1. Appendix B
STATISTICAL TABLES
OVERVIEW
Table B.1: Proportions of the Area Under the Normal Curve
Table B.2: 1200 Two-Digit Random Numbers
Table B.3: Critical Values for Studentβs t-TEST
Table B.4: Power of Studentβs Single Sample t-Ratio
Table B.5: Power of Studentβs Two Sample t-Ratio, One-Tailed Tests
Table B.6: Power of Studentβs Two Sample t-Ratio, Two-Tailed Tests
Table B.7: Critical Values for Pearsonβs Correlation Coefο¬cient
Table B.8 Critical Values for Spearmanβs Rank Order Correlation
Coefο¬cient
Table B.9: r to z Transformation
Table B.10: Power of Pearsonβs Correlation Coefο¬cient
Table B.11: Critical Values for the F-Ratio
Table B.12: Critical Values for the Fmax Test
Table B.13: Critical Values for the Studentized Range Test
Table B.14: Power of Anova
Table B.15: Critical Values for Chi-Squared
Table B.16: Critical Values for MannβWhitney u-Test
Understanding Business Research, First Edition. Bart L. Weathington, Christopher J.L. Cunningham,
and David J. Pittenger.
ο 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.
435
2. 436 STATISTICAL TABLES
TABLE B.1: PROPORTIONS OF THE AREA UNDER THE NORMAL CURVE
Using Table B.1
Table B.1 is used to convert the raw score to a z-score using the equation below (also
discussed in Appendix A), where X is the observed score, M is the mean of the data,
and SD is the standard deviation of the data.
z =
(X β M )
SD
The z-score is a standard deviate that allows you to use the standard normal distri-
bution. The normal distribution has a mean of 0.0 and a standard deviation of 1.0. The
normal distribution is symmetrical. The values in Table B.1 represent the proportion of
area in the standard normal curve that occurs between speciο¬c points. The table contains
z-scores between 0.00 and 3.98. Because the normal distribution is symmetrical, the table
represents z-scores ranging between β3.98 and 3.98.
Column A of the table represents the z-score. Column B represents the proportion
of the curve between the mean and the z-score. Column C represents the proportion of
the curve that extends from to z-score to β.
Example:
Negative z-Score Positive z-Score
z-score = β1.30 z-score = +1.30
0.0
β4.0 β3.0 β2.0 β1.0 0.0 1.0 2.0 3.0 4.0
0.1
0.2
Relative
frequency
x
0.3
0.4
Column B
Column C
Relative
frequency
x
0.0
β4.0 β3.0 β2.0 β1.0 0.0 1.0 2.0 3.0 4.0
0.1
0.2
0.3
0.4
Column B
Column C
Column B Column C
Negative z-Scores
Area between mean and βz 0.4032 β 40.32% of curve
Area less than βz β 0.0968 9.68% of curve
Positive z-Scores
Area between mean and +z 0.4032 β 40.32% of curve
Area greater than +z β 0.0968 9.68% of curve
Area between βz and + z 0.4032 + 0.4032 = 0.8064 or 80.64% of curve
Area below βz and above +z 0.0968 + 0.0968 = 0.1936 or 19.36% of curve
3. TABLE B.1: PROPORTIONS OF THE AREA UNDER THE NORMAL CURVE 437
TABLE B.1. Proportions of the Area Under the Normal Curve
A B C A B C A B C
Area Area Area
between Area between Area between Area
z M and z beyond z z M and z beyond z z M and z beyond z
