11-8 ADEQUACY OF THE REGRESSION MODEL 399(b) Prepare a normal probability plot of the residuals and
interpret this graph.
(c) Plot residuals versus and x. Do the regression assump-
tions appear to be satisfied?
11-45. Refer to the gasoline mileage data in Exercise 11-7.
(a) What proportion of total variability in highway gaso-
line mileage performance is accounted for by engine
displacement?
(b) Plot the residuals versus and x, and comment on the graphs.
(c) Prepare a normal probability plot of the residuals. Does
the normality assumption appear to be satisfied?
11-46. Consider the data in Exercise 11-8 on ygreen
liquor Na 2 S concentration and xpaper machine production.
Suppose that a 14th sample point is added to the original data,
where y 14 59 and x 14 855.
(a) Prepare a scatter diagram of yversus x. Fit the simple lin-
ear regression model to all 14 observations.
(b) Test for significance of regression with 0.05.
(c) Estimate ^2 for this model.
(d) Compare the estimate of ^2 obtained in part (c) above with
the estimate of ^2 obtained from the original 13 points.
Which estimate is larger and why?
(e) Compute the residuals for this model. Does the value of
e 14 appear unusual?
(f ) Prepare and interpret a normal probability plot of the
residuals.
(g) Plot the residuals versus and versus x. Comment on
these graphs.
11-47. Refer to Exercise 11-9, which presented data on
blood pressure rise yand sound pressure level x.
(a) What proportion of total variability in blood pressure rise
is accounted for by sound pressure level?
(b) Prepare a normal probability plot of the residuals from
this least squares model. Interpret this plot.
(c) Plot residuals versus and versus x. Comment on these plots.
11-48. Exercise 11-10 presents data on wear volume yand
oil viscosity x.
(a) Calculate R^2 for this model. Provide an interpretation of
this quantity.
(b) Plot the residuals from this model versus and versus x.
Interpret these plots.
(c) Prepare a normal probability plot of the residuals. Does
the normality assumption appear to be satisfied?
11-49. Refer to Exercise 11-11, which presented data on
chloride concentration yand roadway area x.
(a) What proportion of the total variability in chloride con-
centration is accounted for by the regression model?
(b) Plot the residuals versus and versus x. Interpret these plots.
(c) Prepare a normal probability plot of the residuals. Does
the normality assumption appear to be satisfied?
11-50. Consider the rocket propellant data in Exercise 11-12.
(a) Calculate R^2 for this model. Provide an interpretation of
this quantity.yˆyˆyˆyˆyˆyˆ(b) Plot the residuals on a normal probability scale. Do any
points seem unusual on this plot?
(c) Delete the two points identified in part (b) from the
sample and fit the simple linear regression model to the
remaining 18 points. Calculate the value of R^2 for the new
model. Is it larger or smaller than the value of R^2 com-
puted in part (a)? Why?
(d) Did the value of change dramatically when the two
points identified above were deleted and the model fit to
the remaining points? Why?
11-51. Show that an equivalent way to define the test for
significance of regression in simple linear regression is to base
the test on R^2 as follows: to test H 0 : 1 0 versus H 1 : 1 0,
calculateand to reject H 0 : 1 0 if the computed value f 0
f,1,n 2.
11-52. Suppose that a simple linear regression model has
been fit to n25 observations and R^2 0.90.
(a) Test for significance of regression at 0.05. Use the
results of Exercise 11-51.
(b) What is the smallest value of R^2 that would lead to the
conclusion of a significant regression if 0.05?
11-53. Consider the rocket propellant data in Exercise 11-- Calculate the standardized residuals for these data. Does
this provide any helpful information about the magnitude of
the residuals?
11-54. Studentized Residuals.Show that the variance
of the ith residual is
Hint:The ith studentized residual is defined as(a) Explain why rihas unit standard deviation.
(b) Do the standardized residualshave unit standard deviation?
(c) Discuss the behavior of the studentized residual when the
sample value xiis very close to the middle of the range of x.
(d) Discuss the behavior of the studentized residual when the
sample value xiis very near one end of the range of x.rieiB
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1
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bdcov 1 Yi, Yˆi 2 ^2 c
1
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Sxx
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1
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