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.)

Powerball Jackpots and Tickets Sold Listed below are the same data from Table 10-1 in the Chapter Problem, but an additional pair of values has been added from actual Powerball results. Is there sufficient evidence to conclude that there is a linear correlation between lottery jackpots and numbers of tickets sold? Comment on the effect of the added pair of values in the last column. Compare the results to those obtained in Example 4.

Table showing Powerball jackpots and tickets sold with values: 334, 127, 300, 54, 16, 41, etc.

Accepted Answer

An ecologist records the average daily temperature in degrees Celsius and the number of bird species observed in 7 different locations. We have a set of points. At a significance level of alpha equals 0.05, is there evidence of a linear correlation between temperature and the number of bird species using the critical value approach? I've also provided a table here to help us find our values. Now let's first find a hypothesis. We have our own hypothesis. Where our correlation value row. This equals a 0. Other words, no. Correlation We also have our alternative. Where Ro It's not equal to 0. Or Correlation exists. In particular linear correlation. Now to solve this, we will first find. The Pearson correlation coefficient, which is given by R equals sample size, multiplied. By the sum Of X multiplied by Y. Minus the products of the sum of X and the sum of Y. This is divided by Square root Of sample size, multiplied. By the sum Of X 2 Minus The sum of X squared. All multiplied By a sample size multiplied by the sum of Y2. Minus the sum. Of Y itself squared. Now we can plug in some values. From the table we have here. We have The sum in X, which is the total of our X column. Which is 105. The sum of way The sum of our Y column, which is 50. Sum of XY, it's a sum of XY color. Which is 799 Then we have the sum of our X2 column. Which in this case, It's 1645. And the sum of our y squared column. Which is 392. Sample size is also equals to 7. Let's plug in all of these values into our equation now. We get 7 multiplied. By 799 -105, multiplied by 50. Divided by the square root. Of 7 Multiply. By 1645 minus. 105 squared All of that is multiplied. By 7 multiplied by 392. Minus 50 squared We can simplify this in a calculator. This gives 0.992. As our correlation coefficient. No. We'll use our tests. We need a critical value. Critical value Depends on our degrees of freedom. And our significance level. Because of freedom will be sample size minus 2, which is 7 minus 2 or 5. Significance level, it tells us, is 0.05. So now, from the table, we can find the R critical value. Are critical will be in our case, approximately + or minus 0.754. We'll get a plus or minus because this is a two-tailed test. And we'll compare this value to our value for R. We have the absolute value of our. Is greater than Are critical in this case, 0.992 is greater than. 0.754. Because our value R is greater than our critical value, we will reject. Our hypothesis. And we will say yes, there is a correlation between our two variables. In particular? The coefficient Exceeds A critical value. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Key Concepts

Linear Correlation Coefficient (r)

Scatterplot

P-value and Significance Level

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