STAT 211 Business Statistics I – Term 103
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
DEPARTMENT OF MATHEMATICS & STATISTICS
DHAHRAN, SAUDI ARABIA
STAT 211: BUSINESS STATISTICS I
Semester 103
Final Exam
Wednesday August 17, 2011
Allowed time 9:00 am – 11:30 am
Instructor
section number 02
Sec 2: (09:20 –10:20)
Musawar Malik
[
Name:
Student ID#:
Serial #:
Directions:
1) You must show all work to obtain full credit for questions on this exam.
2) DO NOT round your answers at each step. Round answers only if necessary at your final
step to 4 decimal places.
3) You are allowed to use electronic calculators and other reasonable writing accessories that
help write the exam. Try to define events, formulate problem and solve.
4) Do not keep your mobile with you during the exam, turn off your mobile and leave it aside
Question No Full Marks Marks Obtained
Q1
20
Q2
6
Q3
6
Q4
6
Q5
5
Q6
17
Q7
7
Q8
5
Q9
8
Q10
6
Q11
4
Q12
5
Total
95
1
STAT 211 Business Statistics I – Term 103
2
Question One (15 points)
1. The standard error of the mean
a) is never larger than the standard deviation of the population.
b) decreases as the sample size increases.
c) measures the variability of the mean from sample to sample.
d) All of the above.
2. The Central Limit Theorem is important in statistics because
a) for a large n, it says the population is approximately normal.
b) for any population, it says the sampling distribution of the sample mean is approximately
normal, regardless of the sample size.
c) for a large n, it says the sampling distribution of the sample mean is approximately normal,
regardless of the shape of the population.
d) for any sized sample, it says the sampling distribution of the sample mean is approximately
normal.
3. For air travelers, one of the biggest complaints is of the waiting time between when the airplane taxis
away from the terminal until the flight takes off. This waiting time is known to have a skewed-right
distribution with a mean of 10 minutes and a standard deviation of 8 minutes. Suppose 100 flights
have been randomly sampled. Describe the sampling distribution of the mean waiting time between
when the airplane taxis away from the terminal until the flight takes off for these 100 flights.
a) Distribution is skewed-right with mean = 10 minutes and standard error = 0.8 minutes.
b) Distribution is skewed-right with mean = 10 minutes and standard error = 8 minutes.
c) Distribution is approximately normal with mean = 10 minutes and standard error = 0.8
minutes.
d) Distribution is approximately normal with mean = 10 minutes and standard error = 8
minutes.
4. Which of the following statements about the sampling distribution of the sample mean is incorrect?
a) The sampling distribution of the sample mean is approximately normal whenever the sample
size is sufficiently large ( n  30 ).
b) The sampling distribution of the sample mean is generated by repeatedly taking samples of
size n and computing the sample means.
c) The mean of the sampling distribution of the sample mean is equal to  .
d) The standard deviation of the sampling distribution of the sample mean is equal to  .
5. Since the population is always larger than the sample, the population mean:
a. is always larger than the sample mean
b. is always smaller than the sample mean
