Devore J.L., Berk K.N. Modern Mathematical Statistics with Applications

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85. The difference is significant at the .05, .01, and .001

levels.

89. b. No, given that the 95% CI includes 0, the test at the

.05 level does not reject equality of means.

91. (299.2, 1517.8)

93. (1020.2, 1339.9). Because 0 is not in the CI, we would

reject equality of means at the .01 level.

95. Because t ¼ 2.61 and the one-tailed P-value is .007, the

difference is significant at the .05 level using either a

one-tailed or a two-tailed test.

97. a. Because t ¼ 3.04 and the two-tailed P-value is .008,

the difference is significant at the .05 level.

b. No, the mean of the concentration distribution

depends on both the mean and standard deviation

of the log concentration distribution.

99. Because t ¼ 7.50 and the one-tailed P-value is .0000001,

the difference is highly significant, assuming normality.

101. The two-sample t is inappropriate for paired data. The

paired t gives a mean difference .3, t ¼ 2.67, and the

two-tailed P-value is .045, so the means are significantly

different at the .05 level. We are concluding tentatively

that the label understates the alcohol percentage.

103. Because paired t ¼ 3.88 and the two-tailed P-value is

.008, the difference is significant at the .05 and .01

levels, but not at the .001 level.

105. Because z ¼ 2.63 and the two-tailed P-value is .009,

there is a significant difference at the .01 level,

suggesting better survival at the higher temperature.

107. .902, .826, .029, .00000003

109. Because z ¼ 4.25 and the one-tailed P-value is .00001,

the difference is highly significant and companies

appear to discriminate.

111. With Z ¼ð

X 



YÞ=

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

X=n þ



Y=m

, the result is

z ¼5.33, two-tailed P-value ¼ .0000001, so one

should conclude that there is a significant difference in

parameters.

113. (i) not bioequivalent (ii) not bioequivalent (iii)

bioequivalent

Chapter 11

1. a. Reject H

: m

¼ m

in favor of H

, m

not all the same, because

f ¼ 5.57  2.69 ¼ F

.05,4,30

b. Using Table A.9, .001 < P-value < .01. (The P-value

is .0018)

3. Because f ¼ 6.43  2.95 ¼ F

.05,3,28

, there are

significant differences among the means.

5. Because f ¼ 10.85  4.38 ¼ F

.01,3,36

, there are

significant differences among the means.

Source DF SS MS F P

Formation 3 509.1 169.7 10.85 0.000

Error 36 563.1 15.6

Total 39 1072.3

7. a. The Levene test gives f ¼ 1.47, P-value .236, so there

is no reason to doubt equal variances.

b. Because f ¼ 10.48  4.02 ¼ F

.01,4,30

, there are

significant differences among the means.

Source DF SS MS F P

Plate

length

4 43993 10998 10.48 0.000

Error 30 31475 1049

Total 34 75468

11. w ¼ 36.09 31425

Splitting the paints into two groups, {3, 1, 4}, {2, 5},

there are no significant differences within groups but the

paints in the first group differ significantly (they are

lower) from those in the second group.

13. 31425

427.5 462.0 469.3 502.8 532.1

15. w ¼ 5.92; At the 1% level the only significant

differences are between formation 4 and the first two

formations.

2134

24.69 26.08 29.95 33.84

17. (.029, .379)

19. 426

21. a. Because f ¼ 22.60  3.26 ¼ F

.01,5,78

, there are

significant differences among the means.

b. (99.1, 35.7), (29.4, 99.1)

23. The nonsignificant differences are indicated by the

underscores.

10 6 3 1

45.5 50.85 55.40 58.28

25. a. Assume normality and equal variances.

b. Because f ¼ 1.71 < 2.20 ¼ F

.10,3,48

, P-value ¼ .18,

there are no significant differences among the means.

27. a. Because f ¼ 3.75, P-value ¼ .028, there are

significant differences among the means.

b. Because the normal plot looks fairly straight and the

P-value for the Levene test is .68, there is no reason to

doubt the assumptions of normality and constant

variance.

c. The only significant pairwise difference is between

brands 1 and 4:

4321

5.82 6.35 7.50 8.27

31. .63

33. arcsinð

ﬃﬃﬃﬃﬃﬃﬃ

x=n

35. a. Because f ¼ 1.55 < 3.26 ¼ F

.05,4,12

, there are no

significant differences among the means.

b. Because f ¼ 2.98 < 3.49 ¼ F

.05,3,12

, there are no

significant differences among the means.

37. With f ¼ 5.49  4.56 ¼ F

.01,5,15

, there are significant

differences among the stimulus means. Although not all

differences are significant in the multiple comparisons

analysis, the means for combined stimuli were higher.

828 Chapter 11

Differences among the subject means are not very

important here. The normal plot of residuals shows no

reason to doubt normality. However, the plot of residuals

against the fitted values shows some dependence of the

variance on the mean. If logged response is used in place

of response, the plots look good and the F test result is

similar but stronger. Furthermore, the logged response

gives more significant differences in the multiple

comparisons analysis.

Means:

L1 L2 T L1 + L2 L1 + T L2 + T

24.825 27.875 29.1 40.35 41.22 45.05

39. With f ¼ 2.56 < 2.61 ¼ F

.10,3,12

, there are no significant

differences among the angle means.

41. a. With f ¼ 1.04 < 3.28 ¼ F

.05,2,34

, there are no

significant differences among the treatment means.

Source DF SS MS F

Treatment 2 28.78 14.39 1.04

Block 17 2977.67 175.16 12.68

Error 34 469.56 13.81

Total 53 3476.00

b. The very significant f for blocks, which shows that

blocks differ strongly, implies that blocking was

successful.

