Title answers to exercises - mathematical statistics with applications (7th edition).pdf 1.3 MB 20
```                            Answers to Exercises
Front Cover
Title Page
Contents
Preface
Note to the Student
1 What Is Statistics?
1.1 Introduction
Exercises
1.2 Characterizing a Set of Measurements: Graphical Methods
Exercises
1.3 Characterizing a Set of Measurements: Numerical Methods
Exercises
1.5 Theory and Reality
1.6 Summary
Supplementary Exercises
2 Probability
2.1 Introduction
2.2 Probability and Inference
2.3 A Review of Set Notation
Exercises
2.4 A Probabilistic Model for an Experiment: The Discrete Case
Exercises
2.5 Calculating the Probability of an Event: The Sample-Point Method
Exercises
2.6 Tools for Counting Sample Points
Exercises
2.7 Conditional Probability and the Independence of Events
Exercises
2.8 Two Laws of Probability
Exercises
2.9 Calculating the Probability of an Event: The Event-Composition Method
Exercises
2.10 The Law of Total Probability and Bayes’ Rule
Exercises
2.11 Numerical Events and Random Variables
Exercises
2.12 Random Sampling
2.13 Summary
Supplementary Exercises
3 Discrete Random Variables and Their Probability Distributions
3.1 Basic Definition
3.2 The Probability Distribution for a Discrete Random Variable
Exercises
3.3 The Expected Value of a Random Variable or a Function of a Random Variable
Exercises
3.4 The Binomial Probability Distribution
Exercises
3.5 The Geometric Probability Distribution
Exercises
3.6 The Negative Binomial Probability Distribution (Optional)
Exercises
3.7 The Hypergeometric Probability Distribution
Exercises
3.8 The Poisson Probability Distribution
Exercises
3.9 Moments and Moment-Generating Functions
Exercises
3.10 Probability-Generating Functions (Optional)
Exercises
3.11 Tchebysheff’s Theorem
Exercises
3.12 Summary
Supplementary Exercises
4 Continuous Variables and Their Probability Distributions
4.1 Introduction
4.2 The Probability Distribution for a Continuous Random Variable
Exercises
4.3 Expected Values for Continuous Random Variables
Exercises
4.4 The Uniform Probability Distribution
Exercises
4.5 The Normal Probability Distribution
Exercises
4.6 The Gamma Probability Distribution
Exercises
4.7 The Beta Probability Distribution
Exercises
4.8 Some General Comment
4.9 Other Expected Values
Exercises
4.10 Tchebysheff’s Theorem
Exercises
4.11 Expectations of Discontinuous Functions and Mixed Probability Distributions (Optional)
Exercises
4.12 Summary
Supplementary Exercises
5 Multivariate Probability Distributions
5.1 Introduction
5.2 Bivariate and Multivariate Probability Distributions
Exercises
5.3 Marginal and Conditional Probability Distributions
Exercises
5.4 Independent Random Variables
Exercises
5.5 The Expected Value of a Function of Random Variables
5.6 Special Theorems
Exercises
5.7 The Covariance of Two Random Variables
Exercises
5.8 The Expected Value and Variance of Linear Functions of Random Variables
Exercises
5.9 The Multinomial Probability Distribution
Exercises
5.10 The Bivariate Normal Distribution (Optional)
Exercises
5.11 Conditional Expectations
Exercises
5.12 Summary
Supplementary Exercises
6 Functions of Random Variables
6.1 Introduction
6.2 Finding the Probability Distribution of a Function of Random Variables
6.3 The Method of Distribution Functions
Exercises
6.4 The Method of Transformations
Exercises
6.5 The Method of Moment-Generating Functions
Exercises
6.6 Multivariable Transformations Using Jacobians (Optional)
Exercises
6.7 Order Statistics
Exercises
6.8 Summary
Supplementary Exercises
7 Sampling Distributions and the Central Limit Theorem
7.1 Introduction
Exercises
7.2 Sampling Distributions Related to the Normal Distribution
Exercises
7.3 The Central Limit Theorem
Exercises
7.4 A Proof of the Central Limit Theorem (Optional)
7.5 The Normal Approximation to the Binomial Distribution
Exercises
7.6 Summary
Supplementary Exercises
8 Estimation
8.1 Introduction
8.2 The Bias and Mean Square Error of Point Estimators
Exercises
8.3 Some Common Unbiased Point Estimators
8.4 Evaluating the Goodness of a Point Estimator
Exercises
8.5 Confidence Intervals
Exercises
8.6 Large-Sample Confidence Intervals
Exercises
8.7 Selecting the Sample Size
Exercises
8.8 Small-Sample Confidence Intervals for &#956; and &#956;[sub(1)] – &#956;[sub(2)]
Exercises
8.9 Confidence Intervals for &#963;[sup(2)]
Exercises
8.10 Summary
Supplementary Exercises
9 Properties of Point Estimators and Methods of Estimation
9.1 Introduction
9.2 Relative Efficiency
Exercises
9.3 Consistency
Exercises
9.4 Sufficiency
Exercises
9.5 The Rao–Blackwell Theorem and Minimum-Variance Unbiased Estimation
Exercises
9.6 The Method of Moments
Exercises
9.7 The Method of Maximum Likelihood
Exercises
9.8 Some Large-Sample Properties of Maximum-Likelihood Estimators (Optional)
Exercises
9.9 Summary
Supplementary Exercises
10 Hypothesis Testing
10.1 Introduction
10.2 Elements of a Statistical Test
Exercises
10.3 Common Large-Sample Tests
Exercises
10.4 Calculating Type II Error Probabilities and Finding the Sample Size for Z Tests
Exercises
10.5 Relationships Between Hypothesis-Testing Procedures and Confidence Intervals