c. is always larger than or equal to the sample mean
d. is always smaller than or equal to the sample mean
e. can be smaller than, or larger than, or equal to the sample mean
STAT 211 Business Statistics I – Term 103
3
6. Which of the following summary measures is affected most by outliers?
a. The median
b. The geometric mean
c. The range
d. The interquartile range
e. All of the above
7. Which of the following is not a measure of variability?
a. The range
b. The variance
c. The arithmetic mean
d. The standard deviation
e. The interquartile range
8. Expressed in percentiles, the interquartile range is the difference between the
a. 10% and 60% values
b. 45% and 95% values
c. 25% and 75% values
d. 15% and 65% values
9. Which of the following statements is correct in questionnaire design?
a. The questionnaire should be kept as short as possible, and the questions themselves should also
be kept short.
b. A mixture of dichotomous, multiple-choice, and open-ended questions may be used.
c. Leading questions must be avoided
d. All of the above are correct statements
10. Which of the following does not characterize stratified random sampling?
a. The population is divided into strata that are distinct.
b. The population is divided into strata that are mutually exclusive and exhaustive.
c. The population is divided into strata that are homogenous.
d. Nonrandom sampling is used.
11. If P(A) = 0.20, P(B) = 0.30 and P(A and B) = 0.06, then A and B are:
a. dependent events
b. independent events
c. mutually exclusive events
d. complementary events
12. Assume that you invested $10,000 in each of three stocks. Each stock can increase in value, decrease in
value, or remain the same. Drawing a probability tree for this experiment will show that the number
of possible outcomes is:
a. 10,000
b. 3
c. 9
d. 27
STAT 211 Business Statistics I – Term 103
4
13. If A and B are independent events with P(A) = 0.60 and P(B) = 0.70, then the probability that A occurs
or B occurs or both occur is:
a. 1.30
b. 0.88
c. 0.42
d. 0.10
14. If A and B are independent events with P(A) = 0.20 and P(B) =0.60, then P(A/B) is:
a. 0.20
b. 0.60
c. 0.40
d. 0.80
15. (2+2+2 = 6 point). Suppose P ( AC ) = 0.30, P ( B C / A ) = 0.40, and P ( B C / AC ) = 0.50.
a. Find P (A and B).
b. P ( B C ).
c. Find P (A or B).
Write all your choices (answers) in this table:
Question number
1
2
3
11
4
5
6
12
13
14
7
8
9
15(a) 15(b) 15(c)
10
STAT 211 Business Statistics I – Term 103
5
Question Two (4+2 = 6 point)
The life in hours of a 75-watt light bulb is known to be normally distributed with standard deviation 25
hours. A random sample of 20 bulbs has a mean life of 1014 hours.
a. Construct a 92% confidence interval on the mean life.
b. Suppose that we wanted to be 95% confident that the error in estimating the mean life is less than five
hours. What sample size should be used?
Question Three (4+1+1 = 6 points)
A postmix beverage machine is adjusted to release a certain amount of syrup into a chamber where it is
mixed with carbonated water. A random sample of 25 beverages was found to have a mean syrup content of
1.10 fluid ounces and a standard deviation of 0.015 fluid ounces.
a. Find a 99% CI on the mean volume of syrup dispensed.
b. Write the assumption required to calculate the above interval.
c. Provide a practical interpretation of this interval.
