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Introductory Business Statistics

A | Statistical Tables

Introductory Business StatisticsA | Statistical Tables
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  1. Preface
  2. 1 Sampling and Data
    1. Introduction
    2. 1.1 Definitions of Statistics, Probability, and Key Terms
    3. 1.2 Data, Sampling, and Variation in Data and Sampling
    4. 1.3 Levels of Measurement
    5. 1.4 Experimental Design and Ethics
    6. Key Terms
    7. Chapter Review
    8. Homework
    9. References
    10. Solutions
  3. 2 Descriptive Statistics
    1. Introduction
    2. 2.1 Display Data
    3. 2.2 Measures of the Location of the Data
    4. 2.3 Measures of the Center of the Data
    5. 2.4 Sigma Notation and Calculating the Arithmetic Mean
    6. 2.5 Geometric Mean
    7. 2.6 Skewness and the Mean, Median, and Mode
    8. 2.7 Measures of the Spread of the Data
    9. Key Terms
    10. Chapter Review
    11. Formula Review
    12. Practice
    13. Homework
    14. Bringing It Together: Homework
    15. References
    16. Solutions
  4. 3 Probability Topics
    1. Introduction
    2. 3.1 Terminology
    3. 3.2 Independent and Mutually Exclusive Events
    4. 3.3 Two Basic Rules of Probability
    5. 3.4 Contingency Tables and Probability Trees
    6. 3.5 Venn Diagrams
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Bringing It Together: Practice
    12. Homework
    13. Bringing It Together: Homework
    14. References
    15. Solutions
  5. 4 Discrete Random Variables
    1. Introduction
    2. 4.1 Hypergeometric Distribution
    3. 4.2 Binomial Distribution
    4. 4.3 Geometric Distribution
    5. 4.4 Poisson Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  6. 5 Continuous Random Variables
    1. Introduction
    2. 5.1 Properties of Continuous Probability Density Functions
    3. 5.2 The Uniform Distribution
    4. 5.3 The Exponential Distribution
    5. Key Terms
    6. Chapter Review
    7. Formula Review
    8. Practice
    9. Homework
    10. References
    11. Solutions
  7. 6 The Normal Distribution
    1. Introduction
    2. 6.1 The Standard Normal Distribution
    3. 6.2 Using the Normal Distribution
    4. 6.3 Estimating the Binomial with the Normal Distribution
    5. Key Terms
    6. Chapter Review
    7. Formula Review
    8. Practice
    9. Homework
    10. References
    11. Solutions
  8. 7 The Central Limit Theorem
    1. Introduction
    2. 7.1 The Central Limit Theorem for Sample Means
    3. 7.2 Using the Central Limit Theorem
    4. 7.3 The Central Limit Theorem for Proportions
    5. 7.4 Finite Population Correction Factor
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  9. 8 Confidence Intervals
    1. Introduction
    2. 8.1 A Confidence Interval for a Population Standard Deviation, Known or Large Sample Size
    3. 8.2 A Confidence Interval for a Population Standard Deviation Unknown, Small Sample Case
    4. 8.3 A Confidence Interval for A Population Proportion
    5. 8.4 Calculating the Sample Size n: Continuous and Binary Random Variables
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  10. 9 Hypothesis Testing with One Sample
    1. Introduction
    2. 9.1 Null and Alternative Hypotheses
    3. 9.2 Outcomes and the Type I and Type II Errors
    4. 9.3 Distribution Needed for Hypothesis Testing
    5. 9.4 Full Hypothesis Test Examples
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  11. 10 Hypothesis Testing with Two Samples
    1. Introduction
    2. 10.1 Comparing Two Independent Population Means
    3. 10.2 Cohen's Standards for Small, Medium, and Large Effect Sizes
    4. 10.3 Test for Differences in Means: Assuming Equal Population Variances
    5. 10.4 Comparing Two Independent Population Proportions
    6. 10.5 Two Population Means with Known Standard Deviations
    7. 10.6 Matched or Paired Samples
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. Bringing It Together: Homework
    14. References
    15. Solutions
  12. 11 The Chi-Square Distribution
    1. Introduction
    2. 11.1 Facts About the Chi-Square Distribution
    3. 11.2 Test of a Single Variance
    4. 11.3 Goodness-of-Fit Test
    5. 11.4 Test of Independence
    6. 11.5 Test for Homogeneity
    7. 11.6 Comparison of the Chi-Square Tests
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. Bringing It Together: Homework
    14. References
    15. Solutions
  13. 12 F Distribution and One-Way ANOVA
    1. Introduction
    2. 12.1 Test of Two Variances
    3. 12.2 One-Way ANOVA
    4. 12.3 The F Distribution and the F-Ratio
    5. 12.4 Facts About the F Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  14. 13 Linear Regression and Correlation
    1. Introduction
    2. 13.1 The Correlation Coefficient r
    3. 13.2 Testing the Significance of the Correlation Coefficient
    4. 13.3 Linear Equations
    5. 13.4 The Regression Equation
    6. 13.5 Interpretation of Regression Coefficients: Elasticity and Logarithmic Transformation
    7. 13.6 Predicting with a Regression Equation
    8. 13.7 How to Use Microsoft Excel® for Regression Analysis
    9. Key Terms
    10. Chapter Review
    11. Practice
    12. Solutions
  15. A | Statistical Tables
  16. B | Mathematical Phrases, Symbols, and Formulas
  17. Index

F Distribution

.
