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 fare (cost of the ride)?

Accepted Answer

All right, hello, everyone. So this this question says, a botanist records the amount of fertilizer in grams used, and the resulting height in centimeters of tomato plants after 2 months. The data and 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 enough evidence to support a claim that there is a linear correlation between the amount of fertilizer and plant height? All right, so before we go about talking about the, the calculation rather for the linear correlation coefficient, let's focus on the data we already have. For this experiment, the dependent variable was the amount of fertilizer. Excuse me, that was the independent variable, and the dependent variable was the height of the plant itself. So the amount of fertilizer represents X and the height represents Y. Now, the reason why I bring that up is because there's additional information that needs to be calculated for the correlation coefficient. So for each data point, right? First, you want to take each value for X and square it. And then you want to take each value for Y and square that as well. And then, for every XY pair, you want to multiply both X and Y together. Now it's important that you do this for all of the data points given because you want to then calculate the sum of all of these values. So you want the sum of all X values, the sum of all Y's, the sum of all X values squared, and y value squared, and then the sum of the products for all XY pairs. And so once you've calculated all of these values, the sums are as follows. The sum of all X's is 220. The sum of all wise is 260. The sum of all X's squared is 7100. The sum of all Y squared is. 8,818. And the sum of all XY pairs multiplied together is 7,765. So now this leads me to the correlation coefficient itself. So R is equal to N, which is the sample size, multiplied by the sum of all Xy's. Subtracted by The sum of all X's multiplied by the sum of all y's. This is then divided by the square root of N equals Or excuse me, this and multiplied by the sum of all X's squared. Subtracted by the sum of all x values squared. Notice how for the second term. The power of 2 is outside of the parenthesis. So what you're plugging in is the sum of all X values. This is then multiplied by the square root of N multiplied by the sum of all Y squared, subtracted by the sum of all Y values, then squared. Notice again how the power of two is outside of parenthesis. All right, so now when you plug in the information that you have, R is equal to 8 multiplied by 7,765, subtracted by 220 multiplied by 260. This is then divided by the square root of 8 multiplied by 7100, subtracted by 220 squared. Multiplied by the square root of 8, multiplied by 8,818, subtracted by 260 squad. And then evaluating this expression. Gives you an R value of 0.989. That's your correlation coefficient. So now, can you conclude that there is a linear correlation? First, let's check our requirements. Right, recall that here we are to assume that the data is a simple random sample. And we can see from the scatter plot that it follows approximately a straight line, with no outliers present. So that means we can proceed to find the critical values. So for the correlation coefficient at alpha equals 0.05, for a two-tailed test, recall that we have to find the corresponding degrees of freedom. In this case, our sample size was 8, so the degrees of freedom. is equal to and subtracted by 2, which gives you 6. So from the table The critical values for R are positive and negative, 0.707. So now, recall that if the absolute value of our calculated R is greater than or equal to the critical value, you can conclude that there is sufficient evidence to support a linear correlation. So in this case, our calculated R was 0.989, which is in fact greater than the critical value. Which means that there is enough evidence to support the claim of a linear correlation. So now this leads us to our final answer. So R was equal to approximately 0.989, and the critical values were equal to positive and negative 0.707. Altogether there is sufficient evidence to support the claim. That there is a linear correlation between the amount of fertilizer and plant height. And there you have it. So with that being said, thank you so very much for watching, and I hope we found this helpful.

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Key Concepts

Linear Correlation Coefficient (r)

P-value

Scatterplot

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