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Chapter 3: Q.44 (page 191)

Least-squares ideaTrace the graph from Exercise 40on your paper. Show why the line drawn on the plot is called the least-squares line.

Short Answer

Expert verified

The least-squares regression line minimizes the squared prediction errors and no other line would show a lesser sum of squared residuals, which is why it is called the least-squares line.

Step by step solution

01

Given Information

We need to explain why the line drawn on the plot is called the least-squares line.

02

Explanation

Figure shows a scatter plot of the data with the regression line added as the least-squares regression line of yon x.

With the regression line added, the scatter plot is shown in the figure.

As you can see from the figure, the least-squares regression line does not deviate greatly from scattered points.

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Most popular questions from this chapter

Measurements on young children in Mumbai, India, found this least-squares line for predicting height y from the arm span x y⏜=6.4+0.93xMeasurements are in centimeters (cm).

In addition to the regression line, the report on the

Mumbai measurements says that r2=0.95.This suggests that

(a) although arm span and height are correlated, arm span does not predict height very accurately.

(b) height increases by 0.95=0.97cm for each additional centimeter of arm span.

(c) 95% of the relationship between height and arm span is accounted for by the regression line.

(d) 95% of the variation in height is accounted for by the regression line.

(e) 95% of the height measurements are accounted for by the regression line.

The graph at the top right plots the gas mileage (miles per gallon) of various cars from the same model year versus the weight of these cars in thousands of pounds. The points marked with red dots correspond to cars made in Japan. From this plot, we may conclude that

(a) there is a positive association between weight and gas mileage for Japanese cars.

(b) the correlation between weight and gas mileage for all the cars is close to 1.

(c) there is little difference between Japanese cars and cars made in other countries.

(d) Japanese cars tend to be lighter in weight than other cars.

(e) Japanese cars tend to get worse gas mileage than other cars.

What form does the relationship take? Why are there two clusters of points?

Nahya infant weights A study of nutrition in developing countries collected data from the Egyptian

village of Nahya. Here are the mean weights (in kilograms) for 170 infants in Nahya who were

weighed each month during their first year of life:

A hasty user of statistics enters the data into software and computes the least-squares line without plotting the data. The result is weight=4.88+0.267(age) A residual plot is shown below. Would it be appropriate to use this regression line to predict y from x? Justify your answer.

Merlins breeding The percent of an animal species in the wild that survives to breed again is often lower following a successful breeding season. A study of merlins (small falcons) in northern Sweden observed the number of breeding pairs in an isolated area and the percent of males (banded for identification) that returned the next breeding season. Here are data for nine years:

Investigate the relationship between breeding pairs and percent return. Follow the four-step process.
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