UNIVERSITY OF REGINADepartment of Mathematics and Statistics

STATISTICS 100

Final Exam, Winter 2015 (April 24th)

Time: 3 hours Full Name:

Pages: 11 Student Number:

Instructor: (check one)

j Leigh A. MacKnight (100-001, 101)

j Iqbal Husain (100-L01)

j Yang Zhaol (100-002)

INSTRUCTIONS

1. Do not open the exam until you are instructed to do so.

2. All work and answers are to be placed on the right side pages of this exam belowthe question. If you require more space for an answer, work on the left side (facing) pageand indicate on the question there is work to be found there.

3. The left side pages are to be used as scrap paper. They are provided for rough workand checking only and will not be graded unless you expressly indicate there is work tobe found there.

4. Place final answers in blanks at right when provided.

5. To receive full credit for correct answers it is necessary to show all work.

6. The exam has a total of 100 marks. The marks allocated for each question are found tothe left of the question.

7. Do not remove staples or any pages from the exam.

8. Probability tables and formula sheet are at the end of the exam.

For instructor use only:

Page: 2 3 4 5 6 7 8 9 10 11 Total

Marks: 6 10 8 8 12 10 12 14 10 10 100

Score:

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STAT 100 Final Page 2 Winter 2015

Marks

1.(6) Test scores on a 100-point test were recorded for 20 students:

71 93 91 86 75 73 86 82 76 57 84 89 67 62 72 77 68 65 75 84

(a) Can you reasonably assume that these test scores have been selected from anormal population?

Answer: yes / no

(b) Use a stem and leaf plot to justify your answer.

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STAT 100 Final Page 3 Winter 2015

2.(10) Petroleum pollution in seas and oceans stimulates the growth of some types of bacteria.A count of petroleumlytic micro-organisms (bacteria per 100 milliliters) in ten portionsof seawater gave these readings:

49 70 54 67 59 40 61 69 71 52

Construct a box plot for the data (you are required to identify the median, lowerquartile Q1, upper quartile Q3, lower fence, upper fence, the smallest and largest values,and any outliers).

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STAT 100 Final Page 4 Winter 2015

3.(8) In a test of a drugs effectiveness, the drug was administered to one group of people anda placebo to another group of people, with the following results:

Helped Not Helped Harmed Total

Drug 80 30 10 120Placebo 40 60 20 120

Total 120 90 30 240

Now a person is randomly selected from all those in the study.

(a) Find the probability that the person was harmed. Answer:

(b) Find the probability that the person was given thedrug and was helped.

Answer:

(c) Find the probability that the person was helped, giventhat they were given the placebo.

Answer:

(d) Are the events the person was given a placebo and the person washelped independent? Circle your answer at right and support youranswer by calculating the relevant probabilities below.

yes / no

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STAT 100 Final Page 5 Winter 2015

4.(8) The probability that a child with a certain type of disability can successfully solve atest problem is 0.40.

(a) If 10 children with this particular disability, are randomly chosen, what is theprobability that more than 1 will successfully solve the test problem?

(b) If 400 children with this particular disability, are randomly chosen, what is theprobability that between 130 and 150 (inclusive) will successfully solve the testproblem? Calculate the probability using the Normal approximation with thecontinuity correction. Is the sample size large enough for the approximation to beadequate?

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STAT 100 Final Page 6 Winter 2015

5.(12) A school district gave a reading achievement exam. The average score for students was122 with a standard deviation of 20. The reading scores follow a normal distribution.

(a) What is the probability that a randomly selected student from this school divisionwill score between 115 and 140?

(b) The school district decides to implement a new reading program. The bottom 20%of the students in the district are eligible for this program. What is the cut-offscore that would make a student eligible for the program?

(c) For a sample of 64 students from this school district, find the probability that theaverage of this sample is more than 118.