Exercises
10.6 Another Way to Report the Results of a Statistical Test: Attained Significance Levels, or p-Values
Exercises
10.7 Some Comments on the Theory of Hypothesis Testing
Exercises
10.8 Small-Sample Hypothesis Testing for &#956; and &#956;[sub(1)] – &#956;[sub(2)]
Exercises
10.9 Testing Hypotheses Concerning Variances
Exercises
10.10 Power of Tests and the Neyman–Pearson Lemma
Exercises
10.11 Likelihood Ratio Tests
Exercises
10.12 Summary
Supplementary Exercises
11 Linear Models and Estimation by Least Squares
11.1 Introduction
11.2 Linear Statistical Models
11.3 The Method of Least Squares
Exercises
11.4 Properties of the Least-Squares Estimators: Simple Linear Regression
Exercises
11.5 Inferences Concerning the Parameters &#946;[sub(i)]
Exercises
11.6 Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression
Exercises
11.7 Predicting a Particular Value of Y by Using Simple Linear Regression
Exercises
11.8 Correlation
Exercises
11.9 Some Practical Examples
Exercises
11.10 Fitting the Linear Model by Using Matrices
Exercises
11.11 Linear Functions of the Model Parameters: Multiple Linear Regression
11.12 Inferences Concerning Linear Functions of the Model Parameters: Multiple Linear Regression
Exercises
11.13 Predicting a Particular Value of Y by Using Multiple Regression
Exercises
11.14 A Test for H[sub(0)]: &#946;[sub(g+1)] = &#946;[sub(g+2) = · · · = &#946;[sub(k)] = 0
Exercises
11.15 Summary and Concluding Remarks
Supplementary Exercises
12 Considerations in Designing Experiments
12.1 The Elements Affecting the Information in a Sample
12.2 Designing Experiments to Increase Accuracy
Exercises
12.3 The Matched-Pairs Experiment
Exercises
12.4 Some Elementary Experimental Designs
Exercises
12.5 Summary
Supplementary Exercises
13 The Analysis of Variance
13.1 Introduction
13.2 The Analysis of Variance Procedure
Exercises
13.3 Comparison of More Than Two Means: Analysis of Variance for a One-Way Layout
13.4 An Analysis of Variance Table for a One-Way Layout
Exercises
13.5 A Statistical Model for the One-Way Layout
Exercises
13.6 Proof of Additivity of the Sums of Squares and E (MST) for a One-Way Layout (Optional)
13.7 Estimation in the One-Way Layout
Exercises
13.8 A Statistical Model for the Randomized Block Design
Exercises
13.9 The Analysis of Variance for a Randomized Block Design
Exercises
13.10 Estimation in the Randomized Block Design
Exercises
13.11 Selecting the Sample Size
Exercises
13.12 Simultaneous Confidence Intervals for More Than One Parameter
Exercises
13.13 Analysis of Variance Using Linear Models
Exercises
13.14 Summary
Supplementary Exercises
14 Analysis of Categorical Data
14.1 A Description of the Experiment
14.2 The Chi-Square Test
14.3 A Test of a Hypothesis Concerning Specified Cell Probabilities: A Goodness-of-Fit Test
Exercises
14.4 Contingency Tables
Exercises
14.5 r &#215; c Tables with Fixed Row or Column Totals
Exercises
14.6 Other Applications
14.7 Summary and Concluding Remarks
Supplementary Exercises
15 Nonparametric Statistics
15.1 Introduction
15.2 A General Two-Sample Shift Model
15.3 The Sign Test for a Matched-Pairs Experiment
Exercises
15.4 The Wilcoxon Signed-Rank Test for a Matched-Pairs Experiment
Exercises
15.5 Using Ranks for Comparing Two Population Distributions: Independent Random Samples
15.6 The Mann–Whitney U Test: Independent Random Samples
Exercises
15.7 The Kruskal–Wallis Test for the One-Way Layout
Exercises
15.8 The Friedman Test for Randomized Block Designs
Exercises
15.9 The Runs Test: A Test for Randomness
Exercises
15.10 Rank Correlation Coefficient
Exercises
15.11 Some General Comments on Nonparametric Statistical Tests
Supplementary Exercises
16 Introduction to Bayesian Methods for Inference
16.1 Introduction
16.2 Bayesian Priors, Posteriors, and Estimators
Exercises
16.3 Bayesian Credible Intervals
Exercises
16.4 Bayesian Tests of Hypotheses
Exercises
Appendix 1 Matrices and Other Useful Mathematical Results
A1.1 Matrices and Matrix Algebra
A1.3 Multiplication of a Matrix by a Real Number
A1.4 Matrix Multiplication
A1.5 Identity Elements
A1.6 The Inverse of a Matrix
A1.7 The Transpose of a Matrix
A1.8 A Matrix Expression for a System of Simultaneous Linear Equations
A1.9 Inverting a Matrix
A1.10 Solving a System of Simultaneous Linear Equations
A1.11 Other Useful Mathematical Results
Appendix 2 Common Probability Distributions, Means, Variances, and Moment-Generating Functions
Table 1 Discrete Distributions
Table 2 Continuous Distributions
Appendix 3 Tables
Table 1 Binomial Probabilities
Table 2 Table of e[sup(-x)]
Table 3 Poisson Probabilities
Table 4 Normal Curve Areas
Table 5 Percentage Points of the t Distributions

Table 7 Percentage Points of the F Distributions
Table 8 Distribution Function of U
Table 9 Critical Values of T in the Wilcoxon Matched-Pairs, Signed-Ranks Test; n = 5(1)50
Table 10 Distribution of the Total Number of Runs R in Samples of Size (n1, n2); P(R < a)
Table 11 Critical Values of Spearman’s Rank Correlation Coefficient
Table 12 Random Numbers
Index
```
##### Document Text Contents
Page 1