Figure A1 Table entry for p is the critical value F* with probability p lying to its right.
Degrees of freedom in the numerator
Degrees of freedom
in the denominator
p 1 2 3 4 5 6 7 8 9
1 .100 39.86 49.50 53.59 55.83 57.24 58.20 58.91 59.44 59.86
.050 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54
.025 647.79 799.50 864.16 899.58 921.85 937.11 948.22 956.66 963.28
.010 4052.2 4999.5 5403.4 5624.6 5763.6 5859.0 5928.4 5981.1 6022.5
.001 405284 500000 540379 562500 576405 585937 592873 598144 602284
2 .100 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.38
.050 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38
.025 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39
.010 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39
.001 998.50 999.00 999.17 999.25 999.30 999.33 999.36 999.37 999.39
3 .100 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.24
.050 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81
.025 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47
.010 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35
.001 167.03 148.50 141.11 137.10 134.58 132.85 131.58 130.62 129.86
4 .100 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94
.050 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00
.025 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90
.010 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66
.001 74.14 61.25 56.18 53.44 51.71 50.53 49.66 49.00 48.47
5 .100 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.32
.050 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77
.025 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68
.010 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16
.001 47.18 37.12 33.20 31.09 29.75 28.83 28.16 27.65 27.24
6 .100 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.96
.050 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10
.025 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52
.010 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98
.001 35.51 27.00 23.70 21.92 20.80 20.03 19.46 19.03 18.69
7 .100 3.59 3.26 3.07 2.96 2.88 2.83 2.78 2.75 2.72
.050 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68
.025 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82
.010 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72
.001 29.25 21.69 18.77 17.20 16.21 15.52 15.02 14.63 14.33
Table A1 F critical values
Degrees of freedom in the numerator
Degrees of freedom
in the denominator
p 10 12 15 20 25 30 40 50 60 120 1000
1 .100 60.19 60.71 61.22 61.74 62.05 62.26 62.53 62.69 62.79 63.06 63.30
.050 241.88 243.91 245.95 248.01 249.26 250.10 251.14 251.77 252.20 253.25 254.19
.025 968.63 976.71 984.87 993.10 998.08 1001.4 1005.6 1008.1 1009.8 1014.0 1017.7
.010 6055.8 6106.3 6157.3 6208.7 6239.8 6260.6 6286.8 6302.5 6313.0 6339.4 6362.7
.001 605621 610668 615764 620908 624017 626099 628712 630285 631337 633972 636301
2 .100 9.39 9.41 9.42 9.44 9.45 9.46 9.47 9.47 9.47 9.48 9.49
.050 19.40 19.41 19.43 19.45 19.46 19.46 19.47 19.48 19.48 19.49 19.49
.025 39.40 39.41 39.43 39.45 39.46 39.46 39.47 39.48 39.48 39.49 39.50
.010 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.48 99.49 99.50
.001 999.40 999.42 999.43 999.45 999.46 999.47 999.47 999.48 999.48 999.49 999.50
3 .100 5.23 5.22 5.20 5.18 5.17 5.17 5.16 5.15 5.15 5.14 5.13
.050 8.79 8.74 8.70 8.66 8.63 8.62 8.59 8.58 8.57 8.55 8.53
.025 14.42 14.34 14.25 14.17 14.12 14.08 14.04 14.01 13.99 13.95 13.91
.010 27.23 27.05 26.87 26.69 26.58 26.50 26.41 26.35 26.32 26.22 26.14
.001 129.25 128.32 127.37 126.42 125.84 125.45 124.96 124.66 124.47 123.97 123.53
4 .100 3.92 3.90 3.87 3.84 3.83 3.82 3.80 3.80 3.79 3.78 3.76
.050 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.70 5.69 5.66 5.63
.025 8.84 8.75 8.66 8.56 8.50 8.46 8.41 8.38 8.36 8.31 8.26
.010 14.55 14.37 14.20 14.02 13.91 13.84 13.75 13.69 13.65 13.56 13.47
