HW3 - Statistics

150 Part One Simple Linear Regression
*3.13. Refer to Copier maintenance Problem.-I .20.
- -- ___L
a. What are the alternative conclusions when testing for lack of fit of a linear regression
function?
b. Perform the test indicated in part (a). Control the risk of Type I error at .05. State the decision
rule and conclusion.
c. Does the test in part (b) detect other departures from regression model (2.1), such as lack
of constant variance or lack of normality in the error terms? Could the results of the test of
lack of fit be affected by such departures? Discuss.
function; use a = .O1. State the alternatives, decision rule, and conclusion.
b. Is there any advantage of having an equal number of replications at each of the X levels? Is
there any disadvantage? -
c. Does the test in part (a) indicate what regression function is appropriate when it leads to the
conclusion that the regression function is not linear? How would you proceed?
3.1 5. Solution concentration. A chemist studied the concentration of a solution (Y) over time (X).
Fifteen identical solutions were prepared. The 15 solutions were randomly divided into five
sets of three, and the five sets were measured, respectively, after 1, 3, 5, 7, and 9 hours. The
results follow.
3 ... 13
Yi: .07 .09 .08 ... 2.84 2.57 3.10
a. Fit a linear regression function.
function; use a = .025. State the alternatives, decision rule, and conclusion.
c. Does the test in part (b) indicate what regression function is appropriate when it leads to t
conclusion that lack of fit of a linear regression function exists? Explain.
3.16. Refer to Solution concentration Problem 3.1 5.
a. Prepare a scatter plot of the data. What transformation of Y might you try, using the prototy
patterns in Figure 3.1 5 to achieve constant variance and linearity?
suggested?
c. Use the transformation Y' = log,, Y and obtain the estimated linear regression function
the transformed data.
to be a good fit to the transformed data?
plot. What do your plots show?
f. Express the estimated regression function in the original units.

172 Part One Simple Linear Regression
ReferenceS
4.1. Miller, R. G., Jr. Simultaneous Statistical Inference. 2nd ed. New York: Springer-Verlag. 1991
4.2. Fuller, W. A. Measurement Error Models. New York: John Wiley & Sons, 1987.
4.3. Berkson, J. "Are There Two Regressions?" Journal of the American Statistical Association 45
(1 950), pp. 164-80.
4.4. Cox, D. R. Planning of Experiments. New York: John Wiley & Sons, 1958, pp. 141-42.
Problems 4.1. When joint confidence intervals for Bo and 8, are developed by the Bonferroni method
a family confidence coefficient of 90 percent, does this imply that 10 percent of the time th
confidence interval for Bo will be incorrect? That 5 percent of the time the confidence interv
for Do will be incorrect and 5 percent of the time that for B1 will be incorrect? Discuss.
4.2. Refer to Problem 2.1. Suppose the student combines the two confidence intervals into a con
dence set. What can you say about the family confidence coefficient for this set?
*4.3. Refer to Copier maintenance Problem 1.20.
a. Will bo and bl tend to err in the same direction or in opposite directions here? Explain.
b. Obtain Bonferroni joint confidence intervals for Bo and B,, using a 95 percent family conti.
dence coefficient.
c. A consultant has suggested that Bo should be 0 and BI should equal 14.0. Do your join1
confidence intervals in part (b) support this view?
*4.4. Refer to Airfreight breakage Problem 1.21.
a. Will bo and bl tend to err in the same direction or in opposite directions here? Explain.
b. Obtain Bonferroni joint confidence intervals for Po and BI, using a 99 percent family co
dence coefficient. Interpret your confidence intervals.
a. Obtain Bonferroni joint confidence intervals for Do and PI, using a 90 percent family co
fidence coefficient. lnterpret your confidence intervals.
b. Are bo and bl positively or negatively correlated here? Is this reflected in your joint con
dence intervals in part (a)?
c. What is the meaning of the family confidence coefficient in part (a)?
*4.6. Refer to Muscle mass Problem 1.27.
a. Obtain Bonferroni joint confidence intervals for Bo and PI, using a 99 percent family contl-
dence coefficient, lnterpret your confidence intervals.
b. Will bo and bl tend to err in the same direction or in opposite directions here? Explain.
c. A researcher has suggested that Po should equal approximately 160 and that Bl should
between - 1.9 and - 1.5. Do thejoint confidence intervals in part (a) support this expectatio
*4.7. Refer to Copier maintenance Problem 1.20.
a. Estimate the expected number of minutes spent when there are 3, 5, and 7 copiers to
serviced, respectively. Use interval estimates with a 90 percent family confidence coefficicrd
based on the Working-Hotelling procedure.
b. Two service calls for preventive maintenance are scheduled in which the numbers of copien
to be serviced are 4 and 7, respectively. A family of prediction intervals for the times 1
procedure., Scheffk or Bonferroni, will provide tighter prediction limits here?
c. Obtain the family of prediction intervals required in part (b), using the more efficid
procedure.

