1 Combinatorial Analysis 2 Axioms Of Probability 3 Conditional Probability And Independence 4 Random Variables 5 Continuous Random Variables 6 Jointly Distributed Random Variables 7 Properties Of Expectation 8 Limit Theorems 9 Additional Topics In Probability 10 Simulation Chapter1: Combinatorial Analysis
Chapter Questions Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and... Problem 1.2P: How many outcome sequences are possible ten a die is rolled four times, where we say, for instance,... Problem 1.3P: Twenty workers are to be assigned to 20 different jobs, one to each job. How many different... Problem 1.4P: John, Jim, Jay, and Jack have formed a band consisting of 4 instruments if each of the boys can play... Problem 1.5P: For years, telephone area codes in the United States and Canada consisted of a sequence of three... Problem 1.6P: A well-known nursery rhyme starts as follows: As I was going to St. Ives I met a man with 7 wives.... Problem 1.7P: a. In how many ways can 3 boys and 3 girls sit in a row? b. In how many ways can 3 boys and 3 girls... Problem 1.8P: When all letters are used, how many different letter arrangements can be made from the letters a.... Problem 1.9P: A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue. If the child puts... Problem 1.10P: In how many ways can 8 people be seated in a row if a. there are no restrictions on the seating... Problem 1.11P: In how many ways can 3 novels. 2 mathematics books, and 1 chemistry book be arranged on a bookshelf... Problem 1.12P: How many 3 digit numbers zyz, with x, y, z all ranging from 0 to9 have at least 2 of their digits... Problem 1.13P: How many different letter permutations, of any length, can be made using the letters M 0 T T 0. (For... Problem 1.14P: Five separate awards (best scholarship, best leadership qualities, and so on) are to be presented to... Problem 1.15P: Consider a group of 20 people. If everyone shakes hands with everyone else, how many handshakes take... Problem 1.16P: How many 5-card poker hands are there? Problem 1.17P: A dance class consists of 22 students, of which 10 are women and 12 are men. If 5 men and 5 women... Problem 1.18P: A student has to sell 2 books from a collection of 6 math, 7 science, and 4 economics books. How... Problem 1.19P: Seven different gifts are to be distributed among 10 children. How many distinct results are... Problem 1.20P: A committee of 7, consisting of 2 Republicans, 2 Democrats, and 3 Independents, is to be chosen from... Problem 1.21P: From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. How... Problem 1.22P: A person has 8 friends, of whom S will be invited to a party. a. How many choices are there if 2 of... Problem 1.23P: Consider the grid of points shown at the top of the next column. Suppose that, starting at the point... Problem 1.24P: In Problem 23, how many different paths are there from A to B that go through the point circled in... Problem 1.25P: A psychology laboratory conducting dream research contains 3 rooms, with 2 beds in each room. If 3... Problem 1.26P: Show k=0n(nk)2k=3n Simplify k=0n(nk)xk Problem 1.27P: Expand (3x2+y)5. Problem 1.28P: The game of bridge is played by 4 players, each of w1om is dealt 13 cards. How many bridge deals are... Problem 1.29P: Expand (x1+2x2+3x3)4. Problem 1.30P: If 12 people are to be divided into 3 committees of respective sizes 3, 4, and 5, how many divisions... Problem 1.31P: If 8 new teachers are to be divided among 4 schools, how many divisions are possible? What if each... Problem 1.32P: Ten weight lifters are competing in a team weight-lifting contest. Of the lifters, 3 are from the... Problem 1.33P: Delegates from 10 countries, including Russia, France, England, and the United States, are to be... Problem 1.34P: If 8 identical blackboards are to be divided among 4 schools, how many divisions are possible? How... Problem 1.35P: An elevator starts at the basement with 8 people (not including the elevator operator) and... Problem 1.36P: We have 520.000 that must be invested among 4 possible opportunities. Each investment must be... Problem 1.37P: Suppose that 10 fish are caught at a lake that contains 5 distinct types of fish. a. How many... Problem 1.1TE: Prove the generalized version of the basic counting principle. Problem 1.2TE: Two experiments are to be performed. The first can result in any one of m possible outcomes. If the... Problem 1.3TE: In how many ways can r objects be selected from a set of n objects if the order of selection is... Problem 1.4TE: There are (nr) different linear arrangements of n balls of which r are black and nr are white. Give... Problem 1.5TE: Determine the number of vectors (x1,...,xn), such that each x1 is either 0 or 1 andi=1nxiK Problem 1.6TE: How many vectors x1,...,xk are there for which each xi is a positive integer such that1xin and... Problem 1.7TE: Give an analytic proof of Equation (4.1). Problem 1.8TE: Prove that (n+mr)=(n0)(mr)+(n1)(mr1)+...