.001 48.05 47.41 46.76 46.10 45.70 45.43 45.09 44.88 44.75 44.40 44.09
5 .100 3.30 3.27 3.24 3.21 3.19 3.17 3.16 3.15 3.14 3.12 3.11
.050 4.74 4.68 4.62 4.56 4.52 4.50 4.46 4.44 4.43 4.40 4.37
.025 6.62 6.52 6.43 6.33 6.27 6.23 6.18 6.14 6.12 6.07 6.02
.010 10.05 9.89 9.72 9.55 9.45 9.38 9.29 9.24 9.20 9.11 9.03
.001 26.92 26.42 25.91 25.39 25.08 24.87 24.60 24.44 24.33 24.06 23.82
6 .100 2.94 2.90 2.87 2.84 2.81 2.80 2.78 2.77 2.76 2.74 2.72
.050 4.06 4.00 3.94 3.87 3.83 3.81 3.77 3.75 3.74 3.70 3.67
.025 5.46 5.37 5.27 5.17 5.11 5.07 5.01 4.98 4.96 4.90 4.86
.010 7.87 7.72 7.56 7.40 7.30 7.23 7.14 7.09 7.06 6.97 6.89
.001 18.41 17.99 17.56 17.12 16.85 16.67 16.44 16.31 16.21 15.98 15.77
7 .100 2.70 2.67 2.63 2.59 2.57 2.56 2.54 2.52 2.51 2.49 2.47
.050 3.64 3.57 3.51 3.44 3.40 3.38 3.34 3.32 3.30 3.27 3.23
.025 4.76 4.67 4.57 4.47 4.40 4.36 4.31 4.28 4.25 4.20 4.15
.010 6.62 6.47 6.31 6.16 6.06 5.99 5.91 5.86 5.82 5.74 5.66
.001 14.08 13.71 13.32 12.93 12.69 12.53 12.33 12.20 12.12 11.91 11.72
Table A2 F critical values (continued)
Degrees of freedom in the numerator
Degrees of freedom
in the denominator
p 1 2 3 4 5 6 7 8 9
8 .100 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.56
.050 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39
.025 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36
.010 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91
.001 25.41 18.49 15.83 14.39 13.48 12.86 12.40 12.05 11.77
9 .100 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44
.050 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18
.025 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03
.010 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35
.001 22.86 16.39 13.90 12.56 11.71 11.13 10.70 10.37 10.11
10 .100 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.38 2.35
.050 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02
.025 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78
.010 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94
.001 21.04 14.91 12.55 11.28 10.48 9.93 9.52 9.20 8.96
11 .100 3.23 2.86 2.66 2.54 2.45 2.39 2.34 2.30 2.27
.050 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90
.025 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59
.010 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63
.001 19.69 13.81 11.56 10.35 9.58 9.05 8.66 8.35 8.12
12 .100 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.21
.050 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80
.025 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44
.010 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39
.001 18.64 12.97 10.80 9.63 8.89 8.38 8.00 7.71 7.48
13 .100 3.14 2.76 2.56 2.43 2.35 2.28 2.23 2.20 2.16
.050 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71
.025 6.41 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.31
.010 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19
.001 17.82 12.31 10.21 9.07 8.35 7.86 7.49 7.21 6.98
14 .100 3.10 2.73 2.52 2.39 2.31 2.24 2.19 2.15 2.12
.050 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65
.025 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.29 3.21
.010 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03
.001 17.14 11.78 9.73 8.62 7.92 7.44 7.08 6.80 6.58
15 .100 3.07 2.70 2.49 2.36 2.27 2.21 2.16 2.12 2.09
.050 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59
.025 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12
.010 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89
.001 16.59 11.34 9.34 8.25 7.57 7.09 6.74 6.47 6.26
Table A3 F critical values (continued)
Degrees of freedom in the numerator
Degrees of freedom
in the denominator
p 10 12 15 20 25 30 40 50 60 120 1000
8 .100 2.54 2.50 2.46 2.42 2.40 2.38 2.36 2.35 2.34 2.32 2.30
.050 3.35 3.28 3.22 3.15 3.11 3.08 3.04 3.02 3.01 2.97 2.93
.025 4.30 4.20 4.10 4.00 3.94 3.89 3.84 3.81 3.78 3.73 3.68