STAT 211 Business Statistics I – Term 103
6
Question Four (4+2 =6 points)
A random sample of 80 suspension helmets used by motorcycle riders and automobile race-car drivers was
subjected to an impact test, and on 25 of these helmets some damage was observed.
a. Find a 95% two-sided confidence interval on the true proportion of helmets of this type that would show
damage from this test.
b. Using the point estimate of p obtained from the preliminary sample of 80 helmets, how many helmets
must be tested to be 95% confident that the error in estimating the true value of p is less than 0.02?
Question Five (5 points)
Do government employees (code 1) take longer coffee breaks than private sector workers (code 2)? That is a
question that interested a management consultant. To examine the issue, he took a random sample of 40
government employees and another random sample of 40 private sector workers and measured the amount
of time (in minutes) they spent in coffee breaks during the day. The results are listed below.
x1  76
s1  8
x2  72
s2  6.5
Estimate with 90% confidence the difference between the two population means.
STAT 211 Business Statistics I – Term 103
Question six (4+1+2+2+1+1+2+2+2 = 17 points)
Because of the rising costs of industrial accidents, many chemical, mining, and manufacturing firms have
instituted safety courses. Employees are encouraged to take these courses designed to heighten safety
awareness. A company is trying to decide which one of two courses to institute. To help make a decision
eight employees take course 1 and another eight take course 2. Each employee takes a test, which is graded
out of a possible 25. The safety test results are shown below.
Course 1
14 21 17 14 17 19 20 16
Course 2 20 18 22 15 23 21 19 15
(Assume equal population variances).
For course 1: mean = 17.25 and standard deviation = 2.6049
a. Calculate the mean and the standard deviation for course 2.
b. What is the point estimate to construct a 95% confidence interval for the difference between the
marks of course 1 and course 2?
c. What is the pooled variance?
d. Calculate the standard error of the estimate.
e. What is the criticl value?
f. What is the margin of error?
7
STAT 211 Business Statistics I – Term 103
g. Calculate the lower and upper limits of the interval and interpret your result.
h. What assumptions are required to calculate the above interval.
i. Based on the interval can we conclude that the average marks for both courses are same? Why?
Question Seven (5+2 =7 points)
A politician regularly polls her constituency to gauge her level of support among voters. This month, 650
out of 1160 voters support her. Five months ago, 400 out of 980 voters supported her.
a. Find a 99% confidence interval for the difference between the true proportions of the voters who support
her.
a. Based on the result in part a, do you agree that her support has increased. Why?
8
STAT 211 Business Statistics I – Term 103
Question Eight (5 points)
At a computer manufacturing company, the actual size of computer chips has a mean of 1 centimeter and a
standard deviation of 0.2 centimeter. A random sample of 36 computer chips is taken. What is the
probability that the sample mean will be below 0.95 centimeters?
Question Nine (5+3 = 8 points)