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STAT 100 Final Page 7 Winter 2015

6.(10) In order to study the relationship between exam preparation time, x , and the scorereceived on the final exam, y, six students were asked to keep a record of their exampreparation time (in hours). The results are as follows:

Prep time x 1.7 2.1 3.2 3.0 4.0 3.5Score on exam y 65 63 71 69 75 72

Note: Px � 17.5, P y � 415, Px2 � 54.79, P y2 � 28805, Pxy � 1229.

(a) Calculate the correlation coefficient, r.

(b) Find the line-of-best-fit (linear regression line).

(c) Estimate a students score on the exam if they studied 2.5 hours.

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STAT 100 Final Page 8 Winter 2015

7.(6) In a study on the nutritional qualities of fast food, the amount of fat was measured fora random sample of 35 hamburgers of a particular restaurant chain. The sample meanand standard deviation were found to be 30.2 and 3.8 grams, respectively. Find a 95%confidence interval for the mean fat content in hamburgers served in these restaurants.

8.(6) A national safety council wishes to estimate the proportion of automobile accidents thatinvolve pedestrian. How large a sample of accident records must be examined to be 98%certain that the estimate does not differ from the true proportion by more than 0.03?

This page is to be used as scrap paper.

STAT 100 Final Page 9 Winter 2015

9.(7) A five-year old census recorded that 20% of the families in a large community livedbelow the poverty line. To determine if this percentage has changed, a random sampleof 400 families is studied and 70 are found to be living below the poverty level. Doesthis finding indicate that the current percentage of families earning incomes below thepoverty level has changed from what it was five years ago? Calculate the p-value forthe test. Use a 5% level of significance.

10.(7) The numbers of misdirected luggage reported per week, for seven consecutive weeks atthe Regina airport, were 13, 8, 15, 11, 3, 6 and 10. Construct a 95% confidence intervalfor the true mean number of misdirected luggage per week reported at the Reginaairport. State any necessary assumptions.

This page is to be used as scrap paper.

STAT 100 Final Page 10 Winter 2015

11.(10) A group of 140 subjects is used in an experiment to compare two treatments.Treatment 1 is given to 78 subjects selected at random and the remaining 62 are giventreatment 2. The means and standard deviations of the responses are as follows:

Treatment 1 Treatment 2Mean 92 118Standard Deviation 46.2 53.4

(a) Construct a 95% confidence interval for the mean difference of the treatmenteffects.

(b) Suppose the investigator wishes to establish that treatment 2 has a higher meanresponse than treatment 1. Test this hypothesis at α � 0.05.

This page is to be used as scrap paper.

STAT 100 Final Page 11 Winter 2015

12.(10) The data below represents the number of hours of relief provided by 3 different brandsof painkillers administered to 15 subjects experiencing pain. Do the average hours ofrelief provided by the painkillers the same for all 3 brands?

Sample 1 5.2 4.7 8.1 6.2 4.2Sample 2 9.1 7.1 8.2 6.0 7.5 9.1Sample 3 3.2 5.8 2.2 7.2

Perform the ANOVA F test to analyze these data (you are required to complete thefollowing ANOVA table, state the hypothesis and do the test). Use α � 0.05.

Source df Sum of Squares Mean Squares F

Treatment(between)Error(within)

Total

z0.

000.

010.

020.

030.

040.

050.

060.

070.

080.

09

0.0

0.5

000

0.5

040

0.50

80

0.51

200.

5160

0.51

99

0.52

390.

5279

0.53

190.

5359

0.1

0.5

398

0.5

438

0.54

78

0.55

170.

5557

0.55

96

0.56

360.

5675

0.57

140.

5753

0.2

0.5

793

0.5

832

0.58

71

0.59

100.

5948

0.59

87

0.60

260.

6064

0.61

030.

6141

0.3

0.6

179

0.6

217

0.62

55

0.62

930.

6331

0.63

68

0.64

060.

6443

0.64

800.

6517

0.4

0.6

554

0.6

591

0.66

28

0.66

640.

