Ch. 10 - Correlation and Regression

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 10 - Correlation and Regression

Problem 10.1.10d

Chapter 10, Problem 10.1.10d

Clusters Refer to the Minitab-generated scatterplot. The four points in the lower left corner are measurements from women, and the four points in the upper right corner are from men.

Find the value of the linear correlation coefficient using all eight points. What does that value suggest about the relationship between x and y?

Verified step by step guidance

Step 1: Understand the problem. The scatterplot shows two distinct clusters of points: one in the lower left corner (representing measurements from women) and one in the upper right corner (representing measurements from men). The task is to calculate the linear correlation coefficient (r) for all eight points and interpret its meaning.

Step 2: Recall the formula for the linear correlation coefficient (r): r = (Σ((x_i - x̄)(y_i - ȳ))) / sqrt(Σ(x_i - x̄)^2 * Σ(y_i - ȳ)^2). Here, x̄ and ȳ are the means of the x and y values, respectively, and x_i and y_i are the individual data points.

Step 3: Calculate the mean of the x-values (x̄) and the mean of the y-values (ȳ). To do this, sum all x-values and divide by the number of points (8), and repeat the process for the y-values.

Step 4: Compute the deviations from the mean for each x and y value (x_i - x̄ and y_i - ȳ). Then, calculate the product of these deviations for each point and sum them to find Σ((x_i - x̄)(y_i - ȳ)).

Step 5: Calculate the squared deviations for x and y (Σ(x_i - x̄)^2 and Σ(y_i - ȳ)^2). Use these values to compute the denominator of the formula. Finally, divide the numerator by the denominator to find the correlation coefficient (r). Interpret the result: if r is close to 0, it suggests no linear relationship; if r is close to 1 or -1, it suggests a strong positive or negative linear relationship, respectively.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linear Correlation Coefficient

The linear correlation coefficient, often denoted as 'r', quantifies the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1, where -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no correlation. In the context of the scatterplot, calculating 'r' helps determine how closely the data points cluster around a straight line.

Scatterplot

A scatterplot is a graphical representation of two quantitative variables, where each point represents an observation. The position of each point on the horizontal and vertical axes indicates the values of the two variables. In this case, the scatterplot shows two distinct clusters of points, which suggests a potential relationship between the variables, and analyzing these clusters can provide insights into the correlation.

Clusters in Data

Clusters in data refer to groups of data points that are closely positioned together in a scatterplot, indicating similar values for the variables being analyzed. In this scenario, the points representing women and men form two separate clusters, suggesting that there may be different relationships or characteristics between the two groups. Understanding these clusters is essential for interpreting the correlation coefficient and the overall relationship between the variables.

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Related Practice

Textbook Question

Comparing Two Means Treating the data as samples from larger populations, test the claim that there is a significant difference between the mean of presidents and the mean of popes.

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Textbook Question

Notation Using the weights (lb) and highway fuel consumption amounts (mi/gal) of the 48 cars listed in Data Set 35 “Car Data” of Appendix B, we get this regression equation:

y^ = 58.9 - 0.00749x, where x represents weight.

c. What is the predictor variable?

Textbook Question

Notation The author conducted an experiment in which the height of each student was measured in centimeters and those heights were matched with the same students’ scores on the first statistics test.

c. Does r change if the heights are converted from centimeters to inches?

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Textbook Question

Exercises 1–10 are based on the following sample data consisting of costs of dinner (dollars) and the amounts of tips (dollars) left by diners. The data were collected by students of the author.

Predictions Repeat the preceding exercise assuming that the linear correlation coefficient is r = 0.132.

Textbook Question

Sum of Squares Criterion In addition to the value of another measurement used to assess the quality of a model is the sum of squares of the residuals. Recall from Section 10-2 that a residual is (the difference between an observed y value and the value predicted from the model). Better models have smaller sums of squares. Refer to the U.S. population data in Table 10-7.

c. Verify that according to the sum of squares criterion, the quadratic model is better than the linear model.

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Textbook 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 in the last column. Is there sufficient evidence to conclude that there is a linear correlation between lottery jackpot amounts 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.

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