0.00 0.0000 0.5000 0.40 0.1554 0.3446 0.80 0.2881 0.2119
0.01 0.0040 0.4960 0.41 0.1591 0.3409 0.81 0.2910 0.2090
0.02 0.0080 0.4920 0.42 0.1628 0.3372 0.82 0.2939 0.2061
0.03 0.0120 0.4880 0.43 0.1664 0.3336 0.83 0.2967 0.2033
0.04 0.0160 0.4840 0.44 0.1700 0.3300 0.84 0.2995 0.2005
0.05 0.0199 0.4801 0.45 0.1736 0.3264 0.85 0.3023 0.1977
0.06 0.0239 0.4761 0.46 0.1772 0.3228 0.86 0.3051 0.1949
0.07 0.0279 0.4721 0.47 0.1808 0.3192 0.87 0.3078 0.1922
0.08 0.0319 0.4681 0.48 0.1844 0.3156 0.88 0.3106 0.1894
0.09 0.0359 0.4641 0.49 0.1879 0.3121 0.89 0.3133 0.1867
0.10 0.0398 0.4602 0.50 0.1915 0.3085 0.90 0.3159 0.1841
0.11 0.0438 0.4562 0.51 0.1950 0.3050 0.91 0.3186 0.1814
0.12 0.0478 0.4522 0.52 0.1985 0.3015 0.92 0.3212 0.1788
0.13 0.0517 0.4483 0.53 0.2019 0.2981 0.93 0.3238 0.1762
0.14 0.0557 0.4443 0.54 0.2054 0.2946 0.94 0.3264 0.1736
0.15 0.0596 0.4404 0.55 0.2088 0.2912 0.95 0.3289 0.1711
0.16 0.0636 0.4364 0.56 0.2123 0.2877 0.96 0.3315 0.1685
0.17 0.0675 0.4325 0.57 0.2157 0.2843 0.97 0.3340 0.1660
0.18 0.0714 0.4286 0.58 0.2190 0.2810 0.98 0.3365 0.1635
0.19 0.0753 0.4247 0.59 0.2224 0.2776 0.99 0.3389 0.1611
0.20 0.0793 0.4207 0.60 0.2257 0.2743 0.99 0.3413 0.1587
0.21 0.0832 0.4168 0.61 0.2291 0.2709 1.01 0.3438 0.1562
0.22 0.0871 0.4129 0.62 0.2324 0.2676 1.02 0.3461 0.1539
0.23 0.0910 0.4090 0.63 0.2357 0.2643 1.03 0.3485 0.1515
0.24 0.0948 0.4052 0.64 0.2389 0.2611 1.04 0.3508 0.1492
0.25 0.0987 0.4013 0.65 0.2422 0.2578 1.05 0.3531 0.1469
0.26 0.1026 0.3974 0.66 0.2454 0.2546 1.06 0.3554 0.1446
0.27 0.1064 0.3936 0.67 0.2486 0.2514 1.07 0.3577 0.1423
0.28 0.1103 0.3897 0.68 0.2517 0.2483 1.08 0.3599 0.1401
0.29 0.1141 0.3859 0.69 0.2549 0.2451 1.09 0.3621 0.1379
0.30 0.1179 0.3821 0.70 0.2580 0.2420 1.10 0.3643 0.1357
0.31 0.1217 0.3783 0.71 0.2611 0.2389 1.11 0.3665 0.1335
0.32 0.1255 0.3745 0.72 0.2642 0.2358 1.12 0.3686 0.1314
0.33 0.1293 0.3707 0.73 0.2673 0.2327 1.13 0.3708 0.1292
0.34 0.1331 0.3669 0.74 0.2704 0.2296 1.14 0.3729 0.1271
0.35 0.1368 0.3632 0.75 0.2734 0.2266 1.15 0.3749 0.1251
0.36 0.1406 0.3594 0.76 0.2764 0.2236 1.16 0.3770 0.1230
0.37 0.1443 0.3557 0.77 0.2794 0.2206 1.17 0.3790 0.1210
0.38 0.1480 0.3520 0.78 0.2823 0.2177 1.18 0.3810 0.1190
0.39 0.1517 0.3483 0.79 0.2852 0.2148 1.19 0.3830 0.1170
(Continued)
6. 440 STATISTICAL TABLES
In the following examples, we add 0.5000 to the area between the mean and z-
score. The 0.5000 represents the proportion of the curve on the complementary half of
the normal curve.
Area at and below +z = +1.30 0.5000 + 0.4032 = 0.9032 or 90.32% of curve
Area at and above βz = β1.30 0.4032 + 0.5000 = 0.9032 or 90.32% of curve
TABLE B.2: 1200 TWO-DIGIT RANDOM NUMBERS
Using Table B.2
This table consists of two-digit random numbers that can range between 00 and 99
inclusive. To select a series of random numbers, select a column and row at random and
then record the numbers. You may move in any direction to generate the sequence of
numbers.
