Ch. 10 - Correlation and Regression

Triola - Elementary Statistics 14th Edition

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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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Textbook Question

Super Bowl and R^2 Let x represent years coded as 1,1,3,... for years starting in 1980, and let y represent the numbers of points scored in each annual Super Bowl beginning in 1980. Using the data from 1980 to the last Super Bowl at the time of this writing, we obtain the following values of R^2 for the different models: linear: 0.008; quadratic: 0.023; logarithmic: 0.0004; exponential: 0.027; power: 0.007. Based on these results, which model is best? Is the best model a good model? What do the results suggest about predicting the number of points scored in a future Super Bowl game?

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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.

CD Yields The table lists the value y (in dollars) of \$1000 deposited in a certificate of deposit at Bank of New York (based on rates currently in effect).

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Textbook Question

Moore’s Law In 1965, Intel cofounder Gordon Moore initiated what has since become known as Moore’s law: The number of transistors per square inch on integrated circuits will double approximately every 18 months. In the table below, the first row lists different years and the second row lists the number of transistors (in thousands) for different years.

Ignoring the listed data and assuming that Moore’s law is correct and transistors per square inch double every 18 months, which mathematical model best describes this law: linear, quadratic, logarithmic, exponential, power? What specific function describes Moore’s law?

Which mathematical model best fits the listed sample data?

Compare the results from parts (a) and (b). Does Moore’s law appear to be working reasonably well?

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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 16 using all of the distance/tip data from the 703 taxi rides listed in Data Set 32 “Taxis” from Appendix B.

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Textbook Question

Interpreting the Coefficient of Determination

In Exercises 5–8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Times of Taxi Rides and Tips r = 0.298 (x = time in minutes, y = the amount of tip in dollars)

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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.

Oscars Listed below are ages of recent Oscar winners matched by the years in which the awards were won (from Data Set 21 “Oscar Winner Age” in Appendix B). Find the best predicted age of an Oscar-winning actress given that the Oscar winner for best actor is 59 years of age. How does the result compare to the actual actress age of 60 years?

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