Chapter 1

1.5 a 2.45 − 2.65, 2.65 − 2.85
b 7/30
c 16/30

1.9 a Approx. .68
b Approx. .95
c Approx. .815
d Approx. 0

1.13 a ȳ = 9.79; s = 4.14
b k = 1: (5.65, 13.93); k = 2: (1.51,

18.07); k = 3: (−2.63, 22.21)
1.15 a ȳ = 4.39; s = 1.87

b k = 1: (2.52, 6.26); k = 2: (0.65,
8.13); k = 3: (−1.22, 10)

1.17 For Ex. 1.2, range/4 = 7.35; s = 4.14;
for Ex. 1.3, range/4 = 3.04; s = 3.17;
for Ex. 1.4, range/4 = 2.32, s = 1.87.

1.19 ȳ − s = −19 < 0

1.21 .84
1.23 a 16%

b Approx. 95%
1.25 a 177

c ȳ = 210.8; s = 162.17
d k = 1: (48.6, 373); k = 2:

(−113.5, 535.1); k = 3: (−275.7,
697.3)

1.27 68% or 231 scores; 95% or 323 scores
1.29 .05
1.31 .025
1.33 (0.5, 10.5)
1.35 a (172 − 108)/4 = 16

b ȳ = 136.1; s = 17.1
c a = 136.1 − 2(17.1) = 101.9;

b = 136.1 + 2(17.1) = 170.3

Chapter 2

2.7 A = {two males} = {(M1, M2),
(M1,M3), (M2,M3)}
B = {at least one female} = {(M1,W1),
(M2,W1), (M3,W1), (M1,W2), (M2,W2),
(M3,W2), (W1,W2)}
B̄ = {no females} = A; A ∪ B = S;
A ∩ B = null; A ∩ B̄ = A