Chapter 4 Simultaneous Inferences and Other Topics in Regression Analysis 173
*4.8. Refer to Airfreight breakage Problem I .2 I.
a. It is desired to obtain interval estimates of the mean number of broken ampules when there
are 0, I, and 2 transfers for a shipment, using a 95 percent family confidence coefficient.
Obtain the desired confidence intervals, using the Working-Hotelling procedure.
b. Are the confidence intervals obtained in part (a) more efficient than Bonferroni intervals
here? Explain.
c. The next three shipments will make 0, I, and 2 transfers, respectively. Obtain prediction
intervals for the number of broken ampules for each of these three shipments, using the
Scheffi5 procedure and a 95 percent family confidence coefficient.
d. Would the Bonferroni procedure have been more efficient in developing the prediction
intervals in part (c)? Explain.
to Plastic hardness Problem 1.22.
a. Management wishes to obtain interval estimates of the mean hardness when the elapsed time
is 20, 30, and 40 hours, respectively. Calculate the desired confidence intervals, using the
Bonferroni procedure and a 90 percent family confidence coefficient. What is the meaning
of the family confidence coefficient here?
b. Is the Bonferroni procedure employed in part (a) the most efficient one that could be
employed here? Explain.
c. The next two test items will be measured after 30 and 40 hours ofelapsed time, respectively.
Predict the hardness for each of these two items, using the most efficient procedure and a
, 90 percent family confidence coefficient.
-$4.10. Refer to Muscle mass Problem 1.27. .
a. The nutritionist is particularly interested in the mean muscle mass for women aged 45,55, and
65. Obtain joint confidence intervals for the means of interest using the Working-Hotelling
procedure and a 95 percent family confidence coefficient.
b. Is the Working-Hotelling procedure the most efficient one to be employed in part (a)?
Explain.
c. Three additional women aged 48, 59, and 74 have contacted the nutritionist. Predict the
muscle mass for each of the'se three women using the Bonferroni procedure and a95 percent
family confidence coefficient.
d. Subsequently, the nutritionist wishes to predict the muscle mass for a fourth woman aged
64, with a family confidence coefficient of 95 percent for the four predictions. Will the three
prediction intervals in part (c) have to be recalculated? Would this also be true if the Scheffk
procedure had been used in constructing the prediction intervals?
4. I I. A behavioral scientist said, "I am never sure whether the regression line goes through the origin.
Hence, I will not use such a model." Comment.
4.12. npographical errors. Shown below are the number of galleys for a manuscript (X) and
the total dollar cost of correcting typographical errors (Y) in a random sample of recent orders
handled by a firm specializing in technical manuscripts. Since Y involves variable costs only, an
analyst wished to determine whether regression-through-the-origin model (4.10) is appropriate
for studying the relation between the two variables.
a. Fit regression model (4.10) and state the estimated regression function.

210 Part One Simple Linear Regression
*5.4. Flavor deterioration. The results shown below were obtained in a small-scale experiment to
study the relation between "F of storage temperature (X) and number of weeks before flavor
deterioration of a food product begins to occur (Y).
i: 1 2 3 4 5
Xi: 8 4 0 -4 -8
Yi : 7.8 9.0 10.2 11 .O 11.7
Assume that first-order regression model (2.1) is applicable. Using matrix methods, find (1)
Y'Y, (2) X'X, (3) X'Y.
5.5. Consumer finance. The data below show, for a consumer finance company operating in six
cities, the number of competing loan companies operating in the city (X) and the number per
thousand of the company's loans made in that city that are currently delinquent (Y):
i: 1 2 3 4 5 6
xi: 4 1. 2 3 3 4
Yi: 16 5 10 15 13 22
Assume that first-order regression model (2.1) is applicable. Using matrix methods, find (I)
Y'Y, (2) X'X, (3) X'Y.
*5.6. Refer to Airfreight breakage Problem 1.21. Using matrix methods, find (I) Y'Y, (2) X'X,
a. Are the column vectors of B linearly dependent?
b. What is the rank of B?
c. What must be the determinant of B?
5.9. Let A be defined as follows:
a. Are the column vectors of A linearly dependent?
b. Restate definition (5.20) in terms of row vectors. Are the row vectors of A linearly dependent?
c. What is the rank of A?
d. Calculate the determinant of A.
5.10. Find the inverse of each of the following matrices:
4 3
A = [; ;] B = [,: 7 I:]
Check in each case that the resulting matrix is indeed the inverse.

