Appendix B STATISTICAL TABLES OVERVIEW

Transcription

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 Student s t-test Table B.4: Power of Student s Single Sample t-ratio Table B.5: Power of Student s Two Sample t-ratio, One-Tailed Tests Table B.6: Power of Student s Two Sample t-ratio, Two-Tailed Tests Table B.7: Critical Values for Pearson s Correlation Coefficient Table B.8 Critical Values for Spearman s Rank Order Correlation Coefficient Table B.9: r to z Transformation Table B.10: Power of Pearson s Correlation Coefficient Table B.11: Critical Values for the F-Ratio Table B.12: Critical Values for the F max 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 Mann Whitney u-test Understanding Business Research, First Edition. Bart L. Weathington, Christopher J.L. Cunningham, and David J. Pittenger 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 M ) SD The z-score is a standard deviate that allows you to use the standard normal distribution. 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 specific points. The table contains z-scores between 0.00 and Because the normal distribution is symmetrical, the table represents z-scores ranging between 3.98 and 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. Example: Negative z-score Positive z-score z-score = 1.30 z-score = Column B Column B Relative frequency 0.2 Column C Relative frequency 0.2 Column C x x Column B Column C Negative z-scores Area between mean and z % of curve Area less than z % of curve Positive z-scores Area between mean and +z % of curve Area greater than +z % of curve Area between z and + z = or 80.64% of curve Area below z and above +z = 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 (Continued)

4 438 STATISTICAL TABLES TABLE B.1. (Continued) 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

5 TABLE B.1: PROPORTIONS OF THE AREA UNDER THE NORMAL CURVE 439 TABLE B.1. (Continued) 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

6 440 STATISTICAL TABLES In the following examples, we add to the area between the mean and z- score. The represents the proportion of the curve on the complementary half of the normal curve. Area at and below +z =+1.30 Area at and above z = = or 90.32% of curve = 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 five 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 five treatment conditions: Range of Random Numbers Treatment Condition

7 TAB L E B Two-Digit Random Numbers (Continued) 441

8 TABLE B.2. (Continued)

9 TABLE B.3: CRITICAL VALUES FOR STUDENT S t -TEST 443 TABLE B.3: CRITICAL VALUES FOR STUDENT S t-test Using Table B.3 For any given df, the table shows the values of t critical corresponding to various levels of probability. The t observed is statistically significant 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 1. For the two sample t-ratio, df = (n 1 1) + (n 2 1). Examples: Nondirectional Hypothesis H 0 : μ μ = 0 H 1 : μ μ 0 α = 0.05, df = 30 t critical =±2.042 If t observed t critical then reject H 0 Directional Hypothesis H 0 : μ μ 0 H 1 : μ μ > 0 α = 0.05, df = 30 t critical = If t observed t critical then reject H 0 H 0 : μ μ 0 H 1 : μ μ<0 α = 0.05, df = 30 t critical = If t observed t critical then reject H 0

10 444 STATISTICAL TABLES TABLE B.3. Critical Values for Student s t-test Level of Significance of a One-Tailed or Directional Test H 0 : μ μ 0orH 0 : μ μ 0 α = 0.10 α = 0.05 α =.025 α = 0.01 α = α = α = α = α = α = α = α = Level of Significance of a Two-Tailed or Nondirectional Test H 0 : μ μ = 0 α = 0.20 α = 0.10 α = 0.05 α = 0.02 α = 0.01 α = df 1 α = α = α = α = α = α =

11 TABLE B.4: POWER OF STUDENT S SINGLE SAMPLE t -RATIO 445 TABLE B.4: POWER OF STUDENT S SINGLE SAMPLE t-ratio Using Table B.4 This table provides the power (1 β) of the single sample t-ratio given effect size, sample size (n), α, and directionality of the test. Example: A researcher plans to conduct a study for which H 0 :isμ = 12.0 usinga two-tailed t-ratio. The researcher believes that with α = 0.05 and that the effect size is 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 = Note that for Cohen s d, an estimate of effect size is as follows: d = 0.20 = small ; d = 0.50 = medium ; d = 0.80 = large.