43. With f ¼ 8.69  6.01 ¼ F

.01,2,18

, there are significant

differences among the three treatment means.

The normal plot of residuals shows no reason to doubt

normality, and the plot of residuals against the fitted

values shows no reason to doubt constant variance.

There is no significant difference between treatments B

and C, but Treatment A differs (it is lower) significantly

from the others at the .01 level.

Means:

A 29.49 B 31.31 C 31.40

45. Because f ¼ 8.87  7.01 ¼ F

01,4,8

, reject the hypothesis

that the variance for B is 0.

49. a.

Source df SS MS F

A 2 30763.0 15381.5 3.79

B 3 34185.6 11395.2 2.81

Interaction 6 43581.2 7263.5 1.79

Error 24 97436.8 4059.9

Total 35 205966.6

b. Because 1.79 < 2.04 ¼ F

.10,6,24

, there is no

significant interaction.

c. Because 3.79  3.40 ¼ F

.05,2,24

, there is a significant

difference among the A means at the .05 level.

d. Because 2.81 < 3.01 ¼ F

..05,6,24

, there is no

significant difference among the B means at the .05

level.

e. Using w ¼ 64.93,

31 2

3960.2 4010.88 4029.10

(continued)

51. a. With f ¼ 1.55 < 2.81 ¼ F

.10,2,12

, there is no

significant interaction at the .10 level.

b. With f ¼ 376.27  18.64 ¼ F

.001,2,12

, there is a

significant difference between the formulation

means at the .001 level.

With f ¼ 19.27  12.97 ¼ F

.001,1,12

, there is a

significant difference among the speed means at the

.001 level.

c. Main effects Formulation: (1) 11.19, (2) –11.19

Speed: (60) 1.99, (70) –5.03, (80) 3.04

53. Here is the ANOVA table

Source DF SS MS F P

Pen 3 1387.5 462.50 0.68 0.583

surface 2 2888.1 1444.04 2.11 0.164

Interaction 6 8100.3 1350.04 1.97 0.149

Error 12 8216.0 684.67

Total 23 20591.8

With f ¼ 1.97 < 2.33 ¼ F

.10,6,12

, there is no significant

interaction at the .10 level.

With f ¼ .68 < 2.61 ¼ F

.10,3,12

, there is no significant

difference among the pen means at the .10 level.

With f ¼ 2.11 < 2.81 ¼ F

.10,2,12

, there is no significant

difference among the surface means at the .10 level.

57. a. F ¼ MSAB/MSE

b. A: F ¼ MSA/MSAB B: F ¼ MSB/MSAB

59. a. Because f ¼ 3.43  2.61 ¼ F

.05,4,40

, there is a

significant difference among the exam means at the

.05 level.

b. Because f ¼ 1.65 < 2.61 ¼ F

.05,4,40

, there is no

significant difference among the retention means at

the .05 level.

61. a.

Source DF SS MS F

Diet 4 .929 .232 2.15

Error 25 2.690 .108

Total 29 3.619

Because f ¼ 2.15 < 2.76 ¼ F

.05,4,25

, there is no

significant difference among the diet means at the .05

level.

b. (.59, .92) Yes, the interval includes 0.

c. .53

63. a. Test H

: m

¼ m

versus H

: the three means

are not all the same. With f ¼ 4.80 and F

.05,2,16

3.63 < 4.80 < 6.23 ¼ F

.01,2,16

, it follows that

.01 < P-value < .05 (more precisely, P ¼ .023).

Reject H

in favor of H

at the 5% level but not at

the 1% level.

b. Only the first and third means differ significantly at

the 5% level.

123

25.59 26.92

28.17

65. Because f ¼ 1123  4.07 ¼ F

.05,3,8

, there are significant

differences among the means at the .05 level.

For Tukey multiple comparisons, w ¼ 7.12:

Chapter 11 829

PCM OCM RM PIM

29.92 33.96 125.84 129.30

The means split into two groups of two. The means within

each group do not differ significantly, but the means in

the top group differ strongly from the means in the

bottom group.

67. The normal plot is reasonably straight, so there is no

reason to doubt the normality assumption.

69.

Source DF SS MS F

A 1 322.667 322.667 980.5

B 3 35.623 11.874 36.1

AB 3 8.557 2.852 8.7

Error 16 5.266 .329

Total 23 372.113

With f ¼ 8.7  3.24 ¼ F

.05,3,16

, there is significant

interaction at the .05 level.

In the presence of significant interaction, main effects are

not very useful.

Chapter 12

1. a. Temperature

17 0

Stem: hundreds and tens

Leaf: ones

17 23

17 445

17 67

18 0000011

18 2222

18 445

18 6

18 8

The distribution is fairly symmetric and bell-shaped with

a center around 180.

Ratio

0 889

Stem: ones;

Leaf: tenths

1 0000

1 4444

166

1 8889

211

300

The distribution is concentrated between 1 and 2, with

some positive skewness.

b. No, x does not determine y: for a given x there may be

more than one y.

c. No, there is a wide range of y values for a given x; for

example when temperature is 18.2 the ratio ranges

from .9 to 2.68.

3. Yes. Yes.

5. b. Yes

c. The relationship of y to x is roughly quadratic.