2.9 S = {A+, B+, AB+, O+, A−, B−,
AB−, O−}

2.11 a P(E5) = .10; P(E4) = .20
b p = .2

2.13 a E1 = very likely (VL); E2 =
somewhat likely (SL); E3 =
unlikely (U); E4 = other (O)

b No; P(VL) = .24, P(SL) = .24,
P(U) = .40, P(O) = .12

c .48

2.15 a .09
b .19

2.17 a .08
b .16
c .14
d .84

2.19 a (V1, V1), (V1, V2), (V1, V3),
(V2, V1), (V2, V2), (V2, V3),
(V3, V1), (V3, V2), (V3, V3)

b If equally likely, all have
probability of 1/9.

c P(A) = 1/3; P(B) = 5/9;
P(A ∪ B) = 7/9;
P(A ∩ B) = 1/9

2.27 a S = {CC, CR, CL, RC, RR, RL,
LC, LR, LL}

b 5/9
c 5/9

877

Page 2

2.29 c 1/15
2.31 a 3/5; 1/15

b 14/15; 2/5
2.33 c 11/16; 3/8; 1/4
2.35 42
2.37 a 6! = 720

b .5
2.39 a 36

b 1/6
2.41 9(10)6
2.43 504 ways
2.45 408,408
2.49 a 8385

b 18,252
c 8515 required
d Yes

2.51 a 4/19,600
b 276/19,600
c 4140/19,600
d 15180/19,600

2.53 a 60 sample points
b 36/60 = .6

2.55 a
(

90
10

)
b
(

20
4

)(
70
6

)/(
90
10

)
= .111

2.57 (4 × 12)/1326 = .0362
2.59 a .000394

b .00355

2.61 a
364n

365n
b .5005

2.63 1/56
2.65 5/162
2.67 a P(A) = .0605

b .001344
c .00029

2.71 a 1/3
b 1/5
c 5/7
d 1
e 1/7

2.73 a 3/4
b 3/4
c 2/3

2.77 a .40 b .37 c .10
d .67 e .6 f .33
g .90 h .27 i .25

2.93 .364
2.95 a .1

b .9

c .6
d 2/3

2.97 a .999
b .9009

2.101 .05
2.103 a .001

b .000125
2.105 .90
2.109 P(A) ≥ .9833
2.111 .149
2.113 (.98)3(.02)
2.115 (.75)4
2.117 a 4(.5)4 = .25

b (.5)4 = 1/16
2.119 a 1/4

b 1/3
2.121 a 1/n

b
1

n
;

1

n

c
3

7
2.125 1/12
2.127 a .857

c No; .8696
d Yes

2.129 .4
2.133 .9412
2.135 a .57

b .18
c .3158
d .90

2.137 a 2/5
b 3/20

2.139 P(Y = 0) = (.02)3;
P(Y = 1) = 3(.02)2(.98);
P(Y = 2) = 3(.02)(.98)2;
P(Y = 3) = (.98)3

2.141 P(Y = 2) = 1/15; P(Y = 3) = 2/15;
P(Y = 4) = 3/15; P(Y = 5) = 4/15;
P(Y = 6) = 5/15

2.145 18!
2.147 .0083
2.149 a .4

b .6
c .25

2.151 4[p4(1 − p) + p(1 − p)4]
2.153 .313
2.155 a .5

b .15
c .10
d .875

Page 10

5.163 b F(y1, y2) =
y1 y2[1 − α(1 − y1)(1 − y2)]

c f (y1, y2) =
1 − α[(1 − 2y1)(1 − 2y2)],
0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1

d Choose two different values for α
with −1 ≤ α ≤ 1.

