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 10.2.14

Chapter 10, Problem 10.2.14

Regression and Predictions
Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1.

Find the regression equation, letting the first variable be the predictor (x) variable.
Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5.

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. (Jackpot amounts are in millions of dollars, ticket sales are in millions.) Find the best predicted number of tickets sold when the jackpot was actually 345 million dollars. How does the result compare to the value of 55 million tickets that were actually sold?

Verified step by step guidance

Step 1: Organize the data into two variables: the predictor variable (x), which is the jackpot amounts, and the response variable (y), which is the number of tickets sold.

Step 2: Calculate the mean and standard deviation for both the x (jackpot amounts) and y (tickets sold) variables. These values are needed to compute the regression equation.

Step 3: Compute the correlation coefficient (r) using the formula: r = (Σ((x_i - x̄)(y_i - ȳ))) / (sqrt(Σ(x_i - x̄)^2) * sqrt(Σ(y_i - ȳ)^2)). This measures the strength and direction of the linear relationship between x and y.

Step 4: Use the formula for the slope (b) of the regression line: b = r * (s_y / s_x), where s_y and s_x are the standard deviations of y and x, respectively. Then calculate the y-intercept (a) using the formula: a = ȳ - b * x̄.

Step 5: Substitute the given jackpot value (345 million dollars) into the regression equation y = a + b * x to predict the number of tickets sold. Compare the predicted value to the actual value of 55 million tickets sold.

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Key Concepts

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

Regression Equation

A regression equation is a mathematical representation that describes the relationship between a dependent variable and one or more independent variables. In this context, the jackpot amount serves as the independent variable (x), while the number of tickets sold is the dependent variable (y). The equation typically takes the form y = mx + b, where m is the slope and b is the y-intercept, allowing predictions of y based on given x values.

Predictor Variable

The predictor variable, also known as the independent variable, is the variable that is manipulated or controlled to observe its effect on another variable. In this scenario, the jackpot amount is the predictor variable, as it is used to predict the number of tickets sold. Understanding the role of the predictor variable is crucial for establishing the direction and strength of the relationship in regression analysis.

Predicted Value

A predicted value is the outcome generated by a regression equation when a specific value of the predictor variable is inputted. In this case, to find the predicted number of tickets sold when the jackpot is 345 million dollars, one would substitute this value into the regression equation. Comparing this predicted value to the actual number of tickets sold provides insights into the accuracy of the model and the relationship between the variables.

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Related Practice

Textbook Question

Finding the Best Model

In Exercises 5–16, construct a scatterplot and identify the mathematical model that best fits the given data. Assume that the model is to be used only for the scope of the given data, and consider only linear, quadratic, logarithmic, exponential, and power models.

Richter Scale The table lists different amounts (metric tons) of the explosive TNT and the corresponding value measured on the Richter scale resulting from explosions of the TNT.

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Textbook Question

Large Data Sets

Exercises 29–32 use the same Appendix B data sets as Exercises 29–32 in Section 10-1. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted values following the prediction procedure summarized in Figure 10-5.

Taxis Repeat Exercise 15 using all of the time/tip data from the 703 taxi rides listed in Data Set 32 “Taxis” from Appendix B.

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Textbook Question

Response and Predictor Variables Using all of the Tour de France bicycle race results up to a recent year, we get this multiple regression equation: Speed = 29.2-0.00260Distance + 0.540Stages + 0.0570Finishers, where Speed is the mean speed of the winner (km/h), Distance is the length of the race (km), Stages is the number of stages in the race, and Finishers is the number of bicyclists who finished the race. Identify the response and predictor variables.

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Textbook Question

Regression and Predictions

Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1.

Find the regression equation, letting the first variable be the predictor (x) variable.

Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5.

Subway and the CPI Use the subway/CPI data from the preceding exercise. What is the best predicted value of the CPI when the subway fare is \$3.00?

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Textbook Question

Finding the Equation of the Regression Line

In Exercises 9 and 10, use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.

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Textbook Question

Standard Error of Estimate A random sample of 118 different female statistics students is obtained and their weights are measured in kilograms and in pounds. Using the 118 paired weights (weight in kg, weight in lb), what is the value of se? For a female statistics student who weighs 100 lb, the predicted weight in kilograms is 45.4 kg. What is the 95% prediction interval?