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Chapter 3: Q 51. (page 204)

Movie candy Is there a relationship between the amount of sugar (in grams) and the number of calories in movie-theater candy? Here are the data from a sample of 12 types of candy:

a. Sketch a scatterplot of the data using sugar as the explanatory variable.

b. Use technology to calculate the equation of the least-squares regression line for predicting the number of calories based on the amount of sugar. Add the line to the scatterplot from part (a).

c. Explain why the line calculated in part (b) is called the 鈥渓east-squares鈥� regression line.

Short Answer

Expert verified

Part (b) The regression line is: y铃�=2.829x+300.040

Part (c) It's known as a least square regression line.

Part (a)

Step by step solution

01

Part (a) Step 1: Given information

The data set is

02

Part (a) Step 2: Concept

The least-squares regression line reduces the sum of squares of vertical distances between the observed points and the line to zero.

03

Part (a) Step 3: Explanation

The scatter plot is:

04

Part (b) Step 1: Explanation

The output obtained using the Ti-83 calculator is:

Thus, the regression line is: y铃�=2.829x+300.040

The regression line on the scatter plot could be shown as:

05

Part (c) Step 1: Explanation

Because it is a regression line, the regression line generated in the previous section minimizes the errors or residuals. As a result, it's known as a least square regression line.

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Most popular questions from this chapter

Sarah鈥檚 parents are concerned that she seems short for her age. Their doctor has kept the following record of Sarah鈥檚 height:

a. Make a scatterplot of these data using age as the explanatory variable. Describe what you see.

b. Using your calculator, find the equation of the least-squares regression line.

c. Calculate and interpret the residual for the point when Sarah was 48 months old.

d. Would you be confident using the equation from part (b) to predict Sarah鈥檚 height when she is 40 years old? Explain.

Each year, students in an elementary school take a standardized math test at the end of the school year. For a class of fourth-graders, the average score was 55.1 with a standard deviation of 12.3. In the third grade, these same students had an average score of 61.7 with a standard deviation of 14.0. The correlation between the two sets of scores is r = 0.95. Calculate the equation of the least-squares regression line for predicting a fourth-grade score from a third-grade score.

a. y^=3.58+0.835x

b. y^=15.69+0.835x

c. y^=2.19+1.08x

d. y^=鈭�11.54+1.08x

Stand mixers The scatterplot shows the weight (in pounds) and cost (in dollars) of 11 stand mixers.35 The mixer from Walmart (highlighted in red) was much lighter鈥攁nd cheaper than the other mixers.

a. Describe what influence the highlighted point has on the equation of the least-squares regression line.

b. Describe what influence the highlighted point has on the standard deviation of the residuals and r2

Olympic athletes The scatterplot shows the relationship between height (in

inches) and weight (in pounds) for the members of the United States 2016 Olympic Track and Field team.10 Describe the relationship between height and weight for these athletes.

When we standardize the values of a variable, the distribution of standardized values has a mean of 0 and a standard deviation of 1. Suppose we measure two variables X and Y on each of several subjects. We standardize both variables and then compute the least squares regression line. Suppose the slope of the least-squares regression line is 20.44. We may conclude that

a. the intercept will also be 鈭�0.44.

b. the intercept will be 1.0.

c. the correlation will be 1/鈭�0.44.

d. the correlation will be 1.0.

e. the correlation will also be 鈭�0.44.
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