.010 5.81 5.67 5.52 5.36 5.26 5.20 5.12 5.07 5.03 4.95 4.87
.001 11.54 11.19 10.84 10.48 10.26 10.11 9.92 9.80 9.73 9.53 9.36
9 .100 2.42 2.38 2.34 2.30 2.27 2.25 2.23 2.22 2.21 2.18 2.16
.050 3.14 3.07 3.01 2.94 2.89 2.86 2.83 2.80 2.79 2.75 2.71
.025 3.96 3.87 3.77 3.67 3.60 3.56 3.51 3.47 3.45 3.39 3.34
.010 5.26 5.11 4.96 4.81 4.71 4.65 4.57 4.52 4.48 4.40 4.32
.001 9.89 9.57 9.24 8.90 8.69 8.55 8.37 8.26 8.19 8.00 7.84
10 .100 2.32 2.28 2.24 2.20 2.17 2.16 2.13 2.12 2.11 2.08 2.06
.050 2.98 2.91 2.85 2.77 2.73 2.70 2.66 2.64 2.62 2.58 2.54
.025 3.72 3.62 3.52 3.42 3.35 3.31 3.26 3.22 3.20 3.14 3.09
.010 4.85 4.71 4.56 4.41 4.31 4.25 4.17 4.12 4.08 4.00 3.92
.001 8.75 8.45 8.13 7.80 7.60 7.47 7.30 7.19 7.12 6.94 6.78
11 .100 2.25 2.21 2.17 2.12 2.10 2.08 2.05 2.04 2.03 2.00 1.98
.050 2.85 2.79 2.72 2.65 2.60 2.57 2.53 2.51 2.49 2.45 2.41
.025 3.53 3.43 3.33 3.23 3.16 3.12 3.06 3.03 3.00 2.94 2.89
.010 4.54 4.40 4.25 4.10 4.01 3.94 3.86 3.81 3.78 3.69 3.61
.001 7.92 7.63 7.32 7.01 6.81 6.68 6.52 6.42 6.35 6.18 6.02
12 .100 2.19 2.15 2.10 2.06 2.03 2.01 1.99 1.97 1.96 1.93 1.91
.050 2.75 2.69 2.62 2.54 2.50 2.47 2.43 2.40 2.38 2.34 2.30
.025 3.37 3.28 3.18 3.07 3.01 2.96 2.91 2.87 2.85 2.79 2.73
.010 4.30 4.16 4.01 3.86 3.76 3.70 3.62 3.57 3.54 3.45 3.37
.001 7.29 7.00 6.71 6.40 6.22 6.09 5.93 5.83 5.76 5.59 5.44
13 .100 2.14 2.10 2.05 2.01 1.98 1.96 1.93 1.92 1.90 1.88 1.85
.050 2.67 2.60 2.53 2.46 2.41 2.38 2.34 2.31 2.30 2.25 2.21
.025 3.25 3.15 3.05 2.95 2.88 2.84 2.78 2.74 2.72 2.66 2.60
.010 4.10 3.96 3.82 3.66 3.57 3.51 3.43 3.38 3.34 3.25 3.18
.001 6.80 6.52 6.23 5.93 5.75 5.63 5.47 5.37 5.30 5.14 4.99
14 .100 2.10 2.05 2.01 1.96 1.93 1.91 1.89 1.87 1.86 1.83 1.80
.050 2.60 2.53 2.46 2.39 2.34 2.31 2.27 2.24 2.22 2.18 2.14
.025 3.15 3.05 2.95 2.84 2.78 2.73 2.67 2.64 2.61 2.55 2.50
.010 3.94 3.80 3.66 3.51 3.41 3.35 3.27 3.22 3.18 3.09 3.02
.001 6.40 6.13 5.85 5.56 5.38 5.25 5.10 5.00 4.94 4.77 4.62
15 .100 2.06 2.02 1.97 1.92 1.89 1.87 1.85 1.83 1.82 1.79 1.76
.050 2.54 2.48 2.40 2.33 2.28 2.25 2.20 2.18 2.16 2.11 2.07
.025 3.06 2.96 2.86 2.76 2.69 2.64 2.59 2.55 2.52 2.46 2.40
.010 3.80 3.67 3.52 3.37 3.28 3.21 3.13 3.08 3.05 2.96 2.88
.001 6.08 5.81 5.54 5.25 5.07 4.95 4.80 4.70 4.64 4.47 4.33
Table A4 F critical values (continued)
Degrees of freedom in the numerator
Degrees of freedom
in the denominator
p 1 2 3 4 5 6 7 8 9
16 .100 3.05 2.67 2.46 2.33 2.24 2.18 2.13 2.09 2.06
.050 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54
.025 6.12 4.69 4.08 3.73 3.50 3.34 3.22 3.12 3.05
.010 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78
.001 16.12 10.97 9.01 7.94 7.27 6.80 6.46 6.19 5.98
17 .100 3.03 2.64 2.44 2.31 2.22 2.15 2.10 2.06 2.03
.050 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49
.025 6.04 4.62 4.01 3.66 3.44 3.28 3.16 3.06 2.98
.010 8.40 6.11 5.19 4.67 4.34 4.10 3.93 3.79 3.68
.001 15.72 10.66 8.73 7.68 7.02 6.56 6.22 5.96 5.75
18 .100 3.01 2.62 2.42 2.29 2.20 2.13 2.08 2.04 2.00
.050 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46
.025 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.93
.010 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60
.001 15.38 10.39 8.49 7.46 6.81 6.35 6.02 5.76 5.56
19 .100 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44
.050 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18
.025 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03
.010 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35
.001 22.86 16.39 13.90 12.56 11.71 11.13 10.70 10.37 10.11
20 .100 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2.00 1.96
.050 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39
.025 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84
.010 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46
.001 14.82 9.95 8.10 7.10 6.46 6.02 5.69 5.44 5.24
21 .100 2.96 2.57 2.36 2.23 2.14 2.08 2.02 1.98 1.95
