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Chapter 5: Problem 32

Explain why it can be dangerous to use the leastsquares line to obtain predictions for \(x\) values that are substantially larger or smaller than those contained in the sample.

Short Answer

Expert verified
Using the least squares line to make predictions for \(x\) values significantly larger or smaller than those in the sample is risky because the relationship between the variables being modelled may not be the same outside the range of the sample data. Issues such as outliers and changes in variable relationships can lead to incorrect predictions when extrapolating.

Step by step solution

01

Understanding Extrapolation

Extrapolation is the process of estimating, beyond the original observation range, the value of a variable based on its relationship with another variable. In the context of this problem, extrapolation will be trying to use the least squares regression line to predict values of \(x\) that are outside the range in the data set used to create the model. It's akin to making future forecasts or retrospective estimations when no data are available.
02

Risks of Extrapolation

Extrapolating may seem like a good idea while trying to make predictions where data is not available, but it is risky. This is because the relationship between the variables could change or be different for values of \(x\) when you go outside the sample's range. The least squares line is the best fit for the given sample, but this does not guarantee that it will continue to be the best fit outside the range of sampled data.
03

Anomaly of Outliers

It is to be noted that extreme values of \(x\) in the sample, also known as outliers, can greatly affect the least squares line. Predicting outcomes for \(x\) values that are substantially larger or smaller than those in the sample can lead to inaccurate predictions and should be avoided. Using such predictions may lead to unsound decisions and inappropriate conclusions.

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Most popular questions from this chapter

The sales manager of a large company selected a random sample of \(n=10\) salespeople and determined for each one the values of \(x=\) years of sales experience and \(y=\) annual sales (in thousands of dollars). A scatterplot of the resulting \((x, y)\) pairs showed a marked linear pattern. a. Suppose that the sample correlation coefficient is \(r=\) \(.75\) and that the average annual sales is \(\bar{y}=100\). If a particular salesperson is 2 standard deviations above the mean in terms of experience, what would you predict for that person's annual sales? b. If a particular person whose sales experience is \(1.5\) standard deviations below the average experience is predicted to have an annual sales value that is 1 standard deviation below the average annual sales, what is the value of \(r ?\)

For each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to \(0 .\) Explain your choice. a. Maximum daily temperature and cooling costs b. Interest rate and number of loan applications c. Incomes of husbands and wives when both have full- time jobs d. Height and IQ e. Height and shoe size f. Score on the math section of the SAT exam and score on the verbal section of the same test g. Time spent on homework and time spent watching television during the same day by elementary school children h. Amount of fertilizer used per acre and crop yield (Hint: As the amount of fertilizer is increased, yield tends to increase for a while but then tends to start decreasing.)

A sample of 548 ethnically diverse students from Massachusetts were followed over a 19 -month period from 1995 and 1997 in a study of the relationship between TV viewing and eating habits (Pediatrics [2003]: \(1321-\) 1326). For each additional hour of television viewed per day, the number of fruit and vegetable servings per day was found to decrease on average by \(0.14\) serving. a. For this study, what is the dependent variable? What is the predictor variable? b. Would the least-squares line for predicting number of servings of fruits and vegetables using number of hours spent watching TV as a predictor have a positive or negative slope? Explain.

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