10-4. Exercises

1. BASIC

Ex 1. For the Basic and Application exercises in this section use the computations that were done for the exercises with the same number in Section 10.2 "The Linear Correlation Coefficient".

Ex 2. Compute the least squares regression line for the data in Exercise 1 of Section 10.2 "The Linear Correlation Coefficient".

Ex 3. Compute the least squares regression line for the data in Exercise 2 of Section 10.2 "The Linear Correlation Coefficient".

Ex 4. Compute the least squares regression line for the data in Exercise 3 of Section 10.2 "The Linear Correlation Coefficient".

Ex 5. Compute the least squares regression line for the data in Exercise 4 of Section 10.2 "The Linear Correlation Coefficient".

Ex 6. For the data in Exercise 5 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. Compute the sum of the squared errors $SSE$ using the definition $Σ(y−\hat{y})^2$ .

3. Compute the sum of the squared errors $SSE$ using the formula $SSE=SS_{yy}−\hat{β_1}SS_{xy}$ .

Ex 7. For the data in Exercise 6 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. Compute the sum of the squared errors $SSE$ using the definition $Σ(y−\hat{y})^2$.

3. Compute the sum of the squared errors $SSE$ using the formula $SSE=SS_{yy}−\hat{β_1}SS_{xy}$.

Ex 8. Compute the least squares regression line for the data in Exercise 7 of Section 10.2 "The Linear Correlation Coefficient".

Ex 9. Compute the least squares regression line for the data in Exercise 8 of Section 10.2 "The Linear Correlation Coefficient".

Ex 10. For the data in Exercise 9 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. Can you compute the sum of the squared errors $SSE$ using the definition

   $Σ(y−\hat{y})^2$? Explain.
3. Compute the sum of the squared errors $SSE$ using the formula

   $SSE=SS_{yy}−\hat{β_1}SS_{xy}$

Ex 11. For the data in Exercise 10 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. Can you compute the sum of the squared errors $SSE$ using the definition

   $Σ(y−\hat{y})^2$? Explain.
3. Compute the sum of the squared errors $SSE$ using the formula

   $SSE=SS_{yy}−\hat{β_1}SS_{xy}$.

2. APPLICATIONS

Ex 12. For the data in Exercise 11 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. On average, how many new words does a child from 13 to 18 months old learn each month? Explain.

3. Estimate the average vocabulary of all 16-month-old children.

Ex 13. For the data in Exercise 12 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. On average, how many additional feet are added to the braking distance for each additional 100 pounds of weight? Explain.

3. Estimate the average braking distance of all cars weighing 3,000 pounds.

Ex 14. For the data in Exercise 13 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. Estimate the average resting heart rate of all 40-year-old men.

3. Estimate the average resting heart rate of all newborn baby boys. Comment on the validity of the estimate.

Ex 15. For the data in Exercise 14 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. Estimate the average wave height when the wind is blowing at 10 miles per hour.

3. Estimate the average wave height when there is no wind blowing. Comment on the validity of the estimate.

Ex 16. For the data in Exercise 15 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. On average, for each additional thousand dollars spent on advertising, how does revenue change? Explain.

3. Estimate the revenue if $2,500 is spent on advertising next year.

Ex 17. For the data in Exercise 16 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. On average, for each additional inch of height of two-year-old girl, what is the change in the adult height? Explain.

3. Predict the adult height of a two-year-old girl who is 33 inches tall.

Ex 18. For the data in Exercise 17 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. Compute $SSE$ using the formula $SSE=SS_{yy}−\hat{β_1}SS_{xy}$.
3. Estimate the average final exam score of all students whose course average just before the exam is 85.

Ex 19. For the data in Exercise 18 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. Compute $SSE$ using the formula $SSE=SS_{yy}−\hat{β_1}SS_{xy}$.
3. Estimate the number of acres that would be harvested if 90 million acres of corn were planted.

Ex 20. For the data in Exercise 19 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. Interpret the value of the slope of the least squares regression line in the context of the problem.

3. Estimate the average concentration of the active ingredient in the blood in men after consuming 1 ounce of the medication.

Ex 21. For the data in Exercise 20 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. Interpret the value of the slope of the least squares regression line in the context of the problem.

3. Estimate the age of an oak tree whose girth five feet off the ground is 92 inches.

Ex 22. For the data in Exercise 21 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. The 28-day strength of concrete used on a certain job must be at least 3,200 psi. If the 3-day strength is 1,300 psi, would we anticipate that the concrete will be sufficiently strong on the 28th day? Explain fully.

Ex 23. For the data in Exercise 22 of Section 10.2 "The Linear Correlation Coefficient"

1. Compute the least squares regression line.

2. If the power facility is called upon to provide more than 95 million watt-hours tomorrow then energy will have to be purchased from elsewhere at a premium. The forecast is for an average temperature of 42 degrees. Should the company plan on purchasing power at a premium?

3. ADDITIONAL EXERCISES

Ex 24. Verify that no matter what the data are, the least squares regression line always passes through the point with coordinates $(\bar{x}, \bar{y})$. Hint: Find the predicted value of $y$ when $x=\bar{x}$ .

Ex 25. In Exercise 1 you computed the least squares regression line for the data in Exercise 1 of Section 10.2 "The Linear Correlation Coefficient".

1. Reverse the roles of $x$ and $y$ and compute the least squares regression line for the new data set x 2 4 6 5 9 y 0 1 3 5 8
2. Interchanging x and y corresponds geometrically to reflecting the scatter plot in a 45-degree line. Reflecting the regression line for the original data the same way gives a line with the equation $\hat{y}=1.346x−3.600$ . Is this the equation that you got in part (a)? Can you figure out why not? Hint: Think about how $x$ and $y$ are treated differently geometrically in the computation of the goodness of fit.
3. Compute $SSE$ for each line and see if they fit the same, or if one fits the data better than the other.

4. LARGE DATA SET EXERCISES

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data1.xls

Data Set 1

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data12.xls

Data Set 12

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data13.xls

Data Set 13

Ex 26. Large Data Set 1 lists the SAT scores and GPAs of 1,000 students.

1. Compute the least squares regression line with SAT score as the independent variable ($x$) and GPA as the dependent variable ($y$).
2. Interpret the meaning of the slope βˆ1β^1 of regression line in the context of problem.

3. Compute $SSE$ , the measure of the goodness of fit of the regression line to the sample data.
4. Estimate the GPA of a student whose SAT score is 1350.

Ex 27. Large Data Set 12 lists the golf scores on one round of golf for 75 golfers first using their own original clubs, then using clubs of a new, experimental design (after two months of familiarization with the new clubs).

1. Compute the least squares regression line with scores using the original clubs as the independent variable ($x$) and scores using the new clubs as the dependent variable ($y$).
2. Interpret the meaning of the slope $\hat{β_1}$ of regression line in the context of problem.
3. Compute $SSE$ , the measure of the goodness of fit of the regression line to the sample data.
4. Estimate the score with the new clubs of a golfer whose score with the old clubs is 73.

Ex 28. Large Data Set 13 records the number of bidders and sales price of a particular type of antique grandfather clock at 60 auctions.

1. Compute the least squares regression line with the number of bidders present at the auction as the independent variable ($x$) and sales price as the dependent variable ($y$).
2. Interpret the meaning of the slope $\hat{β_1}$ of regression line in the context of problem.
3. Compute $SSE$, the measure of the goodness of fit of the regression line to the sample data.
4. Estimate the sales price of a clock at an auction at which the number of bidders is seven.