The breaking strength of plastic bags used for packaging produce is normally distributed, with a mean of 5
pounds per square inch and a standard deviation of 1.5 pounds per square inch.
a. If all bags having strength less than 4.8 or greater than 5.6 pounds per square inch are scrapped, what
proportion of bags is scrapped?
b. The breaking strength of 90% of samples is below what value?
9
STAT 211 Business Statistics I – Term 103
10
Question Ten (3+3 = 6 points)
A study of the time spent shopping in a supermarket in Singapore for a market basket of 40 specific items
showed an approximately uniform distribution between 30 minutes and 50 minutes.
a. What is the probability that the shopping time will be no more than 45 minutes?
b. What are the mean and standard deviation of the shopping time?
Question Eleven (4 points)
The number of errors in a textbook follows a Poisson distribution with a mean of 0.01 errors per page. What
is the probability that there are two or less errors in 100 pages?
Question Twelve (5 points)
A study at a college in the west coast reveals that, historically, 45% of their students are minority students.
If random samples of size 75 are selected, what is the probability that more than 60% of the samples will
have minority students?
STAT 211 Business Statistics I – Term 103
11
The cumulative Standard Normal distribution
Entry represented area under the cumulative standardized normal
distribution from - ∞ to Z
Cumulative Probabilities
Z
 3.0
 2.9
 2.8
 2.7
 2.6
 2.5
 2.4
 2.3
 2.2
 2.1
 2.0
 1.9
 1.8
 1.7
 1.6
 1.5
 1.4
 1.3
 1.2
 1.1
 1.0
 0.9
 0.8
 0.7
 0.6
 0.5
 0.4
 0.3
 0.2
 0.1
 0.0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0013
0.0013
0.0013
0.0012
0.0012
0.0011
0.0011
0.0011
0.0010
0.0010
0.0019
0.0018
0.0018
0.0017
0.0016
0.0016
0.0015
0.0015
0.0014
0.0014
0.0026
0.0025
0.0024
0.0023
0.0023
0.0022
0.0021
0.0021
0.0020
0.0019
0.0035
0.0034
0.0033
0.0032
0.0031
0.0030
0.0029
0.0028
0.0027
0.0026
0.0047
0.0045
0.0044
0.0043
0.0041
0.0040
0.0039
0.0038
0.0037
0.0036
0.0062
0.0060
0.0059
0.0057
0.0055
0.0054
0.0052
0.0051
0.0049
0.0048
0.0082
0.0080
0.0078
0.0075
0.0073
0.0071
0.0069
0.0068
0.0066
0.0064
0.0107
0.0104
0.0102
0.0099
0.0096
0.0094
0.0091
0.0089
0.0087
0.0084
0.0139
0.0136
0.0132
0.0129
0.0125
0.0122
0.0119
0.0116
0.0113
0.0110
0.0179
0.0174
0.0170
0.0166
0.0162
0.0158
0.0154
0.0150
0.0146
0.0143
0.0228
0.0222
0.0217
0.0212
0.0207
0.0202
0.0197
0.0192
0.0188
0.0183
0.0287
0.0281
0.0274
0.0268
0.0262
0.0256
0.0250
0.0244
0.0239
0.0233
0.0359
0.0351
0.0344
0.0336
0.0329
0.0322
0.0314
0.0307
0.0301
0.0294
0.0446
0.0436
0.0427
0.0418
0.0409
0.0401
0.0392
0.0384
0.0375
0.0367
0.0548
0.0537
0.0526
0.0516
0.0505
0.0495
0.0485
0.0475
0.0465
0.0455
0.0668
0.0655
0.0643
0.0630
0.0618
0.0606
0.0594
0.0582
0.0571
0.0559
0.0808
0.0793
0.0778
0.0764
0.0749
0.0735
0.0721
0.0708
0.0694
0.0681
0.0968
0.0951
0.0934
0.0918
0.0901
0.0885
0.0869
0.0853
0.0838
0.0823
0.1151
0.1131
0.1112
0.1093
0.1075
0.1056
0.1038
0.1020
0.1003
0.0985
0.1357
0.1335
0.1314
0.1292
0.1271
0.1251
0.1230
0.1210
0.1190
0.1170
0.1587
0.1562
0.1539
0.1515
0.1492
0.1469
0.1446
0.1423
0.1401
0.1379
0.1841
0.1814
0.1788
0.1762
0.1736
0.1711
0.1685
0.1660
0.1635
0.1611
0.2119
0.2090
0.2061
0.2033
0.2005
0.1977
0.1949
0.1922
0.1894
0.1867
0.2420
0.2389
0.2358
0.2327
0.2296
0.2266
0.2236
0.2206
0.2177