7. a. 5050 psi b. 1.3 psi c. 130 psi d. 130 psi

9. a. .095 m

/min b. .475 m

/min c. .83 m

/min,

1.305 m

/min d. .4207, .3446 e. .0036

11. a. .01 h, .10 h b. 3.0 h, 2.5 h

c. .3653 d. .4624

13. a. y ¼ .63 + .652x

b. 23.46, 2.46

c. 392, 5.72

d. .956

e. y ¼ 2.29 + .564x, r

¼ .688

15. a. y ¼15.2 + .0942x

b. 1.906

c. 1.006 , 0.096, 0.034, 0.774

d. .451

17. a. Yes

b. slope, .827; intercept, 1.13

c. 40.22

d. 5.24

e. .975

19. a. y ¼ 75.2  .209x 54.274

b. The coefficient of determination is .791, meaning that

the predictor accounts for 79.1% of the variation in y.

c. The value of s is 2.56, so typical deviations from the

regression line will be of this size.

21. b. y ¼2.18 + .660x

c. 7.72

d. 7.72

25.

¼ 1:8

þ 32;

¼ 1:8

29. a. Subtracting



x from each x

shifts the plot



x units to

the left. The slope is left unchanged, but the new

y intercept is



y, the height of the old line at x ¼ x.





Y ¼



x and



31. a. .00189

b. .7101

c. No, because here Sðx

 xÞ

is 24,750, smaller than the

value 70,000 in part (a), so Vð

Þ¼s

=Sðx



xÞ

higher here.

33. a. (.51, 1.40)

b. To test H

: b

¼ 1 vs. H

: b

< 1, we compute

t ¼.2258 > 1.383 ¼t

.10,9

, so there is no

reason to reject the null hypothesis, even at the 10%

level. There is no conflict between the data and the

assertion that the slope is at least 1.

35. a.

¼ 1:536, and a 95% CI is (.632, 2.440)

b. Yes, for the test of H

: b

¼ 0 vs. H

: b

6¼ 0, we find

t ¼ 3.62, with P-value .0025. At the .01 level

conclude that there is a useful linear relationship.

c. Because 5 is beyond the range of the data, predicting

at a dose of 5 might involve too much extrapolation.

830 Chapter 12

¼ 1:683, and a 95% CI is (.531, 2.835).

Eliminating the point causes only moderate change,

so the point is not extremely influential.

37. a. Yes, for the test of H

: b

¼ 0 vs. H

: b

6¼ 0, we find

t ¼6.73, with P-value .00002. At the .01 level

conclude that there is a useful linear relationship.

b. (2.77, –1.42)

43. No, z ¼ .73 and the P-value is .46, so there is no evidence

for a significant impact of age on kyphosis.

45. a. s

increases as the distance of x from



x increases

b. (2.26, 3.19)

c. (1.34, 4.11)

d. At least 90%

47. a. The regression equation is y ¼1:58 þ 2:59x and

¼ .838.

b. A 95% confidence interval for the slope is ( 2.16,

3.01). In repetitions of the whole process of data

collection and calculation of the interval, roughly

95% of the intervals will contain the true slope.

c. When tannin ¼ .6 the estimated mean astringency is

0.0335 and the 95% confidence interval is (0.125,

0.058)

d. When tannin ¼ .6 the predicted astringency is

0.0335 and the 95% prediction interval is

(0.5582, 0.4912)

e. Our null hypothesis is that true average astringency

is 0 when tannin is .7, and the alternative is that the

true average is positive. The t for this test is 4.61, with

P-value ¼ .000035, so yes there is compelling

evidence.

49. (431.2, 628.6)

51. a. Yes, for the test of H

: b

¼ 0 vs. H

: b

6¼ 0,

we find t ¼ 10.62, with P-value .000014. At the

.001 level conclude that there is a useful linear

relationship.

b. (8.24, 12.96) With 95% confidence, when the flow

rate is increased by 1 SCCM, the associated expected

change in etch rate is in the interval.

c. (36.10, 40.41) This is fairly precise.

d. (31.86, 44.65) This is much less precise than the

interval in (c)

e. Because 2.5 is closer to the mean, the intervals will be

narrower.

f. B ecause 6 is outside the range of the data, it is

unknown whether the regression will apply there.

g. Use a 99% CI at each value: (23.88, 31.43), (29.93,

35.98), (35.07, 41.45)

53. a. Yes

b. Yes, for the test of H

: b

¼ 0 vs. H

: b

6¼ 0, we find

t ¼4.39, with P-value < .001. At the .001 level

conclude that there is a useful linear relationship.

c. (403.6, 468.2)

57. a. r ¼ .923, so x and y are strongly correlated.

b. unaffected

c. unaffected

d. The normal plots seem consistent with normality, but

the scatter plot shows a slight curvature.

e. For the test of H

: r ¼ 0 vs. H

: r 6¼ 0, we find

t ¼ 7.59, with P-value .00002. At the .001 level

conclude that there is a useful linear relationship.

59. a. For the test of H

: r ¼ 0 vs. H

: r > 0, we find

r ¼ .760, t ¼ 4.05, with P-value < .001. At the .001

level conclude that there is a positive correlation.

b. Because r

¼ .578 we say that the regression accounts

for 57.8 % of the variation in endurance. This also

applies to prediction of lactate level from endurance.

61. For the test of H

: r ¼ 0 vs. H

: r 6¼ 0, we find r ¼ .773,

t ¼ 2.44, with P-value .072. At the .05 level conclude

that there is not a significant correlation. With such a

small sample size, a high r is needed for significance.

63. a. Reject the null hypothesis in favor of the alternative.

b. No, with a large sample size a small r can be

significant.

c. Because t ¼ 2.200  1.96 ¼ t

.025,9998

the correlation

is statistically (but not necessarily practically)

significant at the .05 level.

