406 CHAPTER 11 SIMPLE LINEAR REGRESSION AND CORRELATION11-58. Show that the t-statistic in Equation 11-46 for testing
H 0 : 0 is identical to the t-statistic for testing H 0 : 1 0.
11-59. A random sample of 50 observations was made on
the diameter of spot welds and the corresponding weld shear
strength.
(a) Given that r0.62, test the hypothesis that 0, using
0.01. What is the P-value for this test?
(b) Find a 99% confidence interval for .
(c) Based on the confidence interval in part (b), can you con-
clude that 0.5 at the 0.01 level of significance?
11-60. Suppose that a random sample of 10,000 (X, Y)
pairs yielded a sample correlation coefficient of r0.02.
(a) What is the conclusion that you would reach if you tested
H 0 :0 using 0.05? What is the P-value for this test?
(b) Comment on the practical significance versus the statisti-
cal significance of your answer.
11-61. The following data gave Xthe water content of
snow on April 1 and Ythe yield from April to July
(in inches) on the Snake River watershed in Wyoming for
1919 to 1935. (The data were taken from an article in
Research Notes,Vol. 61, 1950, Pacific Northwest Forest
Range Experiment Station, Oregon)(a) Draw a scatter diagram of these data. Does a straight-line
relationship seem plausible?
(b) Fit a simple linear regression model to these data.
(c) Test for significance of regression using 0.05. What
is the P-value for this test?
(d) Find a 95% confidence interval estimate on the slope.
(e) Test the hypothesis H 0 : 0 0 versus H 1 : 0 0 using
0.05. What conclusions can you draw?
11-65. The strength of paper used in the manufacture of
cardboard boxes (y) is related to the percentage of hardwood
concentration in the original pulp (x). Under controlled condi-
tions, a pilot plant manufactures 16 samples, each from a dif-
ferent batch of pulp, and measures the tensile strength. The
data are shown in the table that follows:
(a) Fit a simple linear regression model to the data.
(b) Test for significance of regression using 0.05.
(c) Construct a 90% confidence interval on the slope 1.
(d) Construct a 90% confidence interval on the intercept 0.
(e) Construct a 95% confidence interval on the mean strength
at x2.5.
(f ) Analyze the residuals and comment on model adequacy.(c) Test the hypothesis H 0 : 0.8 versus H 1 : 0.8, using
0.05. Find the P-value for this test.Supplemental Exercises11-63. Show that, for the simple linear regression model,
the following statements are true:(a) (b)(c)11-64. An article in the IEEE Transactions on
Instrumentation and Measurement (“Direct, Fast, and
Accurate Measurement of VTand Kof MOS Transistor Using
VT-Sift Circuit,” Vol. 40, 1991, pp. 951–955) described the
use of a simple linear regression model to express drain cur-
rent y(in milliamperes) as a function of ground-to-source
voltage x(in volts). The data are as follows:1
n^ an
i 1yˆiyan
i 1
a^1 yi^ yˆi^2 xi^0n
i 11 yi yˆi 2 0yxyx
0.734 1.1 1.50 1.6
0.886 1.2 1.66 1.7
1.04 1.3 1.81 1.8
1.19 1.4 1.97 1.9
1.35 1.5 2.12 2.0(a) Estimate the correlation between Yand X.
(b) Test the hypothesis that 0, using 0.05.
(c) Fit a simple linear regression model and test for signifi-
cance of regression using 0.05. What conclusions
can you draw? How is the test for significance of regres-
sion related to the test on in part (b)?
(d) Test the hypothesis H 0 : 0 0 versus H 1 : 0 0 and
draw conclusions. Use 0.05.
(e) Analyze the residuals and comment on model adequacy.
11-62. A random sample of n25 observations was made
on the time to failure of an electronic component and the tem-
perature in the application environment in which the compo-
nent was used.
(a) Given that r0.83, test the hypothesis that 0, using
0.05. What is the P-value for this test?
(b) Find a 95% confidence interval on .xyx y
23.1 10.5 37.9 22.8
32.8 16.7 30.5 14.1
31.8 18.2 25.1 12.9
32.0 17.0 12.4 8.8
30.4 16.3 35.1 17.4
24.0 10.5 31.5 14.9
39.5 23.1 21.1 10.5
24.2 12.4 27.6 16.1
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