+(nr)(m0) Hint: Consider a group of n men and m women. How... Problem 1.9TE: Use Theoretical Exercise 8 I to prove that (2nn)=k=0n(nk)2 Problem 1.10TE: From a group of n people, suppose that we want to choose a committee of k,kn, one of whom is to be... Problem 1.11TE: The following identity is known as Fermats combinatorial identity:(nk)=i=kn(i1k1)nk Give a... Problem 1.12TE: Consider the following combinatorial identity: k=0nk(nk)=n2n1 a. Present a combinatorial argument... Problem 1.13TE: Show that, for n0 ,i=0n(1)i(ni)=0 Hint: Use the binomial theorem. Problem 1.14TE: From a set of n people, a committee of size j is to be chosen, and from this committee, a... Problem 1.15TE: Let Hn(n) be the number of vectors x1,...,xk for which each xi is a positive integer satisfying 1xin... Problem 1.16TE: Consider a tournament of n contestants in which the outcome is an ordering of these contestants,... Problem 1.17TE: Present a combinatorial explanation of why (nr)=(nr,nr) Problem 1.18TE: Argue that(nn1,n2,...,nr)=(n1n11,n2,...,nr)+(nn1,n21,...,nr)+...+(nn1,n2,...,nr1) Hint: Use an... Problem 1.19TE: Prove the multinomial theorem. Problem 1.20TE: In how many ways can n identical balls be distributed into r urns so that the ith urn contains at... Problem 1.21TE: Argue that there are exactly (rk)(n1nr+k) solutions of x1+x2+...+xr=n for which exactly k of the xi... Problem 1.22TE Problem 1.23TE: Determine the number of vectors (xi,...,xn) such that each xi, is a nonnegative integer and i=1nxik. Problem 1.1STPE: How many different linear arrangements are there of the letters A, B, C, D, E, F for which a. A and... Problem 1.2STPE: If 4 Americans, 3 French people, and 3 British people are to be seated in a row, how many seating... Problem 1.3STPE: A president. treasurer, and secretary. all different, are to be chosen from a club onsisting of 10... Problem 1.4STPE: A student is to answer 7 out of 10 questions in an examination. How many choices has she? How many... Problem 1.5STPE: In how many ways can a man divide 7 gifts among his 3 children if the eldest is to receive 3 gifts... Problem 1.6STPE: How many different 7-place license plates are possible mien 3 of the entries are letters and 4 are... Problem 1.7STPE: Give a combinatorial explanation of the identity(nr)=(nnr) Problem 1.8STPE: Consider n-digit numbers where each digit is one of the 10 integers 0,1, ... ,9. How many such... Problem 1.9STPE: Consider three classes, each consisting of n students. From this group of 3n students, a group of 3... Problem 1.10STPE: How many 5-digit numbers can be formed from the integers 1,2,... ,9 if no digit can appear more than... Problem 1.11STPE: From 10 married couples, we want to select a group of 6 people that is not allowed to contain a... Problem 1.12STPE: A committee of 6 people is to be chosen from a group consisting of 7 men and 8 women. If the... Problem 1.13STPE: An art collection on auction consisted of 4 Dalis, 5 van Goghs. and 6 Picassos, At the auction were... Problem 1.14STPE Problem 1.15STPE: A total of n students are enrolled in a review course for the actuarial examination in probability.... Problem 1.16STPE Problem 1.17STPE: Give an analytic verification of (n2)=(k2)+k(nk)+(n+k2),1kn. Now, give a combinatorial argument for... Problem 1.18STPE: In a certain community, there are 3 families consisting of a single parent and 1 child, 3 families... Problem 1.19STPE: If there are no restrictions on where the digits and letters are placed, how many 8-place license... Problem 1.20STPE: Verify the identityx1+...+xr=n,xi0n!x1!x2!...xr!=rn a. by a combinatorial argument that first notes... Problem 1.21STPE: Simplify n(n2)+(n3)...+(1)n+1(nn) Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
Related questions
Salaries of 40 college graduates who took a statistics course in college have a mean x, of $ 60,300. Assuming a standard deviation, o, of $12,142, construct a 99% confidence interval for estimating the population mean.