67. a. .184, –.238, –.426

b. The mean that is subtracted is not the mean x

1;n1

, x

,, ..., x

n–1

, or the mean



2;n

of x

, x

,, ..., x

Also, the denominator of r

is not

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

n1

ðx



1;n1

q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðx



2;n

. However, if

n is large then r

is approximately the same as the

correlation. A similar relationship applies to r

c. No

d. After performing one test at the .05 level, doing more

tests raises the probability of at least one type I error to

more than .05.

69. The plot shows no reasons for concern about using the

simple linear regression model.

71. a. The simple linear regression model may not be a

perfect fit because the plot shows some curvature.

b. The plot of standardized residuals is very similar to

the residual plot. The normal probability plot gives

no reason to doubt normality.

73. a. For the test of H

: b

¼ 0 vs. H

: b

6¼ 0, we find

t ¼ 10.97, with P-value .0004. At the .001 level

conclude that there is a useful linear relationship.

b. The residual plot shows curvature, so the linear

relationship of part (a) is questionable.

c. There are no extreme standardized residuals , and the

plot of standardized residuals is similar to the plot of

ordinary residuals.

75. The first data set seems appropriate for a straight-line

model. The second data set shows a quadratic

relationship, so the straight-line relationship is

inappropriate. The third data set is linear except for an

outlier, and removal of the outlier will allow a line to be

fit. The fourth data set has only two values of x, so there is

no way to tell if the relationship is linear.

77. a. To test for lack of fit, we find f ¼ 3.30, with 3

numerator df and 10 denominator df, so the P-value

is .079. At the .05 level we cannot conclude that the

relationship is poor.

b. The scatter plot shows that the relationship is not

linear, in spite of (a). In this case, the plot is more

sensitive than the test.

79. a. 77.3

b. 40.4

c. The coefficient b

is the difference in sales caused by

the window, all other things being equal.

Chapter 12

831

81. a. .686, no

b. We find f ¼ 28.6  2.62 ¼ F

.001,16,186

, so there is a

significant relationship at the .001 level.

c. With all other predictors held constant, the estimated

difference in y between class A and not is .364. In

terms of $/ft

, the effect is multiplicative. Class A

buildings are estimated to be worth 44% more

dollars per square foot, with all other predictors held

constant.

d. The difference in (c) is highly significant because the

two-tailed P-value is .00000013.

83. a. 48.31, 3.69

b. No, because the interaction term will change.

c. Yes, f ¼ 18.92, P-value < .0001.

d. Yes, t ¼ 3.496, P-value ¼ .003  .01

e. (21.6, 41.6)

f. There appear to be no problems with normality or

curvature, but the variance may depend on x

85. a. No

b. With f ¼ 5.03  3.69 ¼ F

.05,5,8

, there is a significant

relationship at the .05 level.

c. Yes, the individual hypotheses deal with the issue of

whether an individual predictor can be deleted, not the

effectiveness of the whole model.

d. 6.2, 3.3, (16.7, 31.9)

e. With f ¼ 3.44 < 4.07 ¼ F

.05,3,8

, there is no reason to

reject the null hypothesis, so the quadratic terms can

be deleted.

87. a. The quadratic terms are important in providing a good

fit to the data.

b. A 95% PI is (.560, .771).

89. a. r

¼ .843 (.000), r

¼ .621 (.001), r

¼ .843

(.000) Here the P-values are given in parentheses to

three decimals.

b. Rating ¼ 2.24 + 0.0419 IBU – 0.166 ABV. Because

the two predictors are highly correlated, one is

redundant.

c. Linearity is an issue.

e. The regression is quite effective, with R

¼ .872. The

ABV coefficient is not significant, so ABV is not

needed. The highly significant positive coefficient

for IBU and negative coefficient for its square show

that Rating increases with IBU, but the rate of increase

is lower at higher IBU.

91. a. X ¼

1

y ¼

400

040

004

b ¼

1:5

y ¼

y 

y ¼

1

SSE ¼ 4, MSE ¼ 4

d. (12.2, 13.2)

e. For the test of H

: b

¼ 0 vs. H

: b

6¼ 0, we

find |t| ¼ .5 < t

.025,1

¼ 12.7, so do not reject H

the .05 level. The x

term does not play a significant

role.

Source DF SS MS F

Regression 2 5 2.5 0.625

Error 1 4 4.0

Total 3 9

With f ¼ .625 < 199.5 ¼ F

.05,2,1

, there is no

significant relationship at the .05 level.

93.



y; s ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðy 



yÞ

=ðn  1Þ

;

¼ 1=n;



y  t

:025;n1

ﬃﬃﬃ

95. a.

m þ n

mþn



mþn

mþ1



; i ¼ 1; ...; m ;



; i ¼ m þ 1; ...; m þ n

SSE ¼

ðy



mþn

mþ1

ðy



s ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

SSE=ðm þ n  2Þ

¼ 4/(m + n)

¼ 128:17;

¼ 14:33

¼ 121; i ¼ 1; ...; 3;

¼ 135:33; i ¼ 4; ...; 6

SSE ¼ 116.67 s ¼ 5.4006 c

¼ 2/3

95% CI for b

(2.09, 26.58)

97. Residual ¼ Dep Var – Predicted Value

Std Error Residual ¼ [MSE – (Std Error Predict)

]

Student Residual ¼ Residual/Std Error Residual

101. a. H

¼ 1=n þðx

 xÞðx

 xÞ=Sðx

 xÞ

Vð

Þ¼s

½1=n þðx

 xÞ

=Sðx

 xÞ



b. VðY



Þ¼s

½1  1=n ðx



xÞ

=Sðx



xÞ



c. The variance of a predicted value is greater for an x

that is farther from



d. The variance of a residual is lower for an x that is

farther from x

e. It is intuitive that the variance of prediction should be

higher with increasing distance. However, points that

are farther away tend to draw the line toward them,

so the residual naturally has lower variance.

