BBUNDERSTAT12 9.4.001.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Use the following linear regression equation to answer the questions.

x1 = 1.7 + 3.0x2 - 7.6x3 + 1.6x4

Suppose x3 and x4 were held at fixed but arbitrary values and x2 increased by 1 unit. What would be the corresponding change in x1?

Suppose x2 increased by 2 units. What would be the expected change in x1?

Suppose x2 decreased by 4 units. What would be the expected change in x1?

(e) Suppose that n = 9 data points were used to construct the given regression equation and that the standard error for the coefficient of x2 is 0.370. Construct a 99% confidence interval for the coefficient of x2. (Use 2 decimal places.)

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(f) Using the information of part (e) and level of significance 10%, test the claim that the coefficient of x2 is different from zero. (Use 2 decimal places.)

t

t critical

2.

[0.17/0.45 Points]

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BBUNDERSTAT12 9.4.002.MI.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Use the following linear regression equation to answer the questions.

x3 = 17.7 + 3.9x1 + 8.8x4 1.0x7

(a)

Which variable is the response variable?

x7

x1

x4

x3

Which variables are the explanatory variables? (Select all that apply.)

x4

x3

x7

x1

(b)

Which number is the constant term? List the coefficients with their corresponding explanatory variables.

constant

x1 coefficient

x4 coefficient

x7 coefficient

(c)

If x1 = 1, x4 =1,

and x7 = 3, what is the predicted value for x3? (Round your answer to one decimal place.)

x3 =

(d)

Explain how each coefficient can be thought of as a "slope" under certain conditions.

If we look at all coefficients together, each one can be thought of as a "slope."

If we hold all explanatory variables as fixed constants, the intercept can be thought of as a "slope."

If we hold all other explanatory variables as fixed constants, then we can look at one coefficient as a "slope."

If we look at all coefficients together, the sum of them can be thought of as the overall "slope" of the regression line.

Suppose x1 and x7 were held at fixed but arbitrary values.

If x4 increased by 1 unit, what would we expect the corresponding change in x3 to be?

If x4 increased by 3 units, what would be the corresponding expected change in x3?

If x4 decreased by 2 units, what would we expect for the corresponding change in x3?

(e)

Suppose that n = 10 data points were used to construct the given regression equation and that the standard error for the coefficient of x4 is 0.966. Construct a 90% confidence interval for the coefficient of x4. (Round your answers to two decimal places.)

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(f)

Using the information of part (e) and level of significance 1%, test the claim that the coefficient of x4 is different from zero. (Round your answers to three decimal places.)

t =

t critical =

3.

[0.04/0.45 Points]

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BBUNDERSTAT12 9.4.003.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

The systolic blood pressure of individuals is thought to be related to both age and weight. For a random sample of 11 men, the following data were obtained.

Systolic Blood pressue

x1 Age (years)

x2 Weight (pounds)

x3

132 52 173

143 59 184

153 67 194

162 73 211

154 64 196

168 74 220

137 54 188

149 61 188

159 65 207

128 46 167

166 72 217

(a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation for each variable. (Use 2 decimal places.)

x s CV

x1

%

x2

%

x3

%

Relative to its mean, which variable has the greatest spread of data values? Which variable has the smallest spread of data values relative to its mean?

x2; x3

x2; x1

x3; x2

x1; x3

(b) For each pair of variables, generate the correlation coefficient r. Compute the corresponding coefficient of determination r2. (Use 3 decimal places.)

r r2

x1, x2

x1, x3

x2, x3

What percent of the variation in x1 can be explained by the corresponding variation in x2? Answer the same question for x3. (Use 1 decimal place.)

x2

%

x3

%

(c) Perform a regression analysis with x1 as the response variable. Use x2 and x3 as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in x1 can be explained by the corresponding variations in x2 and x3 taken together? (Use 1 decimal place.)

%

(d) Look at the coefficients of the regression equation. Write out the regression equation. (Use 3 decimal places.)

x1 =

+

x2 +

x3

If age were held fixed, but a person put on 15 pounds, what would you expect for the corresponding change in systolic blood pressure? (Use 2 decimal places.)

If a person kept the same weight but got 15 years older, what would you expect for the corresponding change in systolic blood pressure? (Use 2 decimal places.)

(e) Test each coefficient to determine if it is zero or not zero. Use level of significance 5%. (Use 2 decimal places for t and 3 decimal places for the P-value.)

t P value

?2

?3

(f) Find a 90% confidence interval for each coefficient. (Use 2 decimal places.)

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?2

?3

(g) Suppose Michael is 68 years old and weighs 192 pounds. Predict his systolic blood pressure, and find a 90% confidence range for your prediction. (Use 1 decimal place.)

prediction

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4.

[0/0.45 Points]

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BBUNDERSTAT12 9.4.004.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Professor Gill has taught General Psychology for many years. During the semester, she gives three multiple-choice exams, each worth 100 points. At the end of the course, Dr. Gill gives a comprehensive final worth 200 points. Let x1, x2, and x3 represent a student's scores on exams 1, 2, and 3, respectively. Let x4 represent the student's score on the final exam. Last semester Dr. Gill had 25 students in her class. The student exam scores are shown below.

x1 x2 x3 x4

73 80 75 152

93 88 93 185

89 91 90 180

96 98 100 196

73 66 70 142

53 46 55 101

69 74 77 149

47 56 60 115

87 79 90 175

79 70 88 164

69 70 73 141

70 65 74 141

93 95 91 184

79 80 73 152

70 73 78 148

93 89 96 192

78 75 68 147

81 90 93 183

88 92 86 177

78 83 77 159

82 86 90 177

86 82 89 175

78 83 85 175

76 83 71 149

96 93 95 192

Since Professor Gill has not changed the course much from last semester to the present semester, the preceding data should be useful for constructing a regression model that describes this semester as well.