.050 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37
.025 5.83 4.42 3.82 3.48 3.25 3.09 2.97 2.87 2.80
.010 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40
.001 14.59 9.77 7.94 6.95 6.32 5.88 5.56 5.31 5.11
22 .100 2.95 2.56 2.35 2.22 2.13 2.06 2.01 1.97 1.93
.050 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34
.025 5.79 4.38 3.78 3.44 3.22 3.05 2.93 2.84 2.76
.010 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35
.001 14.38 9.61 7.80 6.81 6.19 5.76 5.44 5.19 4.99
23 .100 2.94 2.55 2.34 2.21 2.11 2.05 1.99 1.95 1.92
.050 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32
.025 5.75 4.35 3.75 3.41 3.18 3.02 2.90 2.81 2.73
.010 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30
.001 14.20 9.47 7.67 6.70 6.08 5.65 5.33 5.09 4.89
Table A5 F critical values (continued)
Degrees of freedom in the numerator
Degrees of freedom
in the denominator
p 10 12 15 20 25 30 40 50 60 120 1000
16 .100 2.03 1.99 1.94 1.89 1.86 1.84 1.81 1.79 1.78 1.75 1.72
.050 2.49 2.42 2.35 2.28 2.23 2.19 2.15 2.12 2.11 2.06 2.02
.025 2.99 2.89 2.79 2.68 2.61 2.57 2.51 2.47 2.45 2.38 2.32
.010 3.69 3.55 3.41 3.26 3.16 3.10 3.02 2.97 2.93 2.84 2.76
.001 5.81 5.55 5.27 4.99 4.82 4.70 4.54 4.45 4.39 4.23 4.08
17 .100 2.00 1.96 1.91 1.86 1.83 1.81 1.78 1.76 1.75 1.72 1.69
.050 2.45 2.38 2.31 2.23 2.18 2.15 2.10 2.08 2.06 2.01 1.97
.025 2.92 2.82 2.72 2.62 2.55 2.50 2.44 2.41 2.38 2.32 2.26
.010 3.59 3.46 3.31 3.16 3.07 3.00 2.92 2.87 2.83 2.75 2.66
.001 5.58 5.32 5.05 4.78 4.60 4.48 4.33 4.24 4.18 4.02 3.87
18 .100 1.98 1.93 1.89 1.84 1.80 1.78 1.75 1.74 1.72 1.69 1.66
.050 2.41 2.34 2.27 2.19 2.14 2.11 2.06 2.04 2.02 1.97 1.92
.025 2.87 2.77 2.67 2.56 2.49 2.44 2.38 2.35 2.32 2.26 2.20
.010 3.51 3.37 3.23 3.08 2.98 2.92 2.84 2.78 2.75 2.66 2.58
.001 5.39 5.13 4.87 4.59 4.42 4.30 4.15 4.06 4.00 3.84 3.69
19 .100 1.96 1.91 1.86 1.81 1.78 1.76 1.73 1.71 1.70 1.67 1.64
.050 2.38 2.31 2.23 2.16 2.11 2.07 2.03 2.00 1.98 1.93 1.88
.025 2.82 2.72 2.62 2.51 2.44 2.39 2.33 2.30 2.27 2.20 2.14
.010 3.43 3.30 3.15 3.00 2.91 2.84 2.76 2.71 2.67 2.58 2.50
.001 5.22 4.97 4.70 4.43 4.26 4.14 3.99 3.90 3.84 3.68 3.53
20 .100 1.94 1.89 1.84 1.79 1.76 1.74 1.71 1.69 1.68 1.64 1.61
.050 2.35 2.28 2.20 2.12 2.07 2.04 1.99 1.97 1.95 1.90 1.85
.025 2.77 2.68 2.57 2.46 2.40 2.35 2.29 2.25 2.22 2.16 2.09
.010 3.37 3.23 3.09 2.94 2.84 2.78 2.69 2.64 2.61 2.52 2.43
.001 5.08 4.82 4.56 4.29 4.12 4.00 3.86 3.77 3.70 3.54 3.40
21 .100 1.92 1.87 1.83 1.78 1.74 1.72 1.69 1.67 1.66 1.62 1.59
.050 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.94 1.92 1.87 1.82
.025 2.73 2.64 2.53 2.42 2.36 2.31 2.25 2.21 2.18 2.11 2.05
.010 3.31 3.17 3.03 2.88 2.79 2.72 2.64 2.58 2.55 2.46 2.37
.001 4.95 4.70 4.44 4.17 4.00 3.88 3.74 3.64 3.58 3.42 3.28
22 .100 1.90 1.86 1.81 1.76 1.73 1.70 1.67 1.65 1.64 1.60 1.57
.050 2.30 2.23 2.15 2.07 2.02 1.98 1.94 1.91 1.89 1.84 1.79
.025 2.70 2.60 2.50 2.39 2.32 2.27 2.21 2.17 2.14 2.08 2.01
.010 3.26 3.12 2.98 2.83 2.73 2.67 2.58 2.53 2.50 2.40 2.32
.001 4.83 4.58 4.33 4.06 3.89 3.78 3.63 3.54 3.48 3.32 3.17
23 .100 1.89 1.84 1.80 1.74 1.71 1.69 1.66 1.64 1.62 1.59 1.55
.050 2.27 2.20 2.13 2.05 2.00 1.96 1.91 1.88 1.86 1.81 1.76
.025 2.67 2.57 2.47 2.36 2.29 2.24 2.18 2.14 2.11 2.04 1.98
.010 3.21 3.07 2.93 2.78 2.69 2.62 2.54 2.48 2.45 2.35 2.27
.001 4.73 4.48 4.23 3.96 3.79 3.68 3.53 3.44 3.38 3.22 3.08
Table A6 F critical values (continued)
Degrees of freedom in the numerator
Degrees of freedom
in the denominator
p 1 2 3 4 5 6 7 8 9
24 .100 2.93 2.54 2.33 2.19 2.10 2.04 1.98 1.94 1.91
.050 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30
.025 5.72 4.32 3.72 3.38 3.15 2.99 2.87 2.78 2.70
.010 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26
.001 14.03 9.34 7.55 6.59 5.98 5.55 5.23 4.99 4.80
25 .100 2.92 2.53 2.32 2.18 2.09 2.02 1.97 1.93 1.89
.050 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28