103. a. With f ¼ 12.04  9.55 ¼ F

.01,2,7

, there is a

significant relationship at the .01 level.

To test H

: b

¼ 0 vs. H

: b

6¼ 0, |t| ¼

2.96  t

.025,7

¼ 2.36, so reject H

at the .05 level.

The foot term is needed.

To test H

: b

¼ 0 vs. H

: b

6¼ 0, |t| ¼

0.02 < t

.025,7

¼ 2.36, so do not reject H

at the .05

level. The height term is not needed.

b. The highest leverage is .88 for the fifth point. The

height for this student is given as 54 inches, too low

to be correct for this group of students. Also this

value differs by 8

from the wingspan, an extreme

difference.

c. Point 1 has leverage .55, and this student has height

75, foot length 13, both quite high.

Point 2 has leverage .31, and this student has height

66 and foot length 8.5, at the low end.

Point 7 has leverage .31 and this student has both

height and foot length at the high end.

d. Point 2 has the most extreme residual. This student

has a height of 66

and a wingspan of 56

differing

by 10

, so the extremely low wingspan is probably

wrong.

e. For this data set it would make sense to eliminate

points 2 and 5 because they seem to be wrong.

However, outliers are not always mistakes and one

needs to be careful about eliminating them.

832 Chapter 12

105. a. .507% b. .7122

c. To test H

: b

¼ 0 vs. H

: b

6¼ 0, we have t ¼ 3.93,

with P-value .0013. At the .01 level conclude that

there is a useful linear relationship.

d. (1.056, 1.275)

y ¼ 1:014 y 

y ¼:214

107. –36.18, (64.43, –7.94)

109. No, if the relationship of y to x is linear, then the

relationship of y

to x is quadratic.

111. a. Yes

y ¼ 98:293 y 

y ¼ :117

c. s ¼ .155

d. .794

e. 95% CI for b

: (.0613, .0901)

f. The new observation is an outlier, and has a major

impact:

The equation of the line changes from

y ¼ 97.50 + .0757 x to y ¼ 97.28 + .1603 x

s changes from .155 to .291

changes from .794 to .616

113. a. The paired t procedure gives t ¼ 3.54 with a two-

tailed P-value of .002, so at the .01 level we reject the

hypothesis of equal means.

b. The regression line is y ¼ 4.79 + .743x, and the test

of H

: b

¼ 0 vs. H

: b

6¼ 0, gives t ¼ 7.41 with a

P-value of <.000001, so there is a significant

relationship. However, prediction is not perfect,

with r

¼ .753, so one variable accounts for only

75% of the variability in the other.

117. a. linear

b. After fitting a line to the data, the residuals show a

lot of curvature.

c. Yes. The residuals from the logged model show some

departure from linearity, but the fit is good in terms

of R

¼ .988. We find

a ¼ 411:98,

b ¼:03333:

d. (58.15, 104.18)

119. a. The plot suggests a quadratic model.

b. With f ¼ 25.08 and a P-value of < .0001, there is a

significant relationship at the .0001 level

c. CI: (3282.3, 3581.3), PI: (2966.6, 3897.0). Of course,

the PI is wider, as in simple linear regression,

because it needs to include the variability of a new

observation in addition to the variability of the mean.

d. CI: (3257.6, 3565.6), PI: (2945.0, 3878.2). These are

slightly wider than the intervals in (c),which is

appropriate, given that 25 is slightly closer to the

mean and the vertex.

e. With t ¼6.73 and a two-tailed P-value of

< .0001, the quadratic term is significant at the

.0001 level, so this term is definitely needed.

121. a. With f ¼ 2.4 < 5.86 ¼ F

.05,15,4

, there is no

significant relationship at the .05 level

b. No, especially when k is large compared to n

c. .9565

Chapter 13

1. a. reject H

b. do not reject H

c. do not reject

d. do not reject H

3. Do not reject H

because w

¼ 1:57 < 7:815 ¼ w

:05;3

5. Because w

¼ 6.61 with P-value .68, do not reject H

7. Because w

¼ 4.03 with P-value > .10, do not reject H

9. a. [0, .223), [.223, .510), [.510, .916), [.916, 1.609).

[1.609, 1)

b. Because w

¼ 1.25 with P-value > .10, do not reject

11. a. (1, .967), [.967, .431), [.431, 0), [0, .431),

[.431, .967), [.967, 1)

b. (1,.49806), [.49806, .49914), [.49914, .50), [.50,

.50086), [.50086, .50194), [.50194, 1)

c. Because w

¼ 5.53 with P-value > .10, do not reject

13. Using

p ¼ :0843, w

¼ 280.3 with P-value < .001, so

reject the independence model.

15. The likelihood is proportional to y

233

(1 – y)

367

from

which

y ¼ :3883. This gives estimated probabilities

.1400, .3555, .3385, .1433, .0227 and expected counts

21.00, 53.32, 50.78, 21.49, 3.41. Because 3.41 < 5,

combine the last two categories, giving w

¼ 1.62 with

P-value > .10. Do not reject the binomial model.

17.

l ¼ 3:167 which gives w

¼ 103.9 with P-value < .001,

so reject the assumption of a Poisson model.

19.

¼ :4275;

¼ :2750 which gives w

¼ 29.3 with P-

value < .001, so reject the model.

21. Yes, the test gives no reason to reject the null hypothesis

of a normal distribution.

