Ch. 10 - Correlation and Regression

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 10 - Correlation and Regression

Problem 10.CQQ.3

Chapter 10, Problem 10.CQQ.3

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.
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Fixed Percentage If a restaurant were to change its tipping policy so that a constant tip of 20% of the bill is added to the cost of the dinner, what would be the value of the linear correlation coefficient for the paired amounts of dinners/tips?

Verified step by step guidance

Understand the problem: We need to find the linear correlation coefficient for a fixed percentage tip policy, where the tip is 20% of the dinner cost.

Recognize that when the tip is a fixed percentage of the dinner cost, the relationship between dinner cost and tip is perfectly linear. This means that the tip is directly proportional to the dinner cost.

Recall that the linear correlation coefficient, denoted as $ r $, measures the strength and direction of a linear relationship between two variables. When one variable is a constant multiple of another, the correlation coefficient is either +1 or -1, depending on the direction of the relationship.

Since the tip is 20% of the dinner cost, the relationship is positive and direct. Therefore, the correlation coefficient $ r $ will be +1, indicating a perfect positive linear relationship.

Conclude that the value of the linear correlation coefficient for the paired amounts of dinners/tips, under a fixed 20% tipping policy, is +1.

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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', measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where values close to 1 or -1 indicate a strong linear relationship, and values near 0 suggest a weak linear relationship. In this context, it helps determine how well the tips correlate with the dinner costs.

Fixed Percentage Tipping

Fixed percentage tipping involves adding a constant percentage of the bill as a tip, regardless of the bill's total amount. In this scenario, a 20% tip is added to each dinner cost, creating a direct proportional relationship between the dinner cost and the tip amount. This proportionality implies that the correlation coefficient would be perfect, as the tip is a fixed percentage of the dinner cost.

Proportional Relationships

A proportional relationship is one where two quantities increase or decrease at the same rate, maintaining a constant ratio. In the case of a fixed percentage tip, the tip amount is always a constant fraction of the dinner cost, leading to a perfect linear relationship. This means the correlation coefficient would be exactly 1, indicating a perfect positive linear relationship.

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Related Practice

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.

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Scatterplot Construct a scatterplot and comment on the pattern of points.

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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.

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Change in Scale Exercise 1 stated that for the given paired data, r = 0.846. How does that value change if all of the amounts of dinners are left unchanged but all of the tips are expressed in cents instead of dollars?

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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 The sample data result in a linear correlation coefficient of r = 0.846 and the regression equation y^ = -0.00777 + 0.145x. What is the best predicted amount of tip, given that the cost of dinner was \$84.62? How was the predicted value found?

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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. If we find that r = 0, does that indicate that there is no association between those two variables?

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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 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.

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Textbook Question

Casino Size and Revenue Use the same paired data from the preceding exercise.

b. What is the best predicted amount of revenue for a casino with a size of 200 thousand square feet? Is it likely that the best predicted amount of revenue will be accurate?

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