.025 5.69 4.29 3.69 3.35 3.13 2.97 2.85 2.75 2.68
.010 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22
.001 13.88 9.22 7.45 6.49 5.89 5.46 5.15 4.91 4.71
26 .100 2.91 2.52 2.31 2.17 2.08 2.01 1.96 1.92 1.88
.050 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27
.025 5.66 4.27 3.67 3.33 3.10 2.94 2.82 2.73 2.65
.010 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18
.001 13.74 9.12 7.36 6.41 5.80 5.38 5.07 4.83 4.64
27 .100 2.90 2.51 2.30 2.17 2.07 2.00 1.95 1.91 1.87
.050 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25
.025 5.63 4.24 3.65 3.31 3.08 2.92 2.80 2.71 2.63
.010 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15
.001 13.61 9.02 7.27 6.33 5.73 5.31 5.00 4.76 4.57
28 .100 2.89 2.50 2.29 2.16 2.06 2.00 1.94 1.90 1.87
.050 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24
.025 5.61 4.22 3.63 3.29 3.06 2.90 2.78 2.69 2.61
.010 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12
.001 13.50 8.93 7.19 6.25 5.66 5.24 4.93 4.69 4.50
29 .100 2.89 2.50 2.28 2.15 2.06 1.99 1.93 1.89 1.86
.050 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22
.025 5.59 4.20 3.61 3.27 3.04 2.88 2.76 2.67 2.59
.010 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09
.001 13.39 8.85 7.12 6.19 5.59 5.18 4.87 4.64 4.45
30 .100 2.88 2.49 2.28 2.14 2.05 1.98 1.93 1.88 1.85
.050 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21
.025 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.57
.010 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07
.001 13.29 8.77 7.05 6.12 5.53 5.12 4.82 4.58 4.39
40 .100 2.84 2.44 2.23 2.09 2.00 1.93 1.87 1.83 1.79
.050 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12
.025 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.45
.010 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89
.001 12.61 8.25 6.59 5.70 5.13 4.73 4.44 4.21 4.02
Table A7 F critical values (continued)
Degrees of freedom in the numerator
Degrees of freedom
in the denominator
p 10 12 15 20 25 30 40 50 60 120 1000
24 .100 1.88 1.83 1.78 1.73 1.70 1.67 1.64 1.62 1.61 1.57 1.54
.050 2.25 2.18 2.11 2.03 1.97 1.94 1.89 1.86 1.84 1.79 1.74
.025 2.64 2.54 2.44 2.33 2.26 2.21 2.15 2.11 2.08 2.01 1.94
.010 3.17 3.03 2.89 2.74 2.64 2.58 2.49 2.44 2.40 2.31 2.22
.001 4.64 4.39 4.14 3.87 3.71 3.59 3.45 3.36 3.29 3.14 2.99
25 .100 1.87 1.82 1.77 1.72 1.68 1.66 1.63 1.61 1.59 1.56 1.52
.050 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.84 1.82 1.77 1.72
.025 2.61 2.51 2.41 2.30 2.23 2.18 2.12 2.08 2.05 1.98 1.91
.010 3.13 2.99 2.85 2.70 2.60 2.54 2.45 2.40 2.36 2.27 2.18
.001 4.56 4.31 4.06 3.79 3.63 3.52 3.37 3.28 3.22 3.06 2.91
26 .100 1.86 1.81 1.76 1.71 1.67 1.65 1.61 1.59 1.58 1.54 1.51
.050 2.22 2.15 2.07 1.99 1.94 1.90 1.85 1.82 1.80 1.75 1.70
.025 2.59 2.49 2.39 2.28 2.21 2.16 2.09 2.05 2.03 1.95 1.89
.010 3.09 2.96 2.81 2.66 2.57 2.50 2.42 2.36 2.33 2.23 2.14
.001 4.48 4.24 3.99 3.72 3.56 3.44 3.30 3.21 3.15 2.99 2.84
27 .100 1.85 1.80 1.75 1.70 1.66 1.64 1.60 1.58 1.57 1.53 1.50
.050 2.20 2.13 2.06 1.97 1.92 1.88 1.84 1.81 1.79 1.73 1.68
.025 2.57 2.47 2.36 2.25 2.18 2.13 2.07 2.03 2.00 1.93 1.86
.010 3.06 2.93 2.78 2.63 2.54 2.47 2.38 2.33 2.29 2.20 2.11
.001 4.41 4.17 3.92 3.66 3.49 3.38 3.23 3.14 3.08 2.92 2.78
28 .100 1.84 1.79 1.74 1.69 1.65 1.63 1.59 1.57 1.56 1.52 1.48
.050 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.79 1.77 1.71 1.66
.025 2.55 2.45 2.34 2.23 2.16 2.11 2.05 2.01 1.98 1.91 1.84
.010 3.03 2.90 2.75 2.60 2.51 2.44 2.35 2.30 2.26 2.17 2.08
.001 4.35 4.11 3.86 3.60 3.43 3.32 3.18 3.09 3.02 2.86 2.72
29 .100 1.83 1.78 1.73 1.68 1.64 1.62 1.58 1.56 1.55 1.51 1.47
.050 2.18 2.10 2.03 1.94 1.89 1.85 1.81 1.77 1.75 1.70 1.65
.025 2.53 2.43 2.32 2.21 2.14 2.09 2.03 1.99 1.96 1.89 1.82
.010 3.00 2.87 2.73 2.57 2.48 2.41 2.33 2.27 2.23 2.14 2.05
.001 4.29 4.05 3.80 3.54 3.38 3.27 3.12 3.03 2.97 2.81 2.66