23. The P-values are both .243.

25. Let p

¼ the probability that a fruit given treatment i

matures and p

¼ the probability that a fruit given

treatment i aborts, so H

: p

¼ p

for i ¼ 1, 2, 3, 4, 5.

We find w

¼ 24.82 with P-value < .001, so reject the

null hypothesis and conclude that maturation is affected

by leaf removal.

27. If p

denotes the probability of a type j response when

treatment i is applied, then H

¼ p

for

j ¼ 1, 2, 3, 4. With w

¼ 27:66  23:587 ¼ w

:005;9

reject H

at the .005 level. The treatment does affect the

response.

29. With w

¼ 64:65  13:277 ¼ w

:01;4

, reject H

at the .001

level. Political views are related to marijuana usage. In

particular, liberals are more likely to be users.

31. Compute the expected counts by

ijk

¼ n

ijk

¼ n

i

j

k

¼ n

i

j

k

. For the

statistic df ¼ 20.

33. a. With w

¼ :681 < 4:605 ¼ w

:10;2

, do not reject

independence at the .10 level.

b. With w

¼ 6:81  4:605 ¼ w

:10;2

, reject independence

at the .10 level.

c. 677

35. a. With w

¼ 6.45 and P-value .040, reject independence

at the .05 level.

b. With z ¼2.29 and P-value .022, reject

independence at the .05 level.

Chapter 13

833

c. Because the logistic regression takes into account the

order in the professorial ranks, it should be more

sensitive, so it should give a lower P-value.

d. There are few female professors but many assistant

professors, and the assistant professors will be the

professors of the future.

37. With w

¼ 13:005  9:210 ¼ w

:01;2

, reject the null

hypothesis of no effect at the .01 level. Oil does make a

difference (more parasites).

39. a. H

: The population proportion of Late Game

Leader Wins is the same for all four sports; H

: The

proportion of Late Game Leader Wins is not the same

for all four sports. With w

¼ 10:518  7:815 ¼w

:05;3

reject the null hypothesis at level .05. Sports differ in

terms of coming from behind late in the game.

b. Yes (baseball)

41. With w

¼ 197:6  16:812 ¼ w

:01;6

, reject the null

hypothesis at the .01 level. The aged are more likely to

die in a chronic-care facility.

43. With w

¼ :763 < 7:779 ¼ w

:10;4

, do not reject the

hypothesis of independence at the .10 level. There is no

evidence that age influences the need for item pricing.

45. a. No, w

¼ 9:02  7:815 ¼ w

:05;3

b. With w

¼ :157 < 6:251 ¼ w

:10;3

, there is no reason to

say the model does not fit.

47. a. H

: p

¼ p

¼ ... ¼ p

¼ .10 vs. H

: at least one

6¼ .10, with df ¼ 9.

b. H

: p

¼ .01 for i and j ¼ 0,1,2,...,9 vs. H

: at least

one p

6¼ .01, with df ¼ 99.

c. No, there must be more observations than cells to do a

valid chi-square test.

d. The results give no reason to reject randomness.

Chapter 14

1. For a two-tailed test of H

: m ¼ 100 at level .05, we

find that s

¼ 27 and because 14 < s

< 64, we do not

reject H

3. For a two-tailed test of H

: m ¼ 7.39 at level .05, we find

that s

¼ 18 and because s

does not satisfy

21 < s

< 84, we reject H

5. We form the difference and perform a two-tailed test of

: m ¼ 0 at level .05. This gives s

¼ 72 and because

it does not satisfy 14 < s

< 64, we reject H

at the

.05 level.

7. Because s

¼ 162.5 with P-value .044, reject H

: m ¼ 75

in favor of H

: m > 75 at the .05 level.

9. With w ¼ 38, reject H

at the .05 level because the

rejection region is {w  36}.

11. Test H

: m

– m

¼ 1 vs. H

: m

– m

> 1. After

subtracting 1 from the original process measurements,

we get w ¼ 65. Do not reject H

because w < 84.

13. b. Test H

: m

– m

¼ 0 vs. H

: m

– m

< 0. With a

P-value of .002 we reject H

at the .01 level.

15. With w ¼ 135, z ¼ 2.223, and the approximate P-value

is .026, so we would not reject the null hypothesis at the

.01 level.

17. (11.15, 23.80)

19. (.585, .025)

21. (16, 87)

29. a. (.4736, .6669)

b. (.4736, .6669)

33. For a two-tailed test at level .05, we find that s

¼ 24 and

because 4 < s

< 32, we do not reject the hypothesis of

equal means.

35. a. a ¼ .0207; Bin(20, .5)

b. c ¼ 14; because y ¼ 12, do not reject H

37. With K ¼ 20:12  13:277 ¼ w

:01;4

, reject the null hypo-

thesis of equal means at the 1% level. Axial strength does

seem to (as an increasing function) depend on plate

length.

39. Because f

¼ 6:45 < 7:815 ¼ w

:05;3

, do not reject the null

hypothesis of equal emotion means at the 5% level.

41. Because w

¼ 26 < 27, do not reject the null hypothesis

at the 5% level.

