Ch. 9 - Correlation and Regression

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 9 - Correlation and Regression

Problem 9.2.1

Chapter 9, Problem 9.2.1

1. What is a residual? Explain when a residual is positive, negative, and zero.

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Understand that a residual is the difference between an observed value and the predicted value from a regression model. Mathematically, it is expressed as

e = y - y

, where

y

is the observed value and

y

is the predicted value.

Recognize that a residual measures the error or deviation of the prediction from the actual data point, helping to assess the accuracy of the regression model.

A residual is positive when the observed value is greater than the predicted value, meaning the model underestimates the actual data point.

A residual is negative when the observed value is less than the predicted value, indicating the model overestimates the actual data point.

A residual is zero when the observed value exactly equals the predicted value, showing a perfect prediction for that data point.

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Key Concepts

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

Residual

A residual is the difference between an observed value and the predicted value from a regression model. It measures the error or deviation of the prediction from the actual data point, indicating how well the model fits the data.

Positive Residual

A residual is positive when the observed value is greater than the predicted value. This means the model underestimates the actual data point, and the error is above the regression line.

Negative and Zero Residuals

A residual is negative when the observed value is less than the predicted value, indicating the model overestimates the data point. A residual is zero when the observed and predicted values are equal, meaning the model perfectly predicts that data point.

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Residuals and Residual Plots

Related Practice

Textbook Question

"Predicting y-Values In Exercises 3-6, use the multiple regression equation to predict the y-values for the values of the independent variables.

3. Cauliflower Yield The equation used to predict the annual cauliflower yield (in pounds

per acre) is y=24,791+4.508x_1-4.723x_2

where x_1 is the number of acres planted and x_2 is the number of acres harvested.(Adapted from United States Department of Agriculture)

a. x_1 = 36,500, x_2 = 36,100

b. x_1 = 38,100, x_2 = 37,800

c. x_1 = 39,000, x_2 = 38,800

d. x_1 = 42,200, x_2 = 42,100"

Textbook Question

"[APPLET] Registered Nurse Salaries In Exercises 27–30, use the table, which shows the years of experience of 14 registered nurses and their annual salaries (in thousands of dollars). (Adapted from Payscale, Inc.)

27. Correlation Using the scatter plot of the registered nurse salary data shown below, what type of correlation, if any, do you think the data have? Explain.

Textbook Question

In Exercise 25, remove the data for the international soccer player with a maximum weight of 170 kilograms and a jump height of 64 centimeters. Describe how this affects the correlation coefficient r.

Textbook Question

"In Exercises 9 and 10, identify the explanatory variable and the response variable.

9. A nutritionist wants to determine whether the amounts of water consumed each day by persons of the same weight and on the same diet can be used to predict individual weight

loss."

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Textbook Question

4. For a set of data and a corresponding regression line, describe all values of x that provide meaningful predictions for y.

Textbook Question

2. Two variables have a positive linear correlation. Is the slope of the regression line for the variables positive or negative?