Ch. 9 - Correlation and Regression

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 9 - Correlation and Regression

Problem 9.2.4

Chapter 9, Problem 9.2.4

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

Verified step by step guidance

Understand that the regression line is a model that predicts values of y based on values of x within the range of observed data.

Identify the range of x-values in the original dataset used to create the regression line; these are the minimum and maximum x-values observed.

Recognize that meaningful predictions for y are typically made only for x-values within this observed range, as predictions outside this range involve extrapolation and may be unreliable.

Note that predictions for x-values far outside the observed range can lead to inaccurate or misleading results because the relationship modeled by the regression may not hold beyond the data.

Therefore, the meaningful x-values for prediction are those within the interval from the minimum to the maximum observed x-values in the dataset.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Regression Line and Prediction

A regression line models the relationship between an independent variable x and a dependent variable y, allowing predictions of y based on x values. It summarizes the trend in the data and is used to estimate y for given x inputs.

Range of Observed Data (Domain of x)

Meaningful predictions are typically limited to x values within the range of the observed data used to create the regression line. Predictions outside this range, called extrapolation, can be unreliable because the relationship may not hold beyond the observed domain.

Extrapolation vs. Interpolation

Interpolation refers to predicting y for x values within the observed data range, which is generally reliable. Extrapolation involves predicting y for x values outside this range and is less reliable because the model’s assumptions may not apply beyond the data.

Recommended video:

Guided course

04:39

Visualizing Qualitative vs. Quantitative Data

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

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

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 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."

views

Textbook Question

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

Textbook Question

1. Interpret the meaning of the coefficient -8.2 in the multiple regression equation y=112.1+0.43x_1-8.2x_2+29.5x_3.