30 .100 1.82 1.77 1.72 1.67 1.63 1.61 1.57 1.55 1.54 1.50 1.46
.050 2.16 2.09 2.01 1.93 1.88 1.84 1.79 1.76 1.74 1.68 1.63
.025 2.51 2.41 2.31 2.20 2.12 2.07 2.01 1.97 1.94 1.87 1.80
.010 2.98 2.84 2.70 2.55 2.45 2.39 2.30 2.25 2.21 2.11 2.02
.001 4.24 4.00 3.75 3.49 3.33 3.22 3.07 2.98 2.92 2.76 2.61
40 .100 1.76 1.71 1.66 1.61 1.57 1.54 1.51 1.48 1.47 1.42 1.38
.050 2.08 2.00 1.92 1.84 1.78 1.74 1.69 1.66 1.64 1.58 1.52
.025 2.39 2.29 2.18 2.07 1.99 1.94 1.88 1.83 1.80 1.72 1.65
.010 2.80 2.66 2.52 2.37 2.27 2.20 2.11 2.06 2.02 1.92 1.82
.001 3.87 3.64 3.40 3.14 2.98 2.87 2.73 2.64 2.57 2.41 2.25
Table A8 F critical values (continued)
Degrees of freedom in the numerator
Degrees of freedom
in the denominator
p 1 2 3 4 5 6 7 8 9
50 .100 2.81 2.41 2.20 2.06 1.97 1.90 1.84 1.80 1.76
.050 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07
.025 5.34 3.97 3.39 3.05 2.83 2.67 2.55 2.46 2.38
.010 7.17 5.06 4.20 3.72 3.41 3.19 3.02 2.89 2.78
.001 12.22 7.96 6.34 5.46 4.90 4.51 4.22 4.00 3.82
60 .100 2.79 2.39 2.18 2.04 1.95 1.87 1.82 1.77 1.74
.050 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04
.025 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.33
.010 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72
.001 11.97 7.77 6.17 5.31 4.76 4.37 4.09 3.86 3.69
100 .100 2.76 2.36 2.14 2.00 1.91 1.83 1.78 1.73 1.69
.050 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.97
.025 5.18 3.83 3.25 2.92 2.70 2.54 2.42 2.32 2.24
.010 6.90 4.82 3.98 3.51 3.21 2.99 2.82 2.69 2.59
.001 11.50 7.41 5.86 5.02 4.48 4.11 3.83 3.61 3.44
200 .100 2.73 2.33 2.11 1.97 1.88 1.80 1.75 1.70 1.66
.050 3.89 3.04 2.65 2.42 2.26 2.14 2.06 1.98 1.93
.025 5.10 3.76 3.18 2.85 2.63 2.47 2.35 2.26 2.18
.010 6.76 4.71 3.88 3.41 3.11 2.89 2.73 2.60 2.50
.001 11.15 7.15 5.63 4.81 4.29 3.92 3.65 3.43 3.26
1000 .100 2.71 2.31 2.09 1.95 1.85 1.78 1.72 1.68 1.64
.050 3.85 3.00 2.61 2.38 2.22 2.11 2.02 1.95 1.89
.025 5.04 3.70 3.13 2.80 2.58 2.42 2.30 2.20 2.13
.010 6.66 4.63 3.80 3.34 3.04 2.82 2.66 2.53 2.43
.001 10.89 6.96 5.46 4.65 4.14 3.78 3.51 3.30 3.13
Table A9 F critical values (continued)
Degrees of freedom in the numerator
Degrees of freedom
in the denominator
p 10 12 15 20 25 30 40 50 60 120 1000
50 .100 1.73 1.68 1.63 1.57 1.53 1.50 1.46 1.44 1.42 1.38 1.33
.050 2.03 1.95 1.87 1.78 1.73 1.69 1.63 1.60 1.58 1.51 1.45
.025 2.32 2.22 2.11 1.99 1.92 1.87 1.80 1.75 1.72 1.64 1.56
.010 2.70 2.56 2.42 2.27 2.17 2.10 2.01 1.95 1.91 1.80 1.70
.001 3.67 3.44 3.20 2.95 2.79 2.68 2.53 2.44 2.38 2.21 2.05
60 .100 1.71 1.66 1.60 1.54 1.50 1.48 1.44 1.41 1.40 1.35 1.30
.050 1.99 1.92 1.84 1.75 1.69 1.65 1.59 1.56 1.53 1.47 1.40
.025 2.27 2.17 2.06 1.94 1.87 1.82 1.74 1.70 1.67 1.58 1.49
.010 2.63 2.50 2.35 2.20 2.10 2.03 1.94 1.88 1.84 1.73 1.62
.001 3.54 3.32 3.08 2.83 2.67 2.55 2.41 2.32 2.25 2.08 1.92
100 .100 1.66 1.61 1.56 1.49 1.45 1.42 1.38 1.35 1.34 1.28 1.22
.050 1.93 1.85 1.77 1.68 1.62 1.57 1.52 1.48 1.45 1.38 1.30
.025 2.18 2.08 1.97 1.85 1.77 1.71 1.64 1.59 1.56 1.46 1.36
.010 2.50 2.37 2.22 2.07 1.97 1.89 1.80 1.74 1.69 1.57 1.45
.001 3.30 3.07 2.84 2.59 2.43 2.32 2.17 2.08 2.01 1.83 1.64
200 .100 1.63 1.58 1.52 1.46 1.41 1.38 1.34 1.31 1.29 1.23 1.16
.050 1.88 1.80 1.72 1.62 1.56 1.52 1.46 1.41 1.39 1.30 1.21
.025 2.11 2.01 1.90 1.78 1.70 1.64 1.56 1.51 1.47 1.37 1.25
.010 2.41 2.27 2.13 1.97 1.87 1.79 1.69 1.63 1.58 1.45 1.30
.001 3.12 2.90 2.67 2.42 2.26 2.15 2.00 1.90 1.83 1.64 1.43
1000 .100 1.61 1.55 1.49 1.43 1.38 1.35 1.30 1.27 1.25 1.18 1.08
.050 1.84 1.76 1.68 1.58 1.52 1.47 1.41 1.36 1.33 1.24 1.11
.025 2.06 1.96 1.85 1.72 1.64 1.58 1.50 1.45 1.41 1.29 1.13
.010 2.34 2.20 2.06 1.90 1.79 1.72 1.61 1.54 1.50 1.35 1.16
.001 2.99 2.77 2.54 2.30 2.14 2.02 1.87 1.77 1.69 1.49 1.22
Table A10 F critical values (continued)

Numerical entries represent the probability that a standard normal random variable is between 00 and zz where z=xμσz=xμσ.