6700

0.67

36

0.67

720.

6808

0.68

440.

6879

0.5

0.6

915

0.6

950

0.69

85

0.70

190.

7054

0.70

88

0.71

230.

7157

0.71

900.

7224

0.6

0.7

257

0.7

291

0.73

24

0.73

570.

7389

0.74

22

0.74

540.

7486

0.75

170.

7549

0.7

0.7

580

0.7

611

0.76

42

0.76

730.

7704

0.77

34

0.77

640.

7794

0.78

230.

7852

0.8

0.7

881

0.7

910

0.79

39

0.79

670.

7995

0.80

23

0.80

510.

8078

0.81

060.

8133

0.9

0.8

159

0.8

186

0.82

12

0.82

380.

8264

0.82

89

0.83

150.

8340

0.83

650.

8389

1.0

0.8

413

0.8

438

0.84

61

0.84

850.

8508

0.85

31

0.85

540.

8577

0.85

990.

8621

1.1

0.8

643

0.8

665

0.86

86

0.87

080.

8729

0.87

49

0.87

700.

8790

0.88

100.

8830

1.2

0.8

849

0.8

869

0.88

88

0.89

070.

8925

0.89

44

0.89

620.

8980

0.89

970.

9015

1.3

0.9

032

0.9

049

0.90

66

0.90

820.

9099

0.91

15

0.91

310.

9147

0.91

620.

9177

1.4

0.9

192

0.9

207

0.92

22

0.92

360.

9251

0.92

65

0.92

790.

9292

0.93

060.

9319

1.5

0.9

332

0.9

345

0.93

57

0.93

700.

9382

0.93

94

0.94

060.

9418

0.94

290.

9441

1.6

0.9

452

0.9

463

0.94

74

0.94

840.

9495

0.95

05

0.95

150.

9525

0.95

350.

9545

1.7

0.9

554

0.9

564

0.95

73

0.95

820.

9591

0.95

99

0.96

080.

9616

0.96

250.

9633

1.8

0.9

641

0.9

649

0.96

56

0.96

640.

9671

0.96

78

0.96

860.

9693

0.96

990.

9706

1.9

0.9

713

0.9

719

0.97

26

0.97

320.

9738

0.97

44

0.97

500.

9756

0.97

610.

9767

2.0

0.9

772

0.9

778

0.97

83

0.97

880.

9793

0.97

98

0.98

030.

9808

0.98

120.

9817

2.1

0.9

821

0.9

826

0.98

30

0.98

340.

9838

0.98

42

0.98

460.

9850

0.98

540.

9857

2.2

0.9

861

0.9

864

0.98

68

0.98

710.

9875

0.98

78

0.98

810.

9884

0.98

870.

9890

2.3

0.9

893

0.9

896

0.98

98

0.99

010.

9904

0.99

06

0.99

090.

9911

0.99

130.

9916

2.4

0.9

918

0.9

920

0.99

22

0.99

250.

9927

0.99

29

0.99

310.

9932

0.99

340.

9936

2.5

0.9

938

0.9

940

0.99

41

0.99

430.

9945

0.99

46

0.99

480.

9949

0.99

510.

9952

2.6

0.9

953

0.9

955

0.99

56

0.99

570.

9959

0.99

60

0.99

610.

9962

0.99

630.

9964

2.7

0.9

965

0.9

966

0.99

67

0.99

680.

9969

0.99

70

0.99

710.

9972

0.99

730.

9974

2.8

0.9

974

0.9

975

0.99

76

0.99

770.

9977

0.99

78

0.99

790.

9979

0.99

800.

9981

2.9

0.9

981

0.9

982

0.99

82

0.99

830.

9984

0.99

84

0.99

850.

9985

0.99

860.

9986

3.0

0.9

987

0.9

987

0.99

87

0.99

880.

9988

0.99

89

0.99

890.

9989

0.99

900.