Example: A researcher wished to randomly assign participants to one of ο¬ve treatment
conditions. Recognizing that the numbers in Table B.2 range between 00 and 99, the
researcher decided to use the following table to convert the random numbers to the ο¬ve
treatment conditions:
Range of Random Numbers Treatment Condition
00β20 1
21β40 2
41β60 3
61β80 4
81β99 5
9. TABLE B.3: CRITICAL VALUES FOR STUDENTβS t-TEST 443
TABLE B.3: CRITICAL VALUES FOR STUDENTβS t-TEST
Using Table B.3
For any given df, the table shows the values of tcritical corresponding to various levels of
probability. The tobserved is statistically signiο¬cant at a given level when it is equal to or
greater than the value shown in the table.
For the single sample t-ratio, df = N β 1.
For the two sample t-ratio, df = (n1 β 1) + (n2 β 1).
Examples:
Nondirectional Hypothesis
H0: ΞΌ β ΞΌ = 0 H1: ΞΌ β ΞΌ = 0 Ξ± = 0.05, df = 30
tcritical = Β±2.042 If |tobserved| β₯ |tcritical| then reject H0
Directional Hypothesis
H0: ΞΌ β ΞΌ β€ 0 H1: ΞΌ β ΞΌ 0 Ξ± = 0.05, df = 30
tcritical = +1.697 If tobserved β₯ tcritical then reject H0
H0: ΞΌ β ΞΌ β₯ 0 H1: ΞΌ β ΞΌ 0 Ξ± = 0.05, df = 30
tcritical = β1.697 If tobserved β€ tcritical then reject H0
11. TABLE B.4: POWER OF STUDENTβS SINGLE SAMPLE t-RATIO 445
TABLE B.4: POWER OF STUDENTβS SINGLE SAMPLE t-RATIO
Using Table B.4
This table provides the power (1 β Ξ²) of the single sample t-ratio given effect size,
sample size (n), Ξ±, and directionality of the test.
Example: A researcher plans to conduct a study for which H0: is ΞΌ = 12.0 using a
two-tailed t-ratio. The researcher believes that with Ξ± = 0.05 and that the effect size is
0.20. Approximately how many participants should be in the sample for the power to be
approximately 0.80? According to Table B.4, if the researcher uses 200 participants, the
power will be 1 β Ξ² = 0.83.
Note that for Cohenβs d, an estimate of effect size is as follows:
d = 0.20 = βsmallβ; d = 0.50 = βmediumβ; d = 0.80 = βlarge.β
13. TABLE B.5: POWER OF STUDENTβS TWO SAMPLE t-RATIO, ONE-TAILED TESTS 447
TABLE B.5: POWER OF STUDENTβS TWO SAMPLE t-RATIO, ONE-TAILED
TESTS
0.4
Reject null
Ξ±
0.3
0.2
Relative
frequency
0.1
0.0
β3 β2 β1 0
t
1
Fail to reject null
2 3
Reject null
Ξ±
Fail to reject null
0.4
0.3
0.2
Relative
frequency
0.1
0.0
β3 β1
β2
t
1
0 2 3
Using Table B.5
This table provides the power (1 β Ξ²) of the two sample t-ratio given effect size, sample
size (n), and Ξ± when the researcher uses a directional test.
Example: A researcher plans to conduct a study for which H0: is ΞΌ1 β€ ΞΌ2 using a
one-tailed t-ratio. The researcher believes that with Ξ± = 0.05 and that the effect size is
0.20. Approximately how many participants should be in the sample for power to be
approximately 0.80? According to Table B.5, if the researcher uses 300 participants in
each sample, the power will be 1 β Ξ² = 0.81.