.
Figure A2

Standard Normal Probability Distribution: Z Table

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549
0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015
1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177
1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441
1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857
2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890
2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916
2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936
2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964
2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990
3.1 0.4990 0.4991 0.4991 0.4991 0.4992 0.4992 0.4992 0.4992 0.4993 0.4993
3.2 0.4993 0.4993 0.4994 0.4994 0.4994 0.4994 0.4994 0.4995 0.4995 0.4995
3.3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4996 0.4996 0.4997
3.4 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4998
Table A11 Standard Normal Distribution

Student's t Distribution

.
Figure A3 Upper critical values of Student's t Distribution with v Degrees of Freedom

For selected probabilities, a, the table shows the values tv,a such that P(tv > tv,a) = a, where tv is a Student’s t random variable with v degrees of freedom. For example, the probability is .10 that a Student’s t random variable with 10 degrees of freedom exceeds 1.372.

NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, September 2011.
v 0.10 0.05 0.025 0.01 0.005 0.001
1 3.078 6.314 12.706 31.821 63.657 318.313
2 1.886 2.920 4.303 6.965 9.925 22.327
3 1.638 2.353 3.182 4.541 5.841 10.215
4 1.533 2.132 2.776 3.747 4.604 7.173
5 1.476 2.015 2.571 3.365 4.032 5.893
6 1.440 1.943 2.447 3.143 3.707 5.208
7 1.415 1.895 2.365 2.998 3.499 4.782
8 1.397 1.860 2.306 2.896 3.355 4.499
9 1.383 1.833 2.262 2.821 3.250 4.296
10 1.372 1.812 2.228 2.764 3.169 4.143
11 1.363 1.796 2.201 2.718 3.106 4.024
12 1.356 1.782 2.179 2.681 3.055 3.929
13 1.350 1.771 2.160 2.650 3.012 3.852
14 1.345 1.761 2.145 2.624 2.977 3.787
15 1.341 1.753 2.131 2.602 2.947 3.733
16 1.337 1.746 2.120 2.583 2.921 3.686
17 1.333 1.740 2.110 2.567 2.898 3.646
18 1.330 1.734 2.101 2.552 2.878 3.610
19 1.328 1.729 2.093 2.539 2.861 3.579
20 1.325 1.725 2.086 2.528 2.845 3.552
21 1.323 1.721 2.080 2.518 2.831 3.527
22 1.321 1.717 2.074 2.508 2.819 3.505
23 1.319 1.714 2.069 2.500 2.807 3.485
24 1.318 1.711 2.064 2.492 2.797 3.467
25 1.316 1.708 2.060 2.485 2.787 3.450
26 1.315 1.706* 2.056 2.479 2.779 3.435
27 1.314 1.703 2.052 2.473 2.771 3.421
28 1.313 1.701 2.048 2.467 2.763 3.408
29 1.311 1.699 2.045 2.462 2.756 3.396
30 1.310 1.697 2.042 2.457 2.750 3.385
40 1.303 1.684 2.021 2.423 2.704 3.307
60 1.296 1.671 2.000 2.390 2.660 3.232
100 1.290 1.660 1.984 2.364 2.626 3.174
1.282 1.645 1.960 2.326 2.576 3.090
Table A12 Probability of Exceeding the Critical Value
.
Figure A4

χ2 Probability Distribution

df 0.995 0.990 0.975 0.950 0.900 0.100 0.050 0.025 0.010 0.005
1 0.000 0.000 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879
2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597
3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838
4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860
5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750
6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548
7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278
8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955
9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589
10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188
11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757
12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300
13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819
14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319
15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801
16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267
17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718
18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156
19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582
20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997
21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401
22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796
23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181
24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559
25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928
26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290
27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645
28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993
29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336
30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672
40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766
50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490
60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952
70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215
80 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321
90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299
100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169
Table A13 Area to the Right of the Critical Value of χ2
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