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 9.2.6

Chapter 9, Problem 9.2.6

6. Why is it not appropriate to use a regression line to predict y-values for x-values that are not in (or close to) the range of x-values found in the data?

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Understand the concept of extrapolation: Using a regression line to predict y-values for x-values outside the observed data range is called extrapolation.

Recognize that the regression line is based on the relationship observed within the range of the data, so predictions outside this range assume the same pattern continues, which may not be true.

Consider that the behavior of the variables outside the observed range can be different due to factors not captured in the data, leading to unreliable or misleading predictions.

Note that the uncertainty of predictions increases as you move further away from the range of observed x-values, making the regression line less trustworthy for those points.

Therefore, it is generally inappropriate to use the regression line for x-values far from the data range because the model's assumptions and accuracy are not guaranteed beyond the observed data.

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Key Concepts

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

Extrapolation

Extrapolation involves using a regression model to predict values outside the range of observed data. It is risky because the relationship established within the data range may not hold beyond it, leading to unreliable or misleading predictions.

Range of Data (Domain of Predictor Variable)

The range of data refers to the span of x-values used to fit the regression line. Predictions are most reliable within this range since the model is based on observed patterns; values far outside this range lack supporting data and increase uncertainty.

Assumptions of Linear Regression

Linear regression assumes a consistent linear relationship between variables within the data range. When predicting outside this range, these assumptions may fail, as the relationship could change, making the model's predictions invalid or inaccurate.

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Textbook Question

"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.

29. New Vehicle Sales Construct a 95% prediction interval for new vehicle sales for General Motors in Exercise 19 when the number of new vehicles sold by Ford is 2028 thousand."

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Textbook Question

2. Compare the numbers of dependent and independent variables in a multiple regression equation and a single regression equation.

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Textbook Question

What is the coefficient of determination for two variables that have perfect positive linear correlation or perfect negative linear correlation? Interpret your answer.

112

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Textbook Question

The coefficient of determination r^2 is the ratio of which two types of variations? What does r^2 measure? What does 1 - r^2 measure?

147

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Textbook Question

"Old Vehicles In Exercises 31–34, use the figure shown at the left.

Error of Estimate Find the standard error of estimate Se and interpret the results."

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Textbook Question

"Graphical Analysis In Exercises 1–3, use the figure.

Describe the unexplained variation about a regression line in words and in symbols."

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