Note that for Cohenβs d, an estimate of effect size:
d = 0.20 = βsmallβ; d = 0.50 = βmediumβ; d = 0.80 = βlarge.β
15. TABLE B.6: POWER OF STUDENTβS TWO SAMPLE t-RATIO, TWO-TAILED TESTS 449
TABLE B.6: POWER OF STUDENTβS TWO SAMPLE t-RATIO,
TWO-TAILED TESTS
0.4
Reject null
a/2
Reject null
a/2
0.3
0.2
Relative
frequency
0.1
0.0
β3 β2 β1 0
t
1
Fail to reject null
2 3
Using Table B.6
This table provides the power (1 β Ξ²) of the two sample t-ratio given effect size, sample
size (n), and Ξ± when the researcher uses a nondirectional test.
Example: A researcher plans to conduct a study for which H0: is ΞΌ1 = ΞΌ2 using a
two-tailed t-ratio. The researcher believes that with Ξ± = 0.05 and that the effect size is
0.20. Approximately how many participants should be in the sample for the power to be
approximately 0.80? According to Table B.6, if the researcher uses 400 participants in
each group, the power will be 1 β Ξ² = 0.82.
Note that for Cohenβs d, an estimate of effect size:
d = 0.20 = βsmallβ; d = 0.50 = βmediumβ; d = 0.80 = βlarge.β
17. TABLE B.7: CRITICAL VALUES FOR PEARSONβS CORRELATION COEFFICIENT 451
TABLE B.7: CRITICAL VALUES FOR PEARSONβS CORRELATION
COEFFICIENT
Using Table B.7
For any given df, this table shows the values of r corresponding to various levels of
probability. The robserved is statistically signiο¬cant at a given level when it is equal to or
greater than the value shown in the table.
Examples:
Nondirectional Hypothesis
H0: Ο = 0 H1: Ο = 0 Ξ± = 0.05, df = 30
rcritical = Β±0.3494 If |robserved| β₯ |rcritical| then reject H0
Directional Hypothesis
H0: Ο β€ 0 H1: Ο 0 Ξ± = 0.05, df = 30
rcritical = +0.2960 If robserved β₯ rcritical then reject H0
H0: Ο β₯ 0 H1: Ο 0 Ξ± = 0.05, df = 30
rcritical = β0.2960 If robserved β€ rcritical then reject H0
Note that the relation between the correlation coefο¬cient and the t-ratio is
rc =
tc
(n β 2) + t2
c
19. TABLE B.8 CRITICAL VALUES FOR SPEARMANβS RANK ORDER CORRELATION 453
TABLE B.8 CRITICAL VALUES FOR SPEARMANβS RANK ORDER
CORRELATION COEFFICIENT
Using Table B.8
For any given df, the table shows the values of rS corresponding to various levels of
probability. The rS,observed is statistically signiο¬cant at a given level when it is equal to
or greater than the value shown in the table.
Examples:
Nondirectional Hypothesis
H0: ΟS = 0 H1: ΟS = 0 Ξ± = 0.05 df = 30
rcritical = Β±0.350 If |robserved| β₯ |rcritical| then reject H0
Directional Hypothesis
H0: ΟS β€ 0 H1: ΟS 0 Ξ± = 0.05 df = 30
rcritical = +0.296 If robserved β₯ rcritical then reject H0
H0: ΟS β₯ 0 H1: ΟS 0 Ξ± = 0.05 df = 30
rcritical = β0.296 If robserved β€ rcritical then reject H0
When df 28, we can convert the rS to a t-ratio and then use Table B.8 for
hypothesis testing.