9990

3.1

0.9

990

0.9

991

0.99

91

0.99

910.

9992

0.99

92

0.99

920.

9992

0.99

930.

9993

3.2

0.9

993

0.9

993

0.99

94

0.99

940.

9994

0.99

94

0.99

940.

9995

0.99

950.

9995

3.3

0.9

995

0.9

995

0.99

95

0.99

960.

9996

0.99

96

0.99

960.

9996

0.99

960.

9997

3.4

0.9

997

0.9

997

0.99

97

0.99

970.

9997

0.99

97

0.99

970.

9997

0.99

970.

9998

Norm

alCurv

eAreas

z1.

230

P(z<

1.23

)=

Are

a=

0.89

07

z0.

000.0

10.

020.

030.

040.

050.

060.

070.

080.

09

-3.4

0.00

030.

0003

0.00

030.

0003

0.00

030.

0003

0.00

030.

0003

0.00

030.

0002

-3.3

0.00

050.

0005

0.00

050.

0004

0.00

040.

0004

0.00

040.

0004

0.00

040.

0003

-3.2

0.00

070.

0007

0.00

060.

0006

0.00

060.

0006

0.00

060.

0005

0.00

050.

0005

-3.1

0.00

100.

0009

0.00

090.

0009

0.00

080.

0008

0.00

080.

0008

0.00

070.

0007

-3.0

0.00

130.

0013

0.00

130.

0012

0.00

120.

0011

0.00

110.

0011

0.00

100.

0010

-2.9

0.00

190.

0018

0.00

180.

0017

0.00

160.

0016

0.00

150.

0015

0.00

140.

0014

-2.8

0.00

260.

0025

0.00

240.

0023

0.00

230.

0022

0.00

210.

0021

0.00

200.

0019

-2.7

0.00

350.

0034

0.00

330.

0032

0.00

310.

0030

0.00

290.

0028

0.00

270.

0026

-2.6

0.00

470.

0045

0.00

440.

0043

0.00

410.

0040

0.00

390.

0038

0.00

370.

0036

-2.5

0.00

620.

0060

0.00

590.

0057

0.00

550.

0054

0.00

520.

0051

0.00

490.

0048

-2.4

0.00

820.

0080

0.00

780.

0075

0.00

730.

0071

0.00

690.

0068

0.00

660.

0064

-2.3

0.01

070.

0104

0.01

020.

0099

0.00

960.

0094

0.00

910.

0089

0.00

870.

0084

-2.2

0.01

390.

0136

0.01

320.

0129

0.01

250.

0122

0.01

190.

0116

0.01

130.

0110

-2.1

0.01

790.

0174

0.01

700.

0166

0.01

620.

0158

0.01

540.

0150

0.01

460.

0143

-2.0

0.02

280.

0222

0.02

170.

0212

0.02

070.

0202

0.01

970.

0192

0.01

880.

0183

-1.9

0.02

870.

0281

0.02

740.

0268

0.02

620.

0256

0.02

500.

0244

0.02

390.

0233

-1.8

0.03

590.

0351

0.03

440.

0336

0.03

290.

0322

0.03

140.

0307

0.03

010.

0294

-1.7

0.04

460.

0436

0.04

270.

0418

0.04

090.

0401

0.03

920.

0384

0.03

750.

0367

-1.6

0.05

480.

0537

0.05

260.

0516

0.05

050.

0495

0.04

850.

0475

0.04

650.

0455

-1.5

0.06

680.

0655

0.06

430.

0630

0.06

180.

0606

0.05

940.

0582

0.05

710.

0559

-1.4

0.08

080.

0793

0.07

780.

0764

0.07

490.

0735

0.07

210.

0708

0.06

940.

0681

-1.3

0.09

680.

0951

0.09

340.

0918

0.09

010.

0885

0.08

690.

0853

0.08

380.

0823

-1.2

0.11

510.

1131

0.11

120.

1093

0.10

750.

1056

0.10

380.