0.2148
0.2743
0.2709
0.2676
0.2643
0.2611
0.2578
0.2546
0.2514
0.2483
0.2451
0.3085
0.3050
0.3015
0.2981
0.2946
0.2912
0.2877
0.2843
0.2810
0.2776
0.3446
0.3409
0.3372
0.3336
0.3300
0.3264
0.3228
0.3192
0.3156
0.3121
0.3821
0.3783
0.3745
0.3707
0.3669
0.3632
0.3594
0.3557
0.3520
0.3483
0.4207
0.4168
0.4129
0.4090
0.4052
0.4013
0.3974
0.3936
0.3897
0.3859
0.4602
0.4562
0.4522
0.4483
0.4443
0.4404
0.4364
0.4325
0.4286
0.4247
0.5000
0.4960
0.4920
0.4880
0.4840
0.4801
0.4761
0.4721
0.4681
0.4641
STAT 211 Business Statistics I – Term 103
12
The cumulative Standard Normal distribution
Entry represented area under the cumulative standardized normal
distribution from - ∞ to Z
Cumulative Probabilities
Z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0
0.5000
0.5040
0.5080
0.5120
0.5160
0.5199
0.5239
0.5279
0.5319
0.5359
0.1
0.5398
0.5438
0.5478
0.5517
0.5557
0.5596
0.5636
0.5675
0.5714
0.5753
0.2
0.5793
0.5832
0.5871
0.5910
0.5948
0.5987
0.6026
0.6064
0.6103
0.6141
0.3
0.6179
0.6217
0.6255
0.6293
0.6331
0.6368
0.6406
0.6443
0.6480
0.6517
0.4
0.6554
0.6591
0.6628
0.6664
0.6700
0.6736
0.6772
0.6808
0.6844
0.6879
0.5
0.6915
0.6950
0.6985
0.7019
0.7054
0.7088
0.7123
0.7157
0.7190
0.7224
0.6
0.7257
0.7291
0.7324
0.7357
0.7389
0.7422
0.7454
0.7486
0.7517
0.7549
0.7
0.7580
0.7611
0.7642
0.7673
0.7704
0.7734
0.7764
0.7794
0.7823
0.7852
0.8
0.7881
0.7910
0.7939
0.7967
0.7995
0.8023
0.8051
0.8078
0.8106
0.8133
0.9
0.8159
0.8186
0.8212
0.8238
0.8264
0.8289
0.8315
0.8340
0.8365
0.8389
1.0
0.8413
0.8438
0.8461
0.8485
0.8508
0.8531
0.8554
0.8577
0.8599
0.8621
1.1
0.8643
0.8665
0.8686
0.8708
0.8729
0.8749
0.8770
0.8790
0.8810
0.8830
1.2
0.8849
0.8869
0.8888
0.8907
0.8925
0.8944
0.8962
0.8980
0.8997
0.9015
1.3
0.9032
0.9049
0.9066
0.9082
0.9099
0.9115
0.9131
0.9147
0.9162
0.9177
1.4
0.9192
0.9207
0.9222
0.9236
0.9251
0.9265
0.9279
0.9292
0.9306
0.9319
1.5
0.9332
0.9345
0.9357
0.9370
0.9382
0.9394
0.9406
0.9418
0.9429
0.9441
1.6
0.9452
0.9463
0.9474
0.9484
0.9495
0.9505
0.9515
0.9525
0.9535
0.9545
1.7
0.9554
0.9564
0.9573
0.9582
0.9591
0.9599
0.9608
0.9616
0.9625
0.9633
1.8
0.9641
0.9649
0.9656
0.9664
0.9671
0.9678
0.9686
0.9693
0.9699
0.9706
1.9
0.9713
0.9719
0.9726
0.9732
0.9738
0.9744
0.9750
0.9756
0.9761
0.9767
2.0
0.9772
0.9778
0.9783
0.9788
0.9793
0.9798
0.9803
0.9808
0.9812
0.9817
2.1
0.9821
0.9826
0.9830
0.9834
0.9838
0.9842
0.9846
0.9850
0.9854
0.9857
2.2
0.9861
0.9864
0.9868
0.9871
0.9875
0.9878
0.9881
0.9884
0.9887
0.9890
2.3
0.9893
0.9896
0.9898
0.9901
0.9904
0.9906
0.9909
0.9911
0.9913
0.9916
2.4
0.9918
0.9920
0.9922
0.9925
0.9927
0.9929
0.9931
0.9932
0.9934
0.9936
2.5
0.9938
0.9940
0.9941
0.9943
0.9945
0.9946
0.9948
0.9949
0.9951
0.9952
2.6
0.9953
0.9955
0.9956
0.9957
0.9959
0.9960
0.9961
0.9962
0.9963
0.9964
2.7
0.9965
0.9966
0.9967
0.9968
0.9969
0.9970
0.9971
0.9972
0.9973
0.9974
2.8
0.9974
0.9975
0.9976
0.9977
0.9977
0.9978
0.9979
0.9979
0.9980
0.9981
2.9
0.9981
0.9982
0.9982
0.9983
0.9984
0.9984
0.9985
0.9985
0.9986
0.9986
3.0
0.9987
0.9987
0.9987
0.9988
0.9988
0.9989
0.9989
0.9989
0.9990
0.9990
STAT 211 Business Statistics I – Term 103
13
For a particular number of degrees of freedom, entry represents the
critical value of corresponding to a specified upper – tail area (  )
Cumulative Probabilities
0.55
0.65
0.75
0.85
0.9
0.95
0.965
0.975
0.99
0.995
Upper Tail Areas
