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Name: ________________________ Class: ___________________ Date: __________

ID: A

Regression

Multiple Choice Identify the choice that best completes the statement or answers the question.

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1. Given the least squares regression line y8 = 5 - 2x:

a. the relationship between x and y is positive. b. the relationship between x and y is negative. c. as x decreases, so does y. d. None of these choices.

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2. A regression analysis between sales (in $1,000) and advertising (in $1,000) resulted in the following least squares line: y8 = 80 + 5x. This implies that:

a. as advertising increases by $1,000, sales increases by $5,000. b. as advertising increases by $1,000, sales increases by $80,000. c. as advertising increases by $5, sales increases by $80. d. None of these choices.

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3. Which of the following techniques is used to predict the value of one variable on the basis of other variables? a. Correlation analysis b. Coefficient of correlation c. Covariance d. Regression analysis

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4. The residual is defined as the difference between: a. the actual value of y and the estimated value of y b. the actual value of x and the estimated value of x c. the actual value of y and the estimated value of x d. the actual value of x and the estimated value of y

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5. Testing whether the slope of the population regression line could be zero is equivalent to testing whether the: a. sample coefficient of correlation could be zero b. standard error of estimate could be zero c. population coefficient of correlation could be zero d. sum of squares for error could be zero

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6. A regression line using 25 observations produced SSR = 118.68 and SSE = 56.32. The standard error of estimate was: a. 2.11 b. 1.56 c. 2.44 d. None of these choices.

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7. If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be: a. 1.0 b. -1.0 c. either 1.0 or -1.0 d. 0.0

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8. If the coefficient of correlation is -0.60, then the coefficient of determination is: a. -0.60 b. -0.36 c. 0.36 d. 0.77

____ 9. The standard error of estimate s is given by: a. SSE / (n - 2) b. SSE / (n - 2) c. SSE / (n - 2)

d. SSE / n - 2

____ 10. If the standard error of estimate s = 20 and n = 10, then the sum of squares for error, SSE, is: a. 400 b. 3,200 c. 4,000 d. 40,000

____ 11. In regression analysis, the coefficient of determination R2 measures the amount of variation in y that is: a. caused by the variation in x. b. explained by the variation in x. c. unexplained by the variation in x. d. None of these choices.

____ 12. In a regression problem, if the coefficient of determination is 0.95, this means that: a. 95% of the y values are positive. b. 95% of the variation in y can be explained by the variation in x. c. 95% of the y values are predicted correctly by the model. d. None of these choices.

____ 13. The sample correlation coefficient between x and y is 0.375. It has been found out that the p-value is 0.256 when testing H0: = 0 against the two-sided alternative H1: 0. To test H0: = 0 against the one-sided alternative H1: > 0 at a significant level of 0.193, the p-value will be equal to a. 0.128 b. 0.512 c. 0.744 d. 0.872

____ 14. In simple linear regression, which of the following statements indicates there is no linear relationship between the variables x and y? a. Coefficient of determination is -1.0. b. Coefficient of correlation is 0.0. c. Sum of squares for error is 0.0. d. None of these choices.

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ID: A

____ 15. In simple linear regression, the coefficient of correlation r and the least squares estimate b1 of the population slope 1: a. must be equal. b. must have the same sign. c. are not related. d. None of these choices.

____ 16. If the coefficient of correlation is 0.90, then the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x is: a. 90% b. 81% c. 95% d. None of these choices.

____ 17. The width of the confidence interval estimate for the predicted value of y depends on a. the standard error of the estimate b. the value of x for which the prediction is being made c. the sample size d. All of these choices are true.

____ 18. In a multiple regression model, the mean of the probability distribution of the error variable is assumed to be: a. 1.0 b. 0.0 c. k, where k is the number of independent variables included in the model. d. None of these choices.

____ 19. In a multiple regression analysis involving 6 independent variables, the total variation in y is 900 and SSR = 600. What is the value of SSE? a. 300 b. 1.50 c. 0.67 d. None of these choices.

____ 20. For a multiple regression model the following statistics are given: Total variation in y = 250, SSE = 50, k = 4, and n = 20. Then, the coefficient of determination adjusted for the degrees of freedom is: a. 0.800 b. 0.747 c. 0.840 d. 0.775

____ 21. A multiple regression model has the form: y8 = 5.25 + 2x1 + 6x2 . As x2 increases by one unit, holding x1 constant, then the value of y will increase by: a. 2 units b. 7.25 units c. 6 units on average d. None of these choices

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____ 22. If all the points for a multiple regression model with two independent variables were right on the regression plane, then the coefficient of determination would equal: a. 0. b. 1. c. 2, since there are two independent variables. d. None of these choices.

____ 23. For a multiple regression model, the total variation in y can be expressed as: a. SSR + SSE. b. SSR - SSE. c. SSE - SSR. d. SSR / SSE.

____ 24. A multiple regression equation includes 5 independent variables, and the coefficient of determination is 0.81. The percentage of the variation in y that is explained by the regression equation is: a. 81% b. 90% c. 86% d. about 16%

____ 25. In a multiple regression model, the value of the coefficient of determination has to fall between a. -1 and +1. b. 0 and +1. c. -1 and 0. d. None of these choices.

True/False Indicate whether the statement is true or false.

____ 26. In a simple linear regression problem, the least squares line is y8 = -3.75 + 1.25x, and the coefficient of determination is 0.81. The coefficient of correlation must be -0.90.

____ 27. A zero correlation coefficient between a pair of random variables means that there is no linear relationship between the random variables.

____ 28. In reference to the equation y8 = -0.80 + 0.12x1 + 0.08x2 , the value -0.80 is the y-intercept.

____ 29. In multiple regression, the standard error of estimate is defined by s = SSE / (n - k) , where n is the sample size and k is the number of independent variables.

____ 30. One of the consequences of multicollinearity in multiple regression is biased estimates on the slope coefficients.

____ 31. Multicollinearity is present when there is a high degree of correlation between the independent variables included in the regression model.

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Name: ________________________

ID: A

Short Answer

Car Speed and Gas Mileage

An economist wanted to analyze the relationship between the speed of a car (x) and its gas mileage (y). As an experiment a car is operated at several different speeds and for each speed the gas mileage is measured. These data are shown below.

Speed Gas Mileage

25 35 45 50 60 65 70 40 39 37 33 30 27 25

32. {Car Speed and Gas Mileage Narrative} Estimate the gas mileage of a car traveling 70 mph.

33. A scatter diagram includes the following data points:

x

3

2

5

4

5

y

8

6 12 10 14

Two regression models are proposed: (1) y8 = 1.2 + 2.5x, and (2) y8 = 4.0x. Using the least squares method, which of these regression models provides the better fit to the data? Why?

Sunshine and Skin Cancer

A medical statistician wanted to examine the relationship between the amount of sunshine (x) in hours, and incidence of skin cancer (y). As an experiment he found the number of skin cancer cases detected per 100,000 of population and the average daily sunshine in eight counties around the country. These data are shown below.

Average Daily Sunshine Skin Cancer per 100,000

5

7

6

7

8

6

4

3

7 11 9 12 15 10 7

5

34. {Sunshine and Skin Cancer Narrative} Draw a scatter diagram of the data and plot the least squares regression line on it.

35. {Sunshine and Skin Cancer Narrative} Estimate the number of skin cancer cases per 100,000 people who live in a state that gets 6 hours of sunshine on average.

36. {Sunshine and Skin Cancer Narrative} Can we conclude at the 1% significance level that there is a linear relationship between sunshine and skin cancer?

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