Ch. 10 - Correlation and Regression

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 10 - Correlation and Regression

Problem 10.q.9

Chapter 10, Problem 10.q.9

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.

Verified step by step guidance

Step 1: Understand the problem. The goal is to analyze the relationship between the cost of dinner and the tip amount using the given data, assuming the linear correlation coefficient r = 0.132. This indicates a weak positive linear relationship between the two variables.

Step 2: Organize the data. The table provides pairs of values for the cost of dinner (independent variable, x) and the tip amount (dependent variable, y). These pairs will be used to calculate predictions based on the weak correlation.

Step 3: Use the formula for the equation of a regression line: y = mx + b, where m is the slope and b is the y-intercept. The slope m can be calculated using the formula: m = r * (sy / sx), where sy is the standard deviation of the tip amounts and sx is the standard deviation of the dinner costs.

Step 4: Calculate the y-intercept b using the formula: b = ȳ - m * x̄, where ȳ is the mean of the tip amounts and x̄ is the mean of the dinner costs. This will allow you to complete the regression equation.

Step 5: Use the regression equation to make predictions for the tip amounts based on the given dinner costs. Substitute each dinner cost into the equation y = mx + b to estimate the corresponding tip amount.

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Key Concepts

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

Linear Correlation Coefficient (r)

The linear correlation coefficient, 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 positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation. In this context, an r value of 0.132 suggests a weak positive correlation between the cost of dinner and the amount of tip left.

Scatter Plot

A scatter plot is a graphical representation that displays the relationship between two quantitative variables. Each point on the plot corresponds to an observation in the dataset, with one variable plotted along the x-axis and the other along the y-axis. Analyzing the scatter plot can help visualize the correlation and identify any trends or patterns in the data, which is essential for interpreting the correlation coefficient.

Regression Analysis

Regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In this case, it could be used to predict the amount of tip based on the cost of dinner. The results of regression analysis can provide insights into how changes in the cost of dinner might affect the tips, allowing for better predictions and understanding of the data.

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?

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Textbook Question

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?

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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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