d.f.
0.45
0.35
0.25
0.15
0.1
0.05
0.035
0.025
0.01
0.005
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.1584
0.1421
0.1366
0.1338
0.1322
0.1311
0.1303
0.1297
0.1293
0.1289
0.1286
0.1283
0.1281
0.1280
0.1278
0.1277
0.1276
0.1274
0.1274
0.1273
0.1272
0.1271
0.1271
0.1270
0.1269
0.1269
0.1268
0.1268
0.1268
0.1267
0.1267
0.1266
0.1266
0.1265
0.1265
0.1264
0.1264
0.1264
0.1263
0.1263
0.1262
0.1261
0.1261
0.1260
0.5095
0.4447
0.4242
0.4142
0.4082
0.4043
0.4015
0.3995
0.3979
0.3966
0.3956
0.3947
0.3940
0.3933
0.3928
0.3923
0.3919
0.3915
0.3912
0.3909
0.3906
0.3904
0.3902
0.3900
0.3898
0.3896
0.3894
0.3893
0.3892
0.3890
0.3888
0.3886
0.3884
0.3882
0.3881
0.3880
0.3878
0.3877
0.3876
0.3875
0.3872
0.3869
0.3867
0.3866
1.0000
0.8165
0.7649
0.7407
0.7267
0.7176
0.7111
0.7064
0.7027
0.6998
0.6974
0.6955
0.6938
0.6924
0.6912
0.6901
0.6892
0.6884
0.6876
0.6870
0.6864
0.6858
0.6853
0.6848
0.6844
0.6840
0.6837
0.6834
0.6830
0.6828
0.6822
0.6818
0.6814
0.6810
0.6807
0.6804
0.6801
0.6799
0.6796
0.6794
0.6786
0.6780
0.6776
0.6772
1.9626
1.3862
1.2498
1.1896
1.1558
1.1342
1.1192
1.1081
1.0997
1.0931
1.0877
1.0832
1.0795
1.0763
1.0735
1.0711
1.0690
1.0672
1.0655
1.0640
1.0627
1.0614
1.0603
1.0593
1.0584
1.0575
1.0567
1.0560
1.0553
1.0547
1.0535
1.0525
1.0516
1.0508
1.0500
1.0494
1.0488
1.0483
1.0478
1.0473
1.0455
1.0442
1.0432
1.0424
3.0777
1.8856
1.6377
1.5332
1.4759
1.4398
1.4149
1.3968
1.3830
1.3722
1.3634
1.3562
1.3502
1.3450
1.3406
1.3368
1.3334
1.3304
1.3277
1.3253
1.3232
1.3212
1.3195
1.3178
1.3163
1.3150
1.3137
1.3125
1.3114
1.3104
1.3086
1.3070
1.3055
1.3042
1.3031
1.3020
1.3011
1.3002
1.2994
1.2987
1.2958
1.2938
1.2922
1.2910
6.3138
2.9200
2.3534
2.1318
2.0150
1.9432
1.8946
1.8595
1.8331
1.8125
1.7959
1.7823
1.7709
1.7613
1.7531
1.7459
1.7396
1.7341
1.7291
1.7247
1.7207
1.7171
1.7139
1.7109
1.7081
1.7056
1.7033
1.7011
1.6991
1.6973
1.6939
1.6909
1.6883
1.6860
1.6839
1.6820
1.6802
1.6787
1.6772
1.6759
1.6706
1.6669
1.6641
1.6620
9.0579
3.5782
2.7626
2.4559
2.2974
2.2011
2.1365
2.0902
2.0554
2.0283
2.0067
1.9889
1.9742
1.9617
1.9509
1.9417
1.9335
1.9264
1.9200
1.9143
1.9092
1.9045
1.9003
1.8965
1.8929
1.8897
1.8867
1.8839
1.8813
1.8789
1.8746
1.8708
1.8674
1.8644
1.8617
1.8593
1.8571
1.8551
1.8532
1.8516
1.8448
1.8401
1.8365
1.8337
12.7062
4.3027
3.1824
2.7764
2.5706
2.4469
2.3646
2.3060
2.2622
2.2281
2.2010
2.1788
2.1604
2.1448
2.1314
2.1199
2.1098
2.1009
2.0930
2.0860
2.0796
2.0739
2.0687
2.0639
2.0595
2.0555
2.0518
2.0484
2.0452
2.0423
2.0369
2.0322
2.0281
2.0244
2.0211
2.0181
2.0154
2.0129
2.0106
2.0086
2.0003
1.9944
1.9901
1.9867
31.8205
6.9646
4.5407
3.7469
3.3649
3.1427
2.9980
2.8965
2.8214
2.7638
2.7181
2.6810
2.6503
2.6245
2.6025
2.5835
2.5669
2.5524
2.5395
2.5280
2.5176
2.5083
2.4999
2.4922
2.4851
2.4786
2.4727
2.4671
2.4620
2.4573
2.4487
2.4411
2.4345
2.4286
2.4233
2.4185
2.4141
2.4102
2.4066
2.4033
2.3901
2.3808
2.3739
2.3685
63.6567
9.9248
5.8409
4.6041
4.0321
3.7074
3.4995
3.3554
3.2498
3.1693
3.1058
3.0545
3.0123
2.9768
2.9467
2.9208
2.8982
2.8784
2.8609
2.8453
2.8314
2.8188
2.8073
2.7969
2.7874
2.7787
2.7707
2.7633
2.7564
2.7500
2.7385
2.7284
2.7195
2.7116
2.7045
2.6981
2.6923
2.6870
2.6822
2.6778
2.6603
2.6479
2.6387
2.6316
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STAT 211 Business Statistics I – Term 103
14
Some Useful Formulas
 S
x
x