1020

0.10

030.

0985

-1.1

0.13

570.

1335

0.13

140.

1292

0.12

710.

1251

0.12

300.

1210

0.11

900.

1170

-1.0

0.15

870.

1562

0.15

390.

1515

0.14

920.

1469

0.14

460.

1423

0.14

010.

1379

-0.9

0.18

410.

1814

0.17

880.

1762

0.17

360.

1711

0.16

850.

1660

0.16

350.

1611

-0.8

0.21

190.

2090

0.20

610.

2033

0.20

050.

1977

0.19

490.

1922

0.18

940.

1867

-0.7

0.24

200.

2389

0.23

580.

2327

0.22

960.

2266

0.22

360.

2206

0.21

770.

2148

-0.6

0.27

430.

2709

0.26

760.

2643

0.26

110.

2578

0.25

460.

2514

0.24

830.

2451

-0.5

0.30

850.

3050

0.30

150.

2981

0.29

460.

2912

0.28

770.

2843

0.28

100.

2776

-0.4

0.34

460.

3409

0.33

720.

3336

0.33

000.

3264

0.32

280.

3192

0.31

560.

3121

-0.3

0.38

210.

3783

0.37

450.

3707

0.36

690.

3632

0.35

940.

3557

0.35

200.

3483

-0.2

0.42

070.

4168

0.41

290.

4090

0.40

520.

4013

0.39

740.

3936

0.38

970.

3859

-0.1

0.46

020.

4562

0.45

220.

4483

0.44

430.

4404

0.43

640.

4325

0.42

860.

4247

-0.0

0.50

000.

4960

0.49

200.

4880

0.48

400.

4801

0.47

610.

4721

0.46

810.

4641

Student’s t Distribution (Critical Values)

t−tConfidence Interval

1− α

−tLeft-tailed Test

α

tRight-tailed Test

α

t−tTwo-tailed Test

α2

α2

df

Confidence Coefficient, 1− α

0.80 0.90 0.95 0.98 0.99 0.999

Level of Significance for One-Tailed Test, α

0.100 0.050 0.025 0.010 0.005 0.0005

Level of Significance for Two-Tailed Test, α

0.20 0.10 0.05 0.02 0.01 0.001

1 3.078 6.314 12.706 31.821 63.657 636.6192 1.886 2.920 4.303 6.965 9.925 31.5993 1.638 2.353 3.182 4.541 5.841 12.9244 1.533 2.132 2.776 3.747 4.604 8.6105 1.476 2.015 2.571 3.365 4.032 6.869

6 1.440 1.943 2.447 3.143 3.707 5.9597 1.415 1.895 2.365 2.998 3.499 5.4088 1.397 1.860 2.306 2.896 3.355 5.0419 1.383 1.833 2.262 2.821 3.250 4.781

10 1.372 1.812 2.228 2.764 3.169 4.587

11 1.363 1.796 2.201 2.718 3.106 4.43712 1.356 1.782 2.179 2.681 3.055 4.31813 1.350 1.771 2.160 2.650 3.012 4.22114 1.345 1.761 2.145 2.624 2.977 4.14015 1.341 1.753 2.131 2.602 2.947 4.073

16 1.337 1.746 2.120 2.583 2.921 4.01517 1.333 1.740 2.110 2.567 2.898 3.96518 1.330 1.734 2.101 2.552 2.878 3.92219 1.328 1.729 2.093 2.539 2.861 3.88320 1.325 1.725 2.086 2.528 2.845 3.850

21 1.323 1.721 2.080 2.518 2.831 3.81922 1.321 1.717 2.074 2.508 2.819 3.79223 1.319 1.714 2.069 2.500 2.807 3.76824 1.318 1.711 2.064 2.492 2.797 3.74525 1.316 1.708 2.060 2.485 2.787 3.725