5.165 a (p1et1 + p2et2 + p3et3)n
b m(t , 0, 0)
c Cov(X1, X2) = −np1 p2

Chapter 6

6.1 a
1 − u

2
, −1 ≤ u ≤ 1

b
u + 1

2
, −1 ≤ u ≤ 1

c
1√
u

− 1, 0 ≤ u ≤ 1
d E(U1) = −1/3, E(U2) =

1/3, E(U3) = 1/6
e E(2Y −1) = −1/3, E(1−2Y ) =

1/3, E(Y 2) = 1/6
6.3 b fU (u) ={

(u + 4)/100, −4 ≤ u ≤ 6
1/10, 6 < u ≤ 11

c 5.5833

6.5 fU (u) =
1

16

(
u − 3

2

)−1/2
,

5 ≤ u ≥ 53
6.7 a fU (u) =

1

π

2
u−1/2e−u/2,

u ≥ 0
b U has a gamma distribution with

α = 1/2 and β = 2 (recall that
�(1/2) = √π).

6.9 a fU (u) = 2u, 0 ≤ u ≤ 1
b E(U ) = 2/3
c E(Y1 + Y2) = 2/3

6.11 a fU (u) = 4ue−2u , u ≥ 0, a gamma
density with α = 2
and β = 1/2

b E(U ) = 1, V (U ) = 1/2
6.13 fU (u) = F ′U (u) =

u

β2
e−u/β , u > 0

6.15 [−ln(1 − U )]1/2
6.17 a f (y) = αy

α−1

θα
, 0 ≤ y ≤ θ

b Y = θU 1/α
c y = 4√u. The values are 2.0785,

3.229, 1.5036, 1.5610, 2.403.
6.25 fU (u) = 4ue−2u for u ≥ 0
6.27 a fY (y) =

2

β
we−w

2/β , w ≥ 0, which
is Weibull density with m = 2.

b E(Y k/2) = �
(

k

2
+ 1
)

βk/2

6.29 a fW (w) =
1

(

3
2

)
(kT )3/2

w1/2e−w/kT w > 0

b E(W ) = 3
2

kT

6.31 fU (u) =
2

(1 + u)3 , u ≥ 0
6.33 fU (u) = 4(80 − 31u + 3u2),

4.5 ≤ u ≤ 5
6.35 fU (u) = − ln(u), 0 ≤ u ≤ 1
6.37 a mY1(t) = 1 − p + pet

b mW (t) = E(etW ) = [1− p + pet ]n
6.39 fU (u) = 4ue−2u , u ≥ 0
6.43 a Ȳ has a normal distribution

with mean µ and variance σ 2/n
b P(|Ȳ − µ| ≤ 1) = .7888
c The probabilities are .8664, .9544,

.9756. So, as the sample size
increases, so does the probability
that P(|Ȳ − µ| ≤ 1)

6.45 c = \$190.27
6.47 P(U > 16.0128) = .025
6.51 The distribution of Y1 + (n2 − Y2) is

binomial with n1 + n2 trials and success
probability p = .2

6.53 a Binomial (nm, p) where
ni = m

b Binomial (n1 = n2 + · · · nn , p)
c Hypergeometric (r = n,

N = n1 + n2 + · · · nn)
6.55 P(Y ≥ 20) = .077
6.65 a f (u1, u2) =

1

e−[u

2
1+(u2−u1)2]/2 =

1

e−(2u

2
1−2u1u2+u22)/2

b E(U1) = E(Z1) = 0,
E(U2) = E(Z1 + Z2) = 0,
V (U1) = V (Z1) = 1,
V (U2) = V (Z1 + Z2) =
V (Z1) + V (Z2) = 2,
Cov(U1, U2) = E(Z 21) = 1

Page 11

c Not independent since
ρ
= 0.

d This is the bivariate normal
distribution with µ1 = µ2 = 0,
σ 21 = 1, σ 22 = 2, and ρ =

1√
2

6.69 a f (y1, y2) =
1

y21 y
2
2

, y1 > 1,

y2 > 1
e No

6.73 a g(2)(u) = 2u, 0 ≤ u ≤ 1
b E(U2) = 2/3, V (U2) = 1/18

6.75 (10/15)5

6.77 a
n!

( j − 1)!(k − 1 − j)!(n − k)!
y

j−1
j [yk − y j ]k−1− j [θ − yk]n−k

θ n
,

0 ≤ y j < yk ≤ θ
b

(n − k + 1) j
(n + 1)2(n + 2) θ

2

c
(n − k + j + 1)(k − j)

(n + 1)2(n + 2) θ
2

6.81 b 1 − e−9
6.83 1 − (.5)n
6.85 .5
6.87 a g(1)(y) = e−(y−4), y ≥ 4

b E(Y(1)) = 5

6.89 fR(r) = n(n − 1)rn−2(1 − r),
0 ≤ r ≤ 1

6.93 f (w) = 2
3

(
1√
w

− w
)

, 0 ≤ w ≤ 1

6.95 a fU1(u) =




1

2
0 ≤ u ≤ 1

1

2u2
u > 1

b fU2(u) = ue−u , 0 ≤ u
c Same as Ex. 6.35.