2

n 1
x
2
 n (x ) 2
n 1

P(A or B) = P(A U B) = P(A) + P(B) – P(A∩B)

P(A ∩ B ' ) = P(A) – P(A ∩ B)



P  A | B 
P  A  B
P  B



PB j | A 
P  A

k
 P A | B PB 
i
P x  
n!
x

x ! n  x !
for j  1,2,..., k
P 0  x  a   1  e
a
σ
n
 (1   )
 μ p   , σp 
,
n
 x  zα/2 σ
n

μ x  μ , σx 
 x1  x 2   z  /2

 x1  x 2   t  ,n1  n2 2 sp
n x
  E  X   n  ,   n 1   
  t x e  t
P x  
,  t ,   t
x!
 N  A  A 
N X
X
 n  x  x 
C
nx C x
 
P x  

N
N 
Cn
n
 
 1
if a  x  b

f  x   b  a
,
0
otherwise

s12 s22

n1 n 2

i
1 
2
σ12 σ22

n1
n2
 x1  x 2   z  /2

PA | B j PB j 
e
σ
z
   /2 
 e 
p (1  p )
n
2
z  (1   )
n   /2 2
e
P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A|B)
PB j  A
, n 1
2
z 2 /2 σ 2
2
s
n
p  z  /2

, P  B  0
a  c  d  b P c  X  d   (d c ) f (x )

n


i 1

x t 

2
1
1
,

n1 n 2
where
sp 

 n1  1 s12   n 2  1 s22
n1  n 2  2
 x1  x 2   t /2, v
s12 s 2 2

, where
n1 n 2
2
 s12 s 2 2 



n1 n 2 

v 
2
2
 s12   s 2 2 
  

 n1    n 2 
n1  1
n2 1

d  t
2
, n 1
sd
n

 p1  p2   z  /2
where
x
p1  1 ,
n1
p2 
x2
n2
p1 (1  p1 ) p 2 (1  p 2 )

n1
n2