can't make that conclusion of linear correlation.

However, this first part is asking us for the critical values.  So we're gonna put those in here.  Remember there are two critical values; one is positive, one is negative.  There's two ways to input the answer.  I just like coming down here and pressing on this plus-or-minus button and then putting in the value that I got from the table.  But if you want you could actually list out the two numbers and separate them out with a comma.  So you could have -0.811, comma, and then 0.811 for the positive.  But I like this because it's just a little bit shorter.  Nice work!

Part 2

And now the second part of our problem is asking us to interpret those critical values with respect to our data set.  So here we're saying that the correlation coefficient R and here they're actually relating it to a distribution, which we'll get into a lot more later on in the course.  But which one do we actually choose?  Well, since the R-value that we got from our data is less than the critical value for the positive — OK?  We're just comparing positives to each other, OK?  So that means we're gonna select “between the critical values.”  If this were a negative value and the R-value from the data set was greater than — well, greater absolute value; it's more negative than the negative, or it's more positive than the positive critical value — then we would say that it is below the negative or above the positive.  But that's not what we have here, so we're just going to select “between the critical values” and because we're in that region there is not sufficient evidence to support the claim of a linear correlation.

If we were in, say, one of those tail regions where we're much less than the negative or greater than the positive critical value, then, yes, we would be able to say that.  But as it is, this is what we have.  So I check my answer.  Good job!

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