6.97 p(W = 0) = p(0) = .0512,
p(1) = .2048, p(2) = .3264,
p(3) = .2656, p(4) = .1186,
p(5) = .0294, p(6) = .0038,
p(7) = .0002

6.101 fU (u) = 1, 0 ≤ u ≤ 1 Therefore, U has
a uniform distribution on (0, 1)

6.103
1

π(1 + u21)
, ∞ < u1 < ∞

6.105
1

B(α, β)
uβ−1(1 − u)α−1, 0 < u < 1

6.107 fU (u) =




1

4

u
0 ≤ u < 1

1

8

u
1 ≤ u ≤ 9

6.109 P(U = C1 − C3) = .4156;
P(U = C2 − C3) = .5844

Chapter 7

7.9 a .7698
b For n = 25, 36, 69, and 64, the

probabilities are (respectively)
.8664, .9284, .9642, .9836.

c The probabilities increase with n.
d Yes

7.11 .8664
7.13 .9876
7.15 a E(X̄ − Ȳ ) = µ1 − µ2

b V (X̄ − Ȳ ) = σ 21 /m + σ 22 /n
c The two sample sizes should be at

least 18.
7.17 P

(∑6
i=1 Z

2
i ≤ 6

)
= .57681

7.19 P(S2 ≥ .065) = .10
7.21 a b = 2.42

b a = .656
c .95

7.27 a .17271
b .23041
d .40312

7.31 a 5.99, 4.89, 4.02, 3.65, 3.48, 3.32
c 13.2767
d 13.2767/3.32 ≈ 4

7.35 a E(F) = 1.029
b V (F) = .076
c 3 is 7.15 standard deviations above

this mean; unlikely value.
7.39 a normal, E(θ̂) = θ =

c1µ1 + c2µ2 + · · · + ckµk
V (θ̂) =

(
c21
n1

+ c
2
2

n2
+ · · · + c

2
k

nk

)
σ 2

b χ2 with n1 + n2 + · · · + nk − k df
c t with n1 + n2 + · · · + nk − k df

7.43 .9544
7.45 .0548
7.47 153
7.49 .0217
7.51 664
7.53 b Ȳ is approximately normal: .0132.
7.55 a random sample; approximately 1.

b .1271

Page 20

15.49 a .0256
b An usually small number of runs

(judged at α = .05) would imply a
clustering of defective items in
time; do not reject.

15.51 R = 13, do not reject
15.53 rS = .911818; yes.
15.55 a rS = −.8449887

b Reject
15.57 rS = .6768, use two-tailed test, reject
15.59 rS = 0; p–value < .005

15.61 a Randomized block design
b No
c p–value = .04076, yes

15.63 T = 73.5, do not reject, consistent with
Ex. 15.62

15.65 U = 17.5, fail to reject H0
15.67 .0159
15.69 H = 7.154, reject
15.71 Fr = 6.21, do not reject
15.73 .10

Chapter 16

16.1 a β(10, 30)
b n = 25
c β(10, 30), n = 25
d Yes
e Posterior for the β(1, 3) prior.

16.3 c Means get closer to .4, std dev
decreases.

e Looks more and more like normal
distribution.

16.7 a
Y + 1
n + 4

b
np + 1
n + 4 ;

np(1 − p)
(n + 4)2

16.9 b
α + 1

α + β + Y ;
(α + 1)(β + Y − 1)

(α + β + Y + 1)(α + β + Y )
16.11 e Ȳ

(

nβ + 1
)

+ αβ
(

1

nβ + 1
)

16.13 a (.099, .710)
b Both probabilities are .025
c P(.099 < p < .710) = .95
h Shorter for larger n.

16.15 (.06064, .32665)
16.17 (.38475, .66183)
16.19 (5.95889, 8.01066)
16.21 Posterior probabilities of null and

alternative are .9526 and .0474,
respectively, accept H0.

16.23 Posterior probabilities of null and
alternative are .1275 and .8725,
respectively, accept Ha .

16.25 Posterior probabilities of null and
alternative are .9700 and .0300,
respectively, accept H0.