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Chapter 3: Q 28. (page 174)

A carpenter sells handmade wooden benches at a craft fair every week. Over the past year, the carpenter has varied the price of the benches from \(80

to \)120 and recorded the average weekly profit he made at each selling price. The prices of the bench and the corresponding average profits are shown in the table.

a. Make a scatterplot to show the relationship between price and profit.

b. The correlation for these data is r=0Explain how this can be true even though there is a strong relationship between price and average profit.

Short Answer

Expert verified

Part (b) The correlation measures only the strength of the relationship between the variables, while the variables possess a quadratic relationship.

Part (a)

Step by step solution

01

Part (a) Step 1: Given information

Prices of the bench and corresponding average profits:

02

Part (a) Step 2: Explanation

For scatterplot:

Horizontal axis: Price

Vertical axis: Average profit

03

Part (b) Step 1: Explanation

At r=0

Only the strength of the linear link between the variables is measured by the correlation.

That implies

No linear relationship exists between price and average profit.

However,

That does not rule out the possibility of a link between variables.

Note that

Since all the points lie on the same parabola that opens downward, variables possess a perfect quadratic relationship between them.

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Most popular questions from this chapter

In a scatterplot of the average price of a barrel of oil and the average retail price of a gallon of gas, you expect to see

a. very little association.

b. a weak negative association.

c. a strong negative association.

d. a weak positive association.

e. a strong positive association.

Dem bones Archaeopteryx is an extinct beast that had feathers like a bird but teeth and a long bony tail like a reptile. Only six fossil specimens are known to exist today. Because these specimens differ greatly in size, some scientists think they are different species rather than individuals from the same species. If the specimens belong to the same species and differ in size because some are younger than others, there should be a positive linear relationship between the lengths of a pair of bones from all individuals. An outlier from

this relationship would suggest a different species. Here are data on the lengths (in centimeters) of the femur (a leg bone) and the humerus (a bone in the upper arm) for the five specimens that preserve both bones:

a. Make a scatterplot. Do you think that all five specimens come from the same species? Explain.

b. Find the correlation r step by step, using the formula on page 166. Explain how your value for r matches your graph in part (a).

Using data from the LPGA tour, a regression analysis was performed using x = average driving distance and y = scoring average. Using the output from the regression analysis shown below, determine the equation of the least-squares regression line.

a. y^=87.974+2.391x

b. y^=87.974+1.01216x

c. y^=87.974−0.060934x

d. y^=−0.060934+1.01216x

e. y^=−0.060934+87.947x

Hot dogs Are hot dogs that are high in calories and also high in salt? The

following scatterplot shows the calories and salt content (measured in milligrams of sodium) in 17 brands of meat hot dogs.

a. The correlation for these data is r=0.87 Interpret this value.

b. What effect does the hot dog brand with the smallest calorie content have on the correlation? Justify your answer.

Who’s got hops? Haley, Jeff, and Nathan measured the height (in inches)

and vertical jump (in inches) of 74 students at their school.34 Here is a scatterplot of the data, along with the least-squares regression line. Jacob (highlighted in red) had a vertical jump of nearly 3 feet!

a. Describe the influence that Jacob’s point has on the equation of the least-squares regression line.

b. Describe the influence that Jacob’s point has on the standard deviation of the residuals and r2
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