834 Chapter 14

Index

Additive model

for ANOVA, 584–6, 589

for linear regression analysis, 624

for multiple regression

analysis, 682

Alternative hypothesis, 426

Analysis of covariance, 699

Analysis of variance (ANOVA)

additive model for, 584–586,

597

data transformation for, 579

definition of, 552

expected value in, 556, 573,

589, 597

fixed vs. random effects, 579

Friedman test, 785

fundamental identity of, 560,

564, 587, 599, 600, 635

interaction model for, 597–606

Kruskal–Wallis test, 784

Levene test, 562–563

linear regression and, 636, 639,

664, 708, 717

mean in, 553, 555, 557

mixed effects model for,

593, 603

multiple comparisons in,

564–571, 578, 589–590, 603

noncentrality parameter for,

574, 582

notation for, 555, 559, 598

power curves for, 574–575

randomized block experiments

and, 590–593

regression identity of, 635–636

sample sizes in, 574–576

single-factor, 553–582

two-factor, 582–608

type I error in, 558–559

type II error in, 574

Ansari–Bradley test, 786

Association, causation and, 251, 671

Asymptotic normal distribution, 298,

371, 375, 377, 671

Asymptotic relative efficiency,

764, 769

Autocorrelation coefficient, 674

Average

definition of, 25

deviation, 33

pairwise, 379, 772–773, 775

rank, 785

weighted (see Weighted average)