$____< u > $____
Transcribed Image Text: л
Critical t value
(negative)
α/2
Left tail
л
Critical t value
(negative)
Right tail
л
Two tails
Critical t value
(positive)
a/2
Critical value
(positive)
Degrees of
Freedom
2
5
8
9
10
11
12
13
14
15
16
17
22888*** *8858838858894882888888888
18
19
20
21
23
24
25
26
27
30
31
33
36
37
40
50
60
70
80
90
100
200
300
400
500
1000
2000
Large
Degrees of
Freedom
0.005
0.01
63.657
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.744
2.738
2.733
2.728
2.724
2.719
2715
2.712
2.708
2.704
2.690
2.678
2.660
2.648
2.639
2.632
2.626
2.601
2.592
2.588
2.586
2.581
2.578
2.576
0.005
0.01
0.01
0.02
31.821
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.453
2.449
2.445
2.441
2.438
2.434
2.431
2.429
2.426
2.423
2.412
2.403
2.390
2.381
2.374
2.368
2.364
2.345
2.339
2.336
2.334
2.330
2,328
2.326
0.01
0.02
Area in One Tail
0.025
Area in Two Tails
0.05
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.040
2.037
2.035
2.032
2.030
2.028
2.026
2.024
2.023
2.021
2014
2.009
2.000
1.994
1.990
1.987
1.984
1.972
1.968
1.966
1.965
1.962
1.961
1.960
Area in One Tail
0.025
Area in Two Tails
0.05
0.05
0.10
6.314
2.920
2.353
2.132
2015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.696
1.694
1.692
1.691
1,690
1.688
1.687
1.686
1.685
1.684
1.679
1.676
1.671
1.667
1.664
1.662
1.660
1.653
1.650
1.649
1.648
1,646
1.646
1.645
0.06
0.10
0.10
0.20
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.309
1.309
1.308
1.307
1.306
1.306
1.305
1.304
1.304
1.303
1.301
1.299
1.296
1.294
1.292
1.291
1.290
1.286
1.284
1.284
1.283
1.282
1.282
1.282
0.10
0.20
Degrees of
Freedom
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
45
50
60
70
80
90
100
200
300
400
500
1000
2000
Large
Degrees of
Freedom
Transcribed Image Text: -833333
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
2222222222=223
1.9
2.0
2.1
2.4
2.5
26
27
2.8
29
3.0
3.1
.00
5000
5300
5790
6179
1999
6915
7257
7580
7881
8150
2413
8643
3049
9032
9192
9332
9452
9654
9641
9713
9772
9021
9061
0013
9018
9938
.9953
9966
9974
9961
9987
9990
9993
9995
9997
9999
00
0
Score
Area
1.645 0.9500
2.575
0.9950
Cumulative Area from the LEFT
01
03
5040
5120
5617
5010
6293
5438
5832
0217
1699
6950
7291
7651
7910
8186
8438
3665
3060
9049
9207
9345
9463
9564
9649
9719
9778
9626
9664
9896
9920
9940
9956
9965
9975
9982
9987
1666
9993
9996
9997
"Use these common values that result from interpolation:
02
5080
5478
5871
6255
0628
0985
7324
7642
7939
8212
8461
0686
8888
9066
9222
9357
9474
9573
9656
9726
9783
9830
9868
9696
9922
9941
3.4
3.50 and up
2
01
NOTE: For values of zabove 3.49, use 0.9999 for the area.
9956
9967
9076
9982
9987
9991
9994
9995
9997
POSITIVE Z Scores
102
1990
7019
7357
7673
7967
8238
8485
8708
0907
2006
82:36
9370
9484
9582
.9664
.9732
9788
9834
9871
9001
1925
9943
9967
9968
9977
9983
9988
9991
9994
9996
.9997
03
04
5160
5657
5048
4331
6700
7054
7369
7704
7995
8264
8508
.8729
8925
9099
9251
9382
9495
9591
9071
9738
9793
9838
9875
9904
9927
9945
9959
9969
9977
9964
9968
.9982
9994
9996
.9997
04
.05
5190
5500
5067
6368
6736
7058
7422
7734
2023
8290
8631
3749
2044
9115
9266
1600
9606
9699
9678
9744
8798
9042
9878
9000
9929
99346
0966
9970
9978
1908
9069
9992
9994
9996
9997
.05
90
5230
5636
6026
6406
6772
7123
7454
7764
8051
8315
8554
8770
2062
9131
9279
9406
9515
9608
9686
9750
9803
9846
9881
9909
9931
9948
9961
9971
9979
9985
9989
9992
9994
9996
9997
90'
07
5279
5675
6064
6443
6808
7157
7486
7794
8078
8340
8577
8790
8900
9147
9292
9418
9525
9616
9693
9756
9608
9850
9634
9911
9932
9949 .
9962
9972
9979
9985
9989
9992
9666
9996
9997
07
06
5319
5714
6103
6480
6844
7190
7517
7823
8106
8365
8509
8810
8997
9162
9008
9429
9535
9625
.9509
9761
9812
9554
9887
9913
9934
9961
9963
9973
9900
9956
9990
9993
9995
9996
9997
90
60
5350
5753
6141
6617
6879
7224
7549
7852
8133
8389
8621
8830
0015
9177
9319
9441
9645
9633
9706
9767
0817
9857
9890
9916
9006
.9952
9964
.9974
9901
9966
.9990
9993
.9995
.9997
.9998
60
Common Critical Values
Confidence
Level
0.90
0.95
0.99
Critical
Value
1.645
1.96
2.575
2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0
1.0
1.1
1.2
1.3
1.4
1.5
1.6
17
1.8
1.9
NENNENNSE
2.0
21
22
2.3
2.4
2.5
26
27
28
29
3.0
3.1
3.2
3.3
3.4
3.50 and up
2
Definition Definition Method in statistics by which an observation’s uncertainty can be quantified. The main use of interval estimating is for describing a range that is made by transforming a point estimate by determining the range of values, or interval within which the population parameter is likely to fall. This range helps in measuring its precision.
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