t = rS
N β 2
1 β r2
S
For example, rS = 0.60, N = 42
t = 0.60
42 β 2
1 β 0.602
, t = 0.60
40
0.64
, t = 0.60
β
62.5
t = 4.74, df = 40
If Ξ± = 0.05, two-tailed,
tcritical = 1.684, Reject H0: Οs = 0
21. TABLE B.9: r TO z TRANSFORMATION 455
TABLE B.9: r TO z TRANSFORMATION
Using Table B.9
This table provides the Fisher r to z transformation. Both positive and negative values
of r may be used. For speciο¬c transformations, use the following equation:
zr =
1
2
loge
1 + r
1 β r
Example:
r = 0.25 β zr = 0.255
TABLE B.9. r to z Transformation
r zr r zr r zr r zr
0.00 0.000 0.25 0.255 0.50 0.549 0.75 0.973
0.01 0.010 0.26 0.266 0.51 0.563 0.76 0.996
0.02 0.020 0.27 0.277 0.52 0.576 0.77 1.020
0.03 0.030 0.28 0.288 0.53 0.590 0.78 1.045
0.04 0.040 0.29 0.299 0.54 0.604 0.79 1.071
0.05 0.050 0.30 0.310 0.55 0.618 0.80 1.099
0.06 0.060 0.31 0.321 0.56 0.633 0.81 1.127
0.07 0.070 0.32 0.332 0.57 0.648 0.82 1.157
0.08 0.080 0.33 0.343 0.58 0.662 0.83 1.188
0.09 0.090 0.34 0.354 0.59 0.678 0.84 1.221
0.10 0.100 0.35 0.365 0.60 0.693 0.85 1.256
0.11 0.110 0.36 0.377 0.61 0.709 0.86 1.293
0.12 0.121 0.37 0.388 0.62 0.725 0.87 1.333
0.13 0.131 0.38 0.400 0.63 0.741 0.88 1.376
0.14 0.141 0.39 0.412 0.64 0.758 0.89 1.422
0.15 0.151 0.40 0.424 0.65 0.775 0.90 1.472
0.16 0.161 0.41 0.436 0.66 0.793 0.91 1.528
0.17 0.172 0.42 0.448 0.67 0.811 0.92 1.589
0.18 0.182 0.43 0.460 0.68 0.829 0.93 1.658
0.19 0.192 0.44 0.472 0.69 0.848 0.94 1.738
0.20 0.203 0.45 0.485 0.70 0.867 0.95 1.832
0.21 0.213 0.46 0.497 0.71 0.887 0.96 1.946
0.22 0.224 0.47 0.510 0.72 0.908 0.97 2.092
0.23 0.234 0.48 0.523 0.73 0.929 0.98 2.298
0.24 0.245 0.49 0.536 0.74 0.950 0.99 2.647
22. 456 STATISTICAL TABLES
TABLE B.10: POWER OF PEARSONβS CORRELATION COEFFICIENT
Using Table B.10
This table provides estimates of the power (1 β Ξ²) of the Pearson correlation coefο¬cient
(r) given effect size, sample size (n), Ξ±, and directionality of the test.
Example: A researcher plans to conduct a study for which H0: is Ο = 0.0 using a two-
tailed test. The researcher believes that with Ξ± = 0.05 and that the effect size is 0.30.
Approximately how many participants should be in the sample for the power to be
approximately 0.80? According to Table B.10, if the researcher uses 90 participants, the
power will be 1 β Ξ² = 0.82.
Note that for effect sizes,
r = 0.10 = βsmallβ; r = 0.30 = βmediumβ; r = 0.50 = βlarge.β
24. 458 STATISTICAL TABLES
TABLE B.11: CRITICAL VALUES FOR THE F-RATIO
Using Table B.11
This table provides the critical values required to reject the null hypothesis for the
analysis of variance. Note that the bold text represents Ξ± = 0.01, whereas the regular
text represents Ξ± = 0.05. To use the table, you will need to identify the degrees of
freedom for the numerator and denominator. The degrees of freedom for numerator are
those used to determine the mean square for the treatment effect or interaction. The
degrees of freedom for denominator are those used to determine the mean square for the
within-groups or error variance.
Example: One Factor ANOVA A researcher conducts a study that produces the fol-
lowing ANOVA summary table.
Source SS df MS F
Between groups 28.00 2 14.00 3.50
Within groups 156.00 39 4.00 β
Total 184.00 41 β β
From the Summary Table
Degrees of freedom, numerator: dfN = 2
Degrees of freedom, denominator: dfd = 39
Fobserved = 3.50
From Table B.11
Because the exact values of the degrees of freedom for the denominator are not listed,
you must interpolate between the two adjacent numbers.