11e experimenl 11
eks before flavol
methods, find I I
y operating in si.
id the number pk.1
lent (Y):
, methods, find ( 1 1
(I) Y'Y, (2) X'S
i, (2) X'X, (3) X')'
r linearly dependem
5.1 1. Find the inverse of the following matrix:
Chapter 5 Malrix Approach lo Simple Linear Regression Analysis 211
Check that the resulting matrix is indeed the inverse.
'"5.1 2. Refer to Flavor deterioration Problem 5.4. Find (XIX)-'
5.13. Refer to Consumer finance Problem 5.5. Find (XIX)-I.
:!:5. 14. Consider the simultaneous equations:
a. Write these equations in matrix notation.
b. Using matrix methods, tind the solutions for yl and y?.
5.15. Consider the simultaneous equations:
a. Write these equations in matrix notation.
b. Using matrix methods, tind the solutions for ?;I and ~ 1 : .
5.16. Consider the estimated linear regression function in the form of (1.15). Write expressions in
this form for the titled values ?, in matrix terms for i = 1, . . . , 5.
Consider the following function> of the random variables YI, Y2, and Y3:
w, = Y, + Y? + Y3
a. State the above in matrix notation.
b. Find the expectation of the random vector W.
c. Find the variance-covariance matrix of W.
':'5.18. Consider the following functions of the random variables YI, Y?, Y,, and Y4:
I I
W? = -
2
a. State the above in matrix notation.
b. Find the expectation of the random vector W.
c. Find the variance-covaria~ice matrix of W
Find the matrix A of the quadrat~c form:
5.20. Find the matrix A of the quadratic form:
,(YI + Y2) - -(Y3+ Y4)

I
1
I
212 Part One Simpk Linear Regression
. .
find the quadratic form of the observations YI and Y2.
5.22. For the matrix:
find the quadratic form of the observations YI, Y2. and Y3.
*5.23. Refer to Flavor deterioration Problems 5.4 and 5.12.
a. Using matrix methods, obtain the following: (1) vector of estimated regression coeffic
(2) vector of residuals, (3) SSR, (4) SSE, (5) estimated variance-covariance matrix
(6) point estimate of E{Yh) when Xh = -6, (7) estimated variance of ?h when XI, = -
b. What simplifications arose from the spacing of the X levels in the experiment?
c. Find the hat matrix H. =
d. ~ind s2(e).
5.24. Refer to Consumer finance Problems 5.5 and 5.13.
a. Using matrix methods, obtain the following: (1) vector of estimated regression coeffic
(2) vector of residuals, (3) SSR, (4) SSE, (5) estimated variance-covariance matrix
(6) point estimate of E(Yh) when Xh = 4, (7) s2(pred) when Xs = 4.
b. From your estimated variance-covariance matrix in part (a5), obtain the follo
(1) s(bo, b11; (2) s2{bo1; (3) s{bjl.
c. Find the hat matrix H.
d. Find s2{e].
*5.25. Refer to Airfreight breakage Problems 1.21 and 5.6.
a. Using matrix methods, obtain the following: (1) (XIX)-I, (2) b, (3) e, (4) H, (5
(6) s2(b), (7) ?h when XI, = 2, (8)sZ{?h] when XI, = 2.
b. From part (a6), obtain the following: ( I ) s2{bl); (2) s(bo, bl); (3) s(b0).
c. Find the matrix of the quadratic form for SSR.
Refer to Plastic hardness Problems 1.22 and 5.7.
a. Using matrix methods, obtain the following: (1) (XiX)-I, (2) b, (3) 9,
(6) s2(b), (7) s2{pred) when Xh = 30.
b. From part (a6), obtain the following: (I) s2(bo); (2) s(bn, bl 1; (3) s(bl ).
Exercises 5.27. Refer toregression-through-the-origin model (4.10). Set up theexpectation vector fore. A
thati= 1, ..., 4.
5.28. Consider model (4.10) for regression through the origin and the estimator bl given in
Obtain (4.14) by utilizing (5.60) with X suitably defined.
5.29. Consider the least squares estimator b given in (5.60). Using matrix methods, show thal
unbiased estimator.
5.30. Show that Eh in (5.96) can be expressed in matrix terms as b'Xh.
in terms of the hat matrix.