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

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 ?x, ?y, ?x

2

, ?y

2

, and ?xy, 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 ? = 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?ei=?(yi- ?).Willthis resultalways be thecase?

f. Find SST.

g. Find SSR.

h. Find SSE.

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

j. Findthecorrelationcoefficient,R.

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

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, andSSR.

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

IE424

4

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 24previous 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 Orings(y)

onthevertical axis.

b. Determine bo and b1 in the SLR model. That is, find ? = bo + b1x and then graph this on

your scatter plot in part (a).

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

launchtemperatureof31?F.

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

theleast squaresregressionline.

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

c. Construct a 95%confidence intervalfor 1. Interprettheinterval.

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

coefficientofcorrelation.

h. Perform a residual analysis on the data.

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

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 leastsquares

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 theANOVAtable 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 performing

a residual analysis on your model. Be very clear in stating your results and performing your

analyses.