(a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation for each variable. (Use 2 decimal places.)

x s CV

x1

%

x2

%

x3

%

x4

%

(b) For each pair of variables, generate the correlation coefficient r. Compute the corresponding coefficient of determination r2. (Use 3 decimal places.)

r r2

x1, x2

x1, x3

x1, x4

x2, x3

x2, x4

x3, x4

Of the three exams 1, 2, and 3, which do you think had the most influence on the final exam 4?

(c) Perform a regression analysis with x4 as the response variable. Use x1, x2, and x3 as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in x4 can be explained by the corresponding variations in x1, x2, and x3 taken together? (Use 1 decimal place.)

%

(d) Write out the regression equation. (Use 2 decimal places.)

x4 =+

x1 +

x2 +

x3

If a student were to study "extra hard" for exam 3 and increase his or her score on that exam by 11 points, what corresponding change would you expect on the final exam? (Assume that exams 1 and 2 remain "fixed" in their scores.) (Use 1 decimal place.)

(e) Test each coefficient in the regression equation to determine if it is zero or not zero. Use level of significance 5%. (Use 2 decimal places for t and 3 decimal places for the P-value.)

t P-value

?1

?2

?3

(f) Find a 90% confidence interval for each coefficient. (Use 2 decimal places.)

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?1

?2

?3

(g) This semester Susan has scores of 68, 72, and 75 on exams 1, 2, and 3, respectively. Make a prediction for Susan's score on the final exam and find a 90% confidence interval for your prediction (if your software supports prediction intervals). (Round all answers to nearest integer.)

prediction

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5.

[0/0.45 Points]

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BBUNDERSTAT12 9.4.005.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

A motion picture industry analyst is studying movies based on epic novels. The following data were obtained for 10 Hollywood movies made in the past five years. Each movie was based on an epic novel. For these data, x1 = first-year box office receipts of the movie, x2 = total production costs of the movie, x3 = total promotional costs of the movie, and x4 = total book sales prior to movie release. All units are in millions of dollars.

x1 x2 x3 x4

85.1 8.5 5.1 4.7

106.3 12.9 5.8 8.8

50.2 5.2 2.1 15.1

130.6 10.7 8.4 12.2

54.8 3.1 2.9 10.6

30.3 3.5 1.2 3.5

79.4 9.2 3.7 9.7

91.0 9.0 7.6 5.9

135.4 15.1 7.7 20.8

89.3 10.2 4.5 7.9

(a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation for each variable. (Use 2 decimal places.)

x s CV

x1

%

x2

%

x3

%

x4

%

Relative to its mean, which variable has the largest spread of data values?

x2

x4

x3

x1

(b) For each pair of variables, generate the correlation coefficient r. Compute the corresponding coefficient of determination r2. (Use 3 decimal places.)

r r2

x1, x2

x1, x3

x1, x4

x2, x3

x2, x4

x3, x4

Which of the three variables x2, x3, and x4 has the least influence on box office receipts?

x3

x2

x4

What percent of the variation in box office receipts can be attributed to the corresponding variation in production costs? (Use 1 decimal place.)

%

(c) Perform a regression analysis with x1 as the response variable. Use x2, x3, and x4 as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in x1 can be explained by the corresponding variations in x2, x3, and x4 taken together? (Use 1 decimal place.)

%

(d) Write out the regression equation. (Use 2 decimal places.)

x1 =

+

x2 +

x3 +

x4

If x2 (production costs) and x4 (book sales) were held fixed but x3 (promotional costs) were increased by 0.7 million dollars, what would you expect for the corresponding change in x1 (box office receipts)? (Use 2 decimal places.)

(e) Test each coefficient in the regression equation to determine if it is zero or not zero. Use level of significance 5%. (Use 2 decimal places for t and 3 decimal places for the P-value.)

t P-value

?2

?3

?4

(f) Find a 90% confidence interval for each coefficient. (Use 2 decimal places.)

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?2

?3

?4

(g) Suppose a new movie (based on an epic novel) has just been released. Production costs were x2 = 11.4 million; promotion costs were x3 = 4.7 million; book sales were x4 = 8.1 million. Make a prediction for x1 = first-year box office receipts and find an 85% confidence interval for your prediction (if your software supports prediction intervals). (Use 1 decimal place.)

prediction

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(h) Construct a new regression model with x3 as the response variable and x1, x2, and x4 as explanatory variables. (Use 2 decimal places.)

x3 =

+

x1 +

x2 +

x4

Suppose Hollywood is planning a new epic movie with projected box office sales x1 = 100 million and production costs x2 = 12 million. The book on which the movie is based had sales of x4 = 9.2 million. Forecast the dollar amount (in millions) that should be budgeted for promotion costs x3 and find an 80% confidence interval for your prediction.

prediction

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