Fcritical (2, 38) = 3.24, Ξ± = 0.05 Fcritical (2, 38) = 5.21, Ξ± = 0.01
Fcritical (2, 40) = 3.23, Ξ± = 0.05 Fcritical (2, 40) = 5.15, Ξ± = 0.01
Therefore,
Fcritical (2, 39) = 3.235, Ξ± = 0.05 Fcritical (2, 39) = 5.18, Ξ± = 0.01
Fobserved = 3.50 Fcritical = 3.235, Fobserved = 3.50 Fcritical = 5.18,
Reject H0 Do not reject H0
Example: Two-Factor ANOVA
Source SS df MS F
Variable A 0.067 1 0.067 0.01
Variable B 80.433 2 40.217 6.859
AB 58.233 2 29.117 4.966
Within groups 316.600 54 5.863 β
Total 455.333 59 β β
25. TABLE B.11: CRITICAL VALUES FOR THE F -RATIO 459
From the Summary Table
Critical Values
Ξ± = 0.05 Ξ± = 0.01
Fcritical (1, 54) = 4.02 Fcritical (1, 54) = 7.12
Fcritical (2, 54) = 3.16 Fcritical (2, 54) = 5.01
Statistical Decision
Result Ξ± = 0.05 Ξ± = 0.01
Variable A dfN = 1, dfd = 54 β Fobserved = 0.01 Do not reject H0 Do not reject H0
Variable B dfN = 2, dfd = 54 β Fobserved = 6.86 Reject H0 Reject H0
Variable AB dfN = 2, dfd = 54 β Fobserved = 4.97 Reject H0 Do not reject H0
30. 464 STATISTICAL TABLES
TABLE B.12: CRITICAL VALUES FOR THE Fmax TEST
Using Table B.12
To use this table, divide the largest variance by the smallest variance to create Fmax. The
column labeled n represents the number of subjects in each group. If the sample sizes for
the two groups are not equal, determine the average n and round up. The other columns
of numbers represent the number of treatment conditions in the study. If the observed
value of Fmax is less than the tabled value then you may assume that the variances are
homogeneous, Οsmallest = Οlargest.
Example: A researcher conducted a study with six groups. The largest variance was 20
and the smallest variance was 10, with 15 participants in each group. Fmax = 2.00. The
critical value of Fmax = 4.70, Ξ± = 0.05. Therefore, we do NOT reject the hypothesis that
the variances are equivalent. The data do not appear to violate the requirement that there
is homogeneity of variance for the ANOVA.
TABLE B.12. Critical Values for the Fmax Test
Number of Variances in Study
n Ξ± 2 3 4 5 6 7 8 9 10
4 0.05 9.60 15.5 20.6 25.2 29.5 33.6 37.5 41.4 44.6
0.01 23.2 37.0 49.0 59.0 69.0 79.0 89.0 97.0 106.0
5 0.05 7.2 10.8 13.7 16.3 18.7 20.8 22.9 24.7 26.5
0.01 14.9 22.0 28.0 33.0 38.0 42.0 46.0 50.0 54.0
6 0.05 5.8 8.4 10.4 12.1 13.7 15.0 16.3 17.5 18.6
0.01 11.1 15.5 19.1 22.0 25.0 27.0 30.0 32.0 34.0
7 0.05 5.0 6.9 8.4 9.7 10.8 11.8 12.7 13.5 14.3
0.01 8.9 12.1 14.5 16.5 18.4 20.0 22.0 23.0 24.0
8 0.05 4.4 6.0 7.2 8.1 9.0 9.8 10.5 11.1 11.7
0.01 7.5 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9
9 0.05 4.0 5.3 6.3 7.1 7.8 8.4 8.9 9.5 9.9
0.01 6.5 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3
10 0.05 3.7 4.9 5.7 6.3 6.9 7.4 7.9 8.3 8.7
0.01 5.9 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9
12 0.05 3.3 4.2 4.8 5.3 5.7 6.1 6.4 6.7 7.0
0.01 4.9 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9
15 0.05 2.7 3.5 4.0 4.4 4.7 4.9 5.2 5.4 5.6
0.01 4.1 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5
20 0.05 2.5 2.9 3.3 3.5 3.7 3.9 4.1 4.2 4.4
0.01 3.3 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6
30 0.05 2.1 2.4 2.6 2.8 2.9 3.0 3.1 3.2 3.3
0.01 2.6 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0
60 0.05 1.7 1.9 1.9 2.0 2.1 2.2 2.2 2.3 2.3
0.01 2.0 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6
β 0.05 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.01 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
31. TABLE B.13: CRITICAL VALUES FOR THE STUDENTIZED RANGE TEST 465
TABLE B.13: CRITICAL VALUES FOR THE STUDENTIZED RANGE TEST
Using Table B.13
This table contains the critical values developed by Tukey for his HSD test. To use
the table, you need the degrees of freedom for the within-groups term in the ANOVA
summary table and the number of means to be compared by the HSD test.