Bar graph, 9, 19

Bartlett’s test, 562

Bayesian approach to inference, 758,

776–782

Bayes’ Theorem, 79–81, 777, 780

Bernoulli distribution, 104, 122, 134,

302 373, 375, 377, 777

Bernoulli random variable

binomial random variable and,

134, 302

Crame

r–Rao inequality for, 375

definition of, 98

expected value, 113

Fisher information on, 372–373,

377

Laplace’s rule of succession

and, 782

mean of, 113

mle for, 377

moment generating function for,

122, 123, 127

pmf of, 103

score function for, 372

in Wilcoxon’s signed-rank

statistic, 314

Beta distribution, 206–208, 777

Beta functions, incomplete, 207

Bias-corrected and accelerated

interval, 415, 417, 538

Bimodal histogram, 18, 19

Binomial distribution

basics of, 128–135

Bayesian approach to, 777–780

multinomial distribution and, 240

normal distribution and,

189–190, 302

Poisson distribution and,

147–149

Binomial experiment, 130–131, 134,

147, 240, 302, 724

Binomial random variable

Bernoulli random variables and,

134, 302

cdf for, 132

definition of, 130

distribution of, 132

expected value of, 134, 135

in hypergeometric experiment,

141

in hypothesis testing, 428–431,

450–454

mean of, 134–135

moment generating function for,

135

multinomial distribution of, 240

in negative binomial experiment,

142

normal approximation of,

189–190, 302

pmf for, 132

and Poisson distribution,

147–149

standard deviation of, 134

unbiased estimation, 335, 337

variance of, 134, 135

Binomial theorem, 135, 142–144

Bioequivalence tests, 551

835

Birth process, pure, 378

Bivariate data, 3, 617, 623, 632,

691, 721

Bivariate normal distribution,

258–260, 310, 318, 477,

667–671

Bonferroni confidence intervals,

424, 657–659, 689

Bootstrap procedure

for confidence intervals,

411–418, 532–534

for paired data, 538–540

for point estimates, 345–346

Bound on the error of estimation, 388

Box–Muller transformation, 271

Boxplot, 37–41

comparative, 40–41

Branching process, 281

Categorical data

classification of, 30

graphs for, 19

in multiple regression analysis,

696–699

Pareto diagram, 24

sample proportion in, 30

Cauchy distribution

mean of, 322, 342

median of, 342

minimal sufficiency for, 367

reciprocals and, 231

standard normal distribution

and, 271

uniform distribution and, 226

variance of sample mean

for, 349

Causation, association and, 251, 671

cdf. See Cumulative distribution

function

Cell counts/frequencies, 725–727,

729–730, 732–740, 744–750

Cell probabilities, 729, 732, 737, 739

Censored experiments, 32, 343–344

Census, 2

Central Limit Theorem

basics of, 298–303

Law of Large Numbers and, 305

proof of, 329–330

sample proportion distribution

and, 190

Wilcoxon rank-sum test and, 770

Wilcoxon signed-rank test

and, 765

Central t distribution, 320–323, 423

Chebyshev’s inequality, 120, 138,

156, 194, 303, 345

Chi-squared distribution

censored experiment and, 421

in confidence intervals,

389–390, 410

critical values for, 317, 389,

409–410, 477, 725, 727,

737–738

definition of, 200

degrees of freedom for, 200, 315

exponential distribution

and, 317

F distribution and, 323–325

gamma distribution and, 200, 315

in goodness-of-fit tests, 720–751

Rayleigh distribution and, 226

standard normal distribution

and, 224, 316–317, 325

of sum of squares, 317, 557

t distribution and, 320, 325

in transformation, 224

Weibull distribution and, 231

Chi-squared random variable

in ANOVA, 557

cdf for, 316

expected value of, 315

in hypothesis testing, 482

in likelihood ratio tests, 477, 480

mean of, 315

moment generating function

of, 315

pdf of, 200, 315

standard normal random

variables and, 224,

316–317, 325

in Tukey’s procedure, 565

variance of, 315

Chi-squared test

degrees of freedom in, 726, 734,

736, 745, 748

for goodness of fit, 724-730,

for homogeneity, 745–747

for independence, 747–749

P-value for, 727–728

for specified distribution,

729–730

z test and, 752

Class interva ls, 15–17, 278, 293,

738–739

Coefficient of determination

definition of, 632–634, 686

F ratio and, 687

in multiple regression, 686

sample correla tion coefficient

and, 664

Coefficient of skewness, 121,

128, 178

Coefficient of variation, 45, 229, 357

Cohort, 281

Combination, 70–72

Comparative boxplot, 40–41, 502,

503, 554

Complement of an event, 53, 60

Compound event, 52, 62

Concentration parameter, 779

Conceptual population, 6, 113,

287, 487

Conditional density, 253

Conditional distribution, 253–263,

361, 369, 667, 735, 758, 777

Conditional mean, 255–262

Conditional probability, 74–81,

84–85, 200, 253–255, 362,

365–366

Conditional probability density

function, 253

Conditional probability mass

function, 253, 255

Conditional variance, 255–262, 367

Confidence bound, 398–399, 403,

440, 494, 500, 513

Confidence interval

adjustment of, 400

in ANOVA, 565, 570–571, 578,

589, 591, 603

based on t distribution, 401–404,

499–501, 505, 513–515,

570–571, 643–646

Bonferroni, 424, 657-659

bootstrap procedure for,

411–418, 538, 540, 532–534

for a contrast, 571

for a correlation coefficient, 671

vs. credibility interval, 777–781

definition of, 382

derivation of, 389

for difference of means, 493–495,

500–501, 505, 513–515,

532–534, 539–540, 565–569,

578, 589, 591, 603

for difference of proportions, 524

distribution-free, 771–776

for exponential distribution

parameter, 389

in linear regression, 643–646,

656–658

for mean, 383–387, 392,

403–404, 411–415

for median, 415–417

in multiple regression, 689, 712

one-sided, 398, 500, 513

for paired data, 513–515, 539

for ratio of variances, 530–531, 537

sample size and, 388

Scheffe

method for, 610

sign, 784

for slope coefficient, 643

for standard deviation, 409–410

for variance, 409–410

width of, 385, 387–388, 394, 397,

404, 417, 495

836 Index

Wilcoxon rank-sum, 774–776

Wilcoxon signed-rank, 772-774

Confidence level

definition of, 382, 385–388

simultaneous, 565–570, 578,

589, 591, 658

in Tukey’s procedure, 565–570,

578, 589, 591

Confidence set, 772

Consistency, 304, 357, 375–377

Consistent estimator, 304, 357,

375–377

Contingency tables, two-way,

744–751

Continuity correction, 189–190

Continuous random variable(s)

conditional pdf for, 254, 789

cumulative distribution function

of, 163–168

definition of, 99, 159

vs. discrete random variable, 162

expected value of, 171–172

joint pdf of (see Joint probability

density functions)

marginal pdf of, 236–238

mean of, 171, 172

moment generating of, 175–177

pdf of (see Probability density

function)

percentiles of, 166–168

standard deviation of, 173–175

transformation of, 220–225,

265–270

variance of, 173–175

Contrast of means, 570–571

Convenience samples, 7

Convergence

in distribution, 153, 329

in mean square, 303

in probability, 304

Convex function, 231

Correction factor, 141, 560, 568,

577, 582

Correction for the mean, 560

Correlation coefficient

autocorrelation coefficient and, 674

in bivariate normal distribution,

258–260, 310, 667

confidence interval for, 671

covariance and, 249

Crame

r–Rao inequality and,

374–375

definition of, 249, 663

estimator for, 666

Fisher transformation, 669

for independent random

variables, 250

in linear regression, 664, 667, 669

measurement error and, 328

paired data and, 515–516

sample (see Sample correlation

coefficient)

Covariance

correlation coefficient and, 249

Crame

r–Rao inequality and,

374–375

definition of, 247

of independent random

variables, 250–251

of linear functions, 249

matrix format for, 711

Covariate, 699

Crame

r–Rao inequality, 374–375

Credibility interval, 777–782

Critical values

chi-squared, 317

F, 324

standard normal (z), 184

studentized range, 565

t, 322, 409

tolerance, 406

Cumulative distribution function

for a continuous random

variable, 163–168

for a discrete random variable,

104–108

inverse function of, 223–224

joint, 282

of order statistics, 272–273

pdf and, 163

percentiles and, 167

pmf and, 105–108

transformation and, 220–225

Cumulative frequency, 24

Cumulative relative frequency, 24

Data

bivariate, 3, 617, 632, 691

categorical (see Categorical data)

censoring of, 32, 343–344

characteristics of, 3

collection of, 7–8

definition of, 2

multivariate, 3, 220

qualitative, 19

univariate, 3

Deductive reasoning, 6

Degrees of freedom (df)

in ANOVA, 557–559,

587, 599

for chi-squared distribution,

200, 315–320

in chi-squared tests, 726, 734,

737, 746

for F distribution, 323

in regression, 631, 685

sample variance and, 35

for Studentized range

distribution, 565

for t distribution, 320, 390,

500, 504

type II error and, 574

Delta method, 174

De Morgan’s laws, 56

Density

conditional, 253–257

curve, 160

function (pdf), 160

joint, 235

marginal, 236

scale, 17

Dependence, 84–88, 238–242, 250,

257, 747

Dependent events, 84-88

Dependent variable, 614

Descriptive statistics, 1–41

Deviation

definition of, 33

minimize absolute deviations

principle, 33, 679

Dichotomous trials, 128

Difference statistic, 347

Discrete random variable(s)

conditional pmf for, 253

cumulative distribution function

of, 104–108

definition of, 99

expected value of, 112

joint pmf of (see Joint probability

mass function)

marginal pmf of, 234

mean of, 112

moment generating of, 122

pmf of (see Probability mass

function)

standard deviation of, 117

transformation of, 225

variance of, 117

Disjoint events, 54

Dotplots, 12

Dummy variable, 696

Dunnett’s method, 571

Efficiency, asymptotic relative,

764, 769

Empirical rule, 187

Erlang distribution, 202, 229

Error(s)

estimated standard, 344, 646, 713

estimation, 334

family vs. individual, 570

measurement, 179, 211, 337, 477

prediction, 405, 658, 683

rounding, 36

standard, 344, 713

Index 837