26 1.315 1.706 2.056 2.479 2.779 3.70727 1.314 1.703 2.052 2.473 2.771 3.69028 1.313 1.701 2.048 2.467 2.763 3.67429 1.311 1.699 2.045 2.462 2.756 3.65930 1.310 1.697 2.042 2.457 2.750 3.646

df

Confidence Coefficient, 1− α

0.80 0.90 0.95 0.98 0.99 0.999

Level of Significance for One-Tailed Test, α

0.100 0.050 0.025 0.010 0.005 0.0005

Level of Significance for Two-Tailed Test, α

0.20 0.10 0.05 0.02 0.01 0.001

31 1.309 1.696 2.040 2.453 2.744 3.63332 1.309 1.694 2.037 2.449 2.738 3.62233 1.308 1.692 2.035 2.445 2.733 3.61134 1.307 1.691 2.032 2.441 2.728 3.60135 1.306 1.690 2.030 2.438 2.724 3.591

36 1.306 1.688 2.028 2.434 2.719 3.58237 1.305 1.687 2.026 2.431 2.715 3.57438 1.304 1.686 2.024 2.429 2.712 3.56639 1.304 1.685 2.023 2.426 2.708 3.55840 1.303 1.684 2.021 2.423 2.704 3.551

41 1.303 1.683 2.020 2.421 2.701 3.54442 1.302 1.682 2.018 2.418 2.698 3.53843 1.302 1.681 2.017 2.416 2.695 3.53244 1.301 1.680 2.015 2.414 2.692 3.52645 1.301 1.679 2.014 2.412 2.690 3.520

46 1.300 1.679 2.013 2.410 2.687 3.51547 1.300 1.678 2.012 2.408 2.685 3.51048 1.299 1.677 2.011 2.407 2.682 3.50549 1.299 1.677 2.010 2.405 2.680 3.50050 1.299 1.676 2.009 2.403 2.678 3.496

51 1.298 1.675 2.008 2.402 2.676 3.49252 1.298 1.675 2.007 2.400 2.674 3.48853 1.298 1.674 2.006 2.399 2.672 3.48454 1.297 1.674 2.005 2.397 2.670 3.48055 1.297 1.673 2.004 2.396 2.668 3.476

60 1.296 1.671 2.000 2.390 2.660 3.46080 1.292 1.664 1.990 2.374 2.639 3.416

100 1.290 1.660 1.984 2.364 2.626 3.390200 1.286 1.653 1.972 2.345 2.601 3.340∞ 1.282 1.645 1.960 2.326 2.576 3.291

The F Distribution (Critical Values)