Example: A researcher conducted a study with four groups. The degrees of freedom for
denominator (df for the within-groups factor) are 12. Using Table B.13,
qcritical = 3.62, Ξ± = 0.10
qcritical = 4.20, Ξ± = 0.05
qcritical = 5.50, Ξ± = 0.01
34. 468 STATISTICAL TABLES
TABLE B.14: POWER OF ANOVA
Using Table B.14
The values in this table help you determine the optimal sample size for an analysis of
variance given the anticipated effect size and Ξ± level.
Example: Single Factor Design A researcher wises to conduct a single factor
design with three levels of the independent variable. How many participants will the
researcher require in each treatment condition to have power equal to 1 β Ξ² = 0.80
when the effect size is moderate, f = 0.25 and Ξ± = 0.05? In this example, dfN = 2.
According to this table, 1 β Ξ² = 0.83 when there are 55 participants in each treatment
condition.
Example: Factorial Design A researcher designed a 3 Γ 4 factorial study. How many
participants should the researcher use in each treatment condition to have power equal
to 1 β Ξ² = 0.80? Also assume that the effect size is moderate, f = 0.25.
First, determine the degrees of freedom for each effect in the ANOVA
dfA = 2 = (3 β 1) j = Levels of factor A
dfB = 3 = (4 β 1) k = Levels of factor B
dfAB = 6 = (3 β 1)(4 β 1)
Next, adjust the degrees of freedom using the following equation. For this example,
assume that the sample size is 10.
nβ²β²
effect =
jk(nij β 1)
dfeffect + 1
+ 1
Adjusted Roundeda
Estimated
dfN Sample Size Sample Size Power
Factor A 2 nβ²
= 12(10β1)
2+1 + 1 nβ²
= 37 nβ²
= 40 1 β Ξ² β 0.68
Factor B 3 nβ² = 12(10β1)
3+1 + 1 nβ² = 28 nβ² = 30 1 β Ξ² β 0.61
Factor AB 6 nβ²
= 12(10β1)
6+1 + 1 nβ²
= 16.429 nβ²
= 16 1 β Ξ² β 0.45
a The adjusted sample size has been rounded to match the closest values in the power tables.
Note that for effect sizes in this type of analysis,
f = 0.10 = βsmallβ; f = 0.25 = βmediumβ; f = 0.40 = βlarge.β
39. TABLE B.15: CRITICAL VALUES FOR CHI-SQUARED 473
TABLE B.15: CRITICAL VALUES FOR CHI-SQUARED
Using Table B.15
For any given df, the table shows the values of Ο2
critical corresponding to various levels
of probability. The Ο2
observed is statistically signiο¬cant at a given level when it is equal to
or greater than the value shown in the table.
The following table lists methods for determining the degrees of freedom for different
types of the Ο2 test.
Goodness-of-ο¬t Test df = k β 1 k represents the number of
categories
Test of independence df = (r β 1)(c β 1) r and c represent the number of
rows and columns
Examples:
Ξ± = 0.05 df = 30
Ο2
critical = 43.773 If Ο2
observed β€ Ο2
critical then reject H0
41. TABLE B.16: CRITICAL VALUES FOR MANNβWHITNEY u-TEST 475
TABLE B.16: CRITICAL VALUES FOR MANNβWHITNEY u-TEST
Using Table B.16
This table provides the critical values for the MannβWhitney U -test. Note that when
calculating this statistic, you can determine the value of U and U β². When calculating
U , its value must be less than or equal to the tabled value to be considered statistically
signiο¬cant at the level of Ξ± selected. When calculating U β²
, its value must be greater than
or equal to the tabled value to be considered statistically signiο¬cant at the level of Ξ±
selected.