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**IE 424 Homework 7 | Complete Solution**

- From Mathematics, Statistics

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**IE 424 Homework 7**

Read Sections: Chapter 7.1 - 7.3 [Regression]

Problems from the Book:

Ch. 7.: 7.1a c-k, 7.2 (determine the regression equation & find the quantities: R, SST, SSE, and SSR), 7.4 a-c Additional Problems i & ii

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Chapter 7:

Problem 7.1a c-k:

The heights (x) and weights (y) of 12 men are given in the following table.

X (in) Y (lbs)

68

135

70

140

71

165

69

155

71

175

72

170

74

190

67

130

69

145

70

158

68

142

72

180

a. Plot a scatter diagram of the data, with x on the horizontal axis and y on the vertical axis.

b. Using your “eyes only,” draw what you think is the “best fitting” line through the data. Determine the equation of this line graphically (recall y = mx + b, where m is the slope of the line and b is the y-intercept).

N / A

c. Evaluate Σx, Σy, Σx2, Σy2, and Σxy, with all summations being from i = 1 to 12. Use Table

as a guide. Also find x-bar and y-bar.

d. Find bo and b1 so that ŷ = bo + b1x . This is, use the least squares to find bo and b1 in the SLR model. How does this regression line compare to your “eye-fit” line?

e. Evaluate Σei = Σ(yi- ŷ). Will this result always be the case?

f. Find SST.

g. Find SSR.

h. Find SSE.

i. Do your findings support the fast that SST = SSR + SSE?

j. Find the correlation coefficient, R.

k. Verify that R2 = SSR/SST. What is the mean of R2?

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Problem 7.2:

The vehicle weight in (100 pounds) and gas mileage data (mpg) given in Section 2.2 is reproduced here. Determine the regression equation and find the quantities R, SST, SSE, and SSR.

x: weight (100 lbs) y: mileage (mpg)

30

18

28

21

21

32

29

17

24

31

33

14

27

21

35

12

25

23

32

14

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Problem 7.4 a-c:

On January 28, 1986, the space shuttle Challenger was launched at a temperature of 31˚F. The ensuing catastrophe was caused by a combustion gas leak through a joint in one of the booster rockets, which was sealed by a device called on O-ring. The Rogers Commission which investigated the accident concluded that O-rings do not seal properly at low temperatures [Journal of the American Statistical Association (JASA), Vol. 84, No. 408, December 1989, pp. 945-957]. The following data are taken from Figure 1.b of the references JASA article. It relates launch temperature to the number of O-rings under thermal distress (i.e., O-rings likely to not seal properly) for 24 previous launches.

Temperature (˚F) # of O-rings under Thermal Distress Temperature (˚F) # of O-rings under Thermal Distress

53

3

69

0

57

1

70

0

58

1

70

0

63

1

72

0

70

1

73

0

70

1

75

0

75

2

76

0

66

0

76

0

67

0

78

0

67

0

79

0

67

0

80

0

68

0

81

0

a. Obtain a scatter plot of the data, with Temp (x) on the horizontal axis and Number of O- rings (y) on the vertical axis.

b. Determine bo and b1 in the SLR model. That is, find ŷ = bo + b1x and then graph this on your scatter plot in part (a).

c. Find ŷ|x=31. That is, predict the number of O-ring failures (i.e., distressed O-rings) for a launch temperature of 31˚F.

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Additional Problems

Additional Problems i.

We are interested in applying a linear regression model to see if the tensile force applied to a steel specimen (in thousands of pounds) has a linear effect on the resulting elongation (in thousandths of an inch).

Tensile Force

1

1

2

3

4

4

5

6 Elongation

14

16

33

40

52

63

76

85

a. Plot the points on a scatter diagram. Does there appear to be a linear trend? Estimate the least squares regression line.

b. Estimate the average elongation for a tensile force of 3.5 thousand pounds.

c. Construct a 95% confidence interval for 1. Interpret the interval.

d. Construct a 95% confidence interval and a 95% prediction interval for the average elongation if x = 3.5 thousand pounds. What is the difference between these two intervals? [Note: We did not calculate these in class but you have the formulas].

e. Test for significance of regression using the two-sided t-test and alpha = 0.05.

f. Construct an ANOVA table and use it to preform the hypothesis test of part (e). Restate the hypotheses and interpret your results.

g. Calculate and interpret the sample coefficient of determination and the sample coefficient of correlation.

h. Perform a residual analysis on the data.

i. Summarize your findings from parts (a) - (h) to asses the appropriateness of the model.

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Additional Problems ii.

[Multiple Regression Example] An experiment to determine the relationship between the amount of ammonia lost and three independent variable produced the data in the following table. The loss of ammonia, y, was measured at various levels of airflow, x1, coolant temperature, x2, and a coded value of the concentration of nitric acid in the absorbing liquid, x3.

x1 x2 x3 y

80

27

89

42

80

27

88

37

75

25

90

37

62

24

87

28

62

22

87

18

62

23

87

18

62

24

93

19

62

24

93

20

58

23

87

15

58

18

80

14

58

18

89

14

58

17

88

13

58

18

82

11

58

19

93

12

50

18

89

8

50

18

86

7

50

19

72

8

50

19

79

8

50

20

80

9

56

20

82

15

70

20

91

15

Use Minitab/Excel to answer the questions below.

a. Assuming the multiple linear regression model to be appropriate, obtain the least- squares regression line.

b. By looking at the p-values, is there any x variable that does not seem significant? State the appropriate hypotheses that would be used in this case.

c. Analyze and interpret the ANOVA table and R2 value.

d. If you found any of the x variables to be insignificant in part (b), rerun the regression with only the significant variables. Perform a complete analysis of the output and perform a residual analysis.

If you found all of the variables to be significant in part (b), complete the regression analysis by per- forming a residual analysis on your model. Be very clear in stating your results and performing your analyses.

**IE 424 Homework 7 | Complete Solution**

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