12 446 STATISTICAL TABLES TABLE B.4. Power of Student s Single Sample t-ratio Power Table: Single Sample t-ratio α = 0.05 Two-Tailed α = 0.01 Two-Tailed n t c t c

13 TABLE B.5: POWER OF STUDENT S TWO SAMPLE t -RATIO, ONE-TAILED TESTS 447 TABLE B.5: POWER OF STUDENT S TWO SAMPLE t-ratio, ONE-TAILED TESTS 0.4 Relative frequency Reject null α Fail to reject null t Relative frequency Fail to reject null Reject null α t Using Table B.5 This table provides the power (1 β) of the two sample t-ratio given effect size, sample size (n), and α when the researcher uses a directional test. Example: A researcher plans to conduct a study for which H 0 :isμ 1 μ 2 using a one-tailed t-ratio. The researcher believes that with α = 0.05 and that the effect size is 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 = Note that for Cohen s d, an estimate of effect size: d = 0.20 = small ; d = 0.50 = medium ; d = 0.80 = large.

14 448 STATISTICAL TABLES TABLE B.5. Power of Student s Two Sample t-ratio, One-Tailed Tests Power Table: Two Sample t-ratio, One-Tailed Tests α = 0.05 One-Tailed α = 0.01 One-Tailed n t c t c

15 TABLE B.6: POWER OF STUDENT S TWO SAMPLE t -RATIO, TWO-TAILED TESTS 449 TABLE B.6: POWER OF STUDENT S TWO SAMPLE t-ratio, TWO-TAILED TESTS 0.4 Relative frequency Reject null a/2 Fail to reject null Reject null a/ t Using Table B.6 This table provides the power (1 β) of the two sample t-ratio given effect size, sample size (n), and α when the researcher uses a nondirectional test. Example: A researcher plans to conduct a study for which H 0 :isμ 1 = μ 2 using a two-tailed t-ratio. The researcher believes that with α = 0.05 and that the effect size is 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 = Note that for Cohen s d, an estimate of effect size: d = 0.20 = small ; d = 0.50 = medium ; d = 0.80 = large.

16 450 STATISTICAL TABLES TABLE B.6. Power of Student s Two Sample t-ratio, Two-Tailed Tests Power Table: Two Sample t-ratio, Two-Tailed Tests α = 0.05 Two-Tailed α = 0.01 Two-Tailed n t c t c

17 TABLE B.7: CRITICAL VALUES FOR PEARSON S CORRELATION COEFFICIENT 451 TABLE B.7: CRITICAL VALUES FOR PEARSON S CORRELATION COEFFICIENT Using Table B.7 For any given df, this table shows the values of r corresponding to various levels of probability. The r observed is statistically significant at a given level when it is equal to or greater than the value shown in the table. Examples: Nondirectional Hypothesis H 0 : ρ = 0 H 1 : ρ 0 α = 0.05, df = 30 r critical =± If r observed r critical then reject H 0 Directional Hypothesis H 0 : ρ 0 H 1 : ρ > 0 α = 0.05, df = 30 r critical = If r observed r critical then reject H 0 H 0 : ρ 0 H 1 : ρ<0 α = 0.05, df = 30 r critical = If r observed r critical then reject H 0 Note that the relation between the correlation coefficient and the t-ratio is r c = t c (n 2) + t 2 c

18 452 STATISTICAL TABLES TABLE B.7. Critical Values for Pearson s Correlation Coefficient Level of Significance of a One-Tailed or Directional Test H 0 : ρ 0orH 0 : ρ 0 α = 0.1 α = 0.05 α = α = 0.01 α = α = Level of Significance of a Two-Tailed or Nondirectional Test H 0 : ρ = 0 df α = 0.2 α = 0.1 α = 0.05 α = 0.02 α = 0.01 α =