F0

α

df2

df1

12

34

5

139.86

161.4

647.8

4052

16211

49.50

199.5

799.5

5000

20000

53.59

215.7

864.2

5403

21615

55.83

224.6

899.6

5625

22500

57.24

230.2

921.8

5764

23056

28.526

18.51

38.51

98.50

198.5

9.000

19.00

39.00

99.00

199.0

9.162

19.16

39.17

99.17

199.2

9.243

19.25

39.25

99.25

199.2

9.293

19.30

39.30

99.30

199.3

35.538

10.13

17.44

34.12

55.55

5.462

9.552

16.04

30.82

49.80

5.391

9.277

15.44

29.46

47.47

5.343

9.117

15.10

28.71

46.19

5.309

9.013

14.88

28.24

45.39

44.545

7.709

12.22

21.20

31.33

4.325

6.944

10.65

18.00

26.28

4.191

6.591

9.979

16.69

24.26

4.107

6.388

9.605

15.98

23.15

4.051

6.256

9.364

15.52

22.46

54.060

6.608

10.01

16.26

22.78

3.780

5.786

8.434

13.27

18.31

3.619

5.409

7.764

12.06

16.53

3.520

5.192

7.388

11.39

15.56

3.453

5.050

7.146

10.97

14.94

63.776

5.987

8.813

13.75

18.63

3.463

5.143

7.260

10.92

14.54

3.289

4.757

6.599

9.780

12.92

3.181

4.534

6.227

9.148

12.03

3.108

4.387

5.988

8.746

11.46

73.589

5.591

8.073

12.25

16.24

3.257

4.737

6.542

9.547

12.40

3.074

4.347

5.890

8.451

10.88

2.961

4.120

5.523

7.847

10.05

2.883

3.972

5.285

7.460

9.522

83.458

5.318

7.571

11.26

14.69

3.113

4.459

6.059

8.649

11.04

2.924

4.066

5.416

7.591

9.596

2.806

3.838

5.053

7.006

8.805

2.726

3.687

4.817

6.632

8.302

93.360

5.117

7.209

10.56

13.61

3.006

4.256

5.715

8.022

10.11

2.813

3.863

5.078

6.992

8.717

2.693

3.633

4.718

6.422

7.956

2.611

3.482

4.484

6.057

7.471

10

3.285

4.965

6.937

10.04

12.83

2.924

4.103

5.456

7.559

9.427

2.728

3.708

4.826

6.552

8.081

2.605

3.478

4.468

5.994

7.343

2.522

3.326

4.236

5.636

6.872

11

3.225

4.844

6.724

9.646

12.23

2.860

3.982

5.256

7.206

8.912

2.660

3.587

4.630

6.217

7.600

2.536

3.357

4.275

5.668

6.881

2.451

3.204

4.044

5.316

6.422

12

3.177

4.747

6.554

9.330

11.75

2.807

3.885

5.096

6.927

8.510

2.606

3.490

4.474

5.953

7.226

2.480

3.259

4.121

5.412

6.521

2.394

3.106

3.891

5.064

6.071

13

3.136

4.667

6.414

9.074

11.37

2.763

3.806

4.965

6.701

8.186

2.560

3.411

4.347

5.739

6.926

2.434

3.179

3.996

5.205

6.233

2.347

3.025

3.767

4.862

5.791

14

3.102

4.600

6.298

8.862

11.06

2.726

3.739

4.857

6.515

7.922

2.522

3.344

4.242

5.564

6.680

2.395

3.112

3.892

5.035

5.998

2.307

2.958

3.663

4.695

5.562

15

3.073

4.543

6.200

8.683

10.80

2.695

3.682

4.765

6.359

7.701

2.490

3.287

4.153

5.417

6.476

2.361

3.056

3.804

4.893

5.803

2.273

2.901

3.576

4.556

5.372

16

3.048

4.494

6.115

8.531

10.58

2.668

3.634

4.687

6.226

7.514

2.462

3.239

4.077

5.292

6.303

2.333

3.007

3.729

4.773

5.638

2.244

2.852

3.502

4.437

5.212

17

3.026

4.451

6.042

8.400

10.38

2.645

3.592

4.619

6.112

7.354

2.437

3.197

4.011

5.185

6.156

2.308

2.965

3.665

4.669

5.497

2.218

2.810

3.438

4.336

5.075

18

3.007

4.414

5.978

8.285

10.22

2.624

3.555

4.560

6.013

7.215

2.416

3.160

3.954

5.092

6.028

2.286

2.928

3.608

4.579

5.375

2.196

2.773

3.382

4.248

4.956

19

2.990

4.381

5.922

