Testing for a Linear Correlation
In Exercises 13–28, con...

Question

Testing for a Linear Correlation

In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of α = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

Taxis Using the data from Exercise 15, is there sufficient evidence to support the claim that there is a linear correlation between the distance of the ride and the tip amount? Does it appear that riders base their tips on the distance of the ride?

Accepted Answer

All right, hello, everyone. So this question says, a study investigates whether there is a linear correlation between the number of cups of coffee consumed per day and the number of hours slept per night among college students. The data collected in the corresponding scatter plot are as follows. Calculate the value of the linear correlation coefficient R and determine the critical values of R at a significant level of alpha equals 0.05. Is there sufficient evidence to support the claim that there is a linear correlation? Between cups of coffee an hour slept. All right, so first, let's begin with that linear correlation coefficient. Now, R is equal to And multiplied by the sum of XY, subtracted by the sum of X multiplied by the sum of Y. And this is all divided by the square root of N multiplied by the sum of X2. Multiplied or excuse me, subtracted by the sum of all X values squared. This is then multiplied by the square root of N. Multiplied by the square root of all Y's squared. And subtracted by the square root of all. This excuse me, the sum of all values of Y. And squared All right, so in order to go ahead and tackle a formula like this one, you're going to have to reference the data points previously given to you. Now, each one of these data points can be treated as an XY pair. Specifically, the number of cups of coffee corresponds to the X axis, that's your X value. And the, the number of hours slept corresponds to your Y value. That's your Y axis. So after taking each data point. And establishing the X and Y of each pair. First, you're going to square every single X value. And also square every y value. So just to clarify, that would be XY. X squared Y squared and then you're also going to multiply. Both X and Y for each pair. This should give you a number of values. And eventually you're going to have to find the sum of all of them. That is the sum of all X values, the sum of all Y values, the sum of all of the squared X values, the sum of all the squared Y values, and the sum of the products of X and Y for each pair. And so after you calculate that data. The result is as follows. The sum of all X's is equal to 16. The sum of all Y is equal to 50. The sum of all of the squared X values is equal to 44. The sum of all squared Y values is equal to 342. And the sum of all XY values X multiplied by Y values equals 106. And the reason why this data is necessary is because this is what you're going to plug into the expression for the linear correlation coefficient. So let's do that. Let's go ahead and solve for art. So first, Recall that N happens to be your sample size. And in this particular case, we have 8. College students. So for your numerator, that's going to be 8. Multiplied by 106. And then subtracted by By 16 multiplied by 50. So that completes your numerator. Now, your denominator. Is equal to 8 multiplied by 44. And then subtracted by. 16 squared. Because notice how for the second term here of the 1st square root in the denominator. The sum of all X values has the power of 2 outside of parentheses, which means that you plug in 16, which is the sum of all X values, and then square 16. So now moving on to the 2nd square root, that's 8. Multiplied by 342. Subtracted by 50 squared. And evaluating this expression equals approximately 0.319. And so now that we have the R value, let's talk about the requirements necessary. For assessing a linear correlation. First and foremost, when considering the scatter plot, the first thing you want to note is whether or not the scatter plot follows a straight lined pattern. And the answer in this case to that question happens to be no. Because notice how the data points themselves are scattered throughout the plot and not following any particular shape. Additionally, you want to ask yourself if there are any outliers present. In this case, there are no outliers, but nonetheless, the data does not form a linear pattern in the scatter plot. So, truthfully, the calculated value of R. Would not be meaningful for assessing a linear correlation. Rather, the critical value of R is used to test the significance of a linear correlation. So if the relationship itself is simply not linear, the test is not appropriate. Which now brings us to our final answer, which I. Which I thought I had pre-written in advance, but I actually didn't, so I went ahead and paused the recording to do so. All right, so now, the answer is as follows, right? First of all, R is approximately 0.319. And so because there's no linear correlation. Noted in the scatter plot, we would say that there is insufficient evidence to support the claim. That there is a linear correlation between cups of coffee and hours slept. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

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Key Concepts

Linear Correlation Coefficient (r)

P-value

Scatterplot

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