8.185

10.07

2.606

3.522

4.508

5.926

7.093

2.397

3.127

3.903

5.010

5.916

2.266

2.895

3.559

4.500

5.268

2.176

2.740

3.333

4.171

4.853

20

2.975

4.351

5.871

8.096

9.944

2.589

3.493

4.461

5.849

6.986

2.380

3.098

3.859

4.938

5.818

2.249

2.866

3.515

4.431

5.174

2.158

2.711

3.289

4.103

4.762

21

2.961

4.325

5.827

8.017

9.830

2.575

3.467

4.420

5.780

6.891

2.365

3.072

3.819

4.874

5.730

2.233

2.840

3.475

4.369

5.091

2.142

2.685

3.250

4.042

4.681

22

2.949

4.301

5.786

7.945

9.727

2.561

3.443

4.383

5.719

6.806

2.351

3.049

3.783

4.817

5.652

2.219

2.817

3.440

4.313

5.017

2.128

2.661

3.215

3.988

4.609

23

2.937

4.279

5.750

7.881

9.635

2.549

3.422

4.349

5.664

6.730

2.339

3.028

3.750

4.765

5.582

2.207

2.796

3.408

4.264

4.950

2.115

2.640

3.183

3.939

4.544

24

2.927

4.260

5.717

7.823

9.551

2.538

3.403

4.319

5.614

6.661

2.327

3.009

3.721

4.718

5.519

2.195

2.776

3.379

4.218

4.890

2.103

2.621

3.155

3.895

4.486

25

2.918

4.242

5.686

7.770

9.475

2.528

3.385

4.291

5.568

6.598

2.317

2.991

3.694

4.675

5.462

2.184

2.759

3.353

4.177

4.835

2.092

2.603

3.129

3.855

4.433

26

2.909

4.225

5.659

7.721

9.406

2.519

3.369

4.265

5.526

6.541

2.307

2.975

3.670

4.637

5.409

2.174

2.743

3.329

4.140

4.785

2.082

2.587

3.105

3.818

4.384

27

2.901

4.210

5.633

7.677

9.342

2.511

3.354

4.242

5.488

6.489

2.299

2.960

3.647

4.601

5.361

2.165

2.728

3.307

4.106

4.740

2.073

2.572

3.083

3.785

4.340

28

2.894

4.196

5.610

7.636

9.284

2.503

3.340

4.221

5.453

6.440

2.291

2.947

3.626

4.568

5.317

2.157

2.714

3.286

4.074

4.698

2.064

2.558

3.063

3.754

4.300

29

2.887

4.183

5.588

7.598

9.230

2.495

3.328

4.201

5.420

6.396

2.283

2.934

3.607

4.538

5.276

2.149

2.701

3.267

4.045

4.659

2.057

2.545

3.044

3.725

4.262

30

2.881

4.171

5.568

7.562

9.180

2.489

3.316

4.182

5.390

6.355

2.276

2.922

3.589

4.510

5.239

2.142

2.690

3.250

4.018

4.623

2.049

2.534

3.026

3.699

4.228

40

2.835

4.085

5.424

7.314

8.828

2.440

3.232

4.051

5.179

6.066

2.226

2.839

3.463

4.313

4.976

2.091

2.606

3.126

3.828

4.374

1.997

2.449

2.904

3.514

3.986

60

2.791

4.001

5.286

7.077

8.495

2.393

3.150

3.925

4.977

5.795

2.177

2.758

3.343

4.126

4.729

2.041

2.525

3.008

3.649

4.140

1.946

2.368

2.786

3.339

3.760

80

2.769

3.960

5.218

6.963

8.335

2.370

3.111

3.864

4.881

5.665

2.154

2.719

3.284

4.036

4.611

2.016

2.486

2.950

3.563

4.029

1.921

2.329

2.730

3.255

3.652

100

2.756

3.936

5.179

6.895

8.241

2.356

3.087

3.828

4.824

5.589

2.139

2.696

3.250

3.984

4.542

2.002

2.463

2.917

3.513

3.963

1.906

2.305

2.696

3.206

3.589

∞2.706

3.841

5.024

6.635

7.879

2.303

2.996

3.689

4.605

5.298

2.084

2.605

3.116

3.782

4.279

1.945

2.372

2.786

3.319

3.715

1.847

2.214

2.567

3.017

3.350

0.100

0.050

0.025

0.010

0.005

0.100

0.050

0.025

0.010

0.005

0.100

0.050

0.025

0.010

0.005

0.100

0.050

0.025

0.010

0.005

0.100

0.050

0.025

0.010

0.005

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