19 TABLE B.8 CRITICAL VALUES FOR SPEARMAN S RANK ORDER CORRELATION 453 TABLE B.8 CRITICAL VALUES FOR SPEARMAN S RANK ORDER CORRELATION COEFFICIENT Using Table B.8 For any given df, the table shows the values of r S corresponding to various levels of probability. The r S,observed is statistically significant at a given level when it is equal to or greater than the value shown in the table. Examples: Nondirectional Hypothesis H 0 : ρ S = 0 H 1 : ρ S 0 α = 0.05 df = 30 r critical =±0.350 If r observed r critical then reject H 0 Directional Hypothesis H 0 : ρ S 0 H 1 : ρ S > 0 α = 0.05 df = 30 r critical = If r observed r critical then reject H 0 H 0 : ρ S 0 H 1 : ρ S < 0 α = 0.05 df = 30 r critical = If r observed r critical then reject H 0 When df > 28, we can convert the r S to a t-ratio and then use Table B.8 for hypothesis testing. For example, r S = 0.60, N = 42 If α = 0.05, two-tailed, t = r S N 2 1 r 2 S t = , t = , t = t = 4.74, df = 40 t critical = 1.684, Reject H 0 : ρ s = 0

20 454 STATISTICAL TABLES TABLE B.8. Critical Values for Spearman s Rank Order Correlation Coefficient Level of Significance of a One-Tailed or Directional Test H 0 : ρ S 0orH 0 : ρ S 0 α = 0.1 α = 0.05 α = α = 0.01 α = α = Level of Significance of a Two-Tailed or Nondirectional Test H 0 : ρ S = 0 df α = 0.2 α = 0.1 α = 0.05 α = 0.02 α = 0.01 α =

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 specific transformations, use the following equation: z r = 1 2 log e ( ) 1 + r 1 r Example: r = 0.25 z r = TABLE B.9. r to z Transformation r z r r z r r z r r z r

22 456 STATISTICAL TABLES TABLE B.10: POWER OF PEARSON S CORRELATION COEFFICIENT Using Table B.10 This table provides estimates of the power (1 β) of the Pearson correlation coefficient (r) given effect size, sample size (n), α, and directionality of the test. Example: A researcher plans to conduct a study for which H 0 :isρ = 0.0 usingatwotailed test. The researcher believes that with α = 0.05 and that the effect size is 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 = Note that for effect sizes, r = 0.10 = small ; r = 0.30 = medium ; r = 0.50 = large.

23 TABLE B.10: POWER OF PEARSON S CORRELATION COEFFICIENT 457 TABLE B.10. Power of Pearson s Correlation Coefficient α = 0.05 One Tailed α = 0.05 Two Tailed Effect Size: r Effect Size: r n n

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 α = 0.01, whereas the regular text represents α = 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 following ANOVA summary table. Source SS df MS F Between groups Within groups Total From the Summary Table Degrees of freedom, numerator: df N = 2 Degrees of freedom, denominator: df d = 39 F observed = 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. F critical (2, 38) = 3.24, α = 0.05 F critical (2, 38) = 5.21, α = 0.01 F critical (2, 40) = 3.23, α = 0.05 F critical (2, 40) = 5.15, α = 0.01 Therefore, F critical (2, 39) = 3.235, α = 0.05 F critical (2, 39) = 5.18, α = 0.01 F observed = 3.50 > F critical = 3.235, F observed = 3.50 < F critical = 5.18, Reject H 0 Do not reject H 0 Example: Two-Factor ANOVA Source SS df MS F Variable A Variable B AB Within groups Total

25 TABLE B.11: CRITICAL VALUES FOR THE F -RATIO 459 From the Summary Table Critical Values α = 0.05 α = 0.01 F critical (1, 54) = 4.02 F critical (1, 54) = 7.12 F critical (2, 54) = 3.16 F critical (2, 54) = 5.01 Statistical Decision Result α = 0.05 α = 0.01 Variable A df N = 1, df d = 54 F observed = 0.01 Do not reject H 0 Do not reject H 0 Variable B df N = 2, df d = 54 F observed = 6.86 Reject H 0 Reject H 0 Variable AB df N = 2, df d = 54 F observed = 4.97 Reject H 0 Do not reject H 0

26 460 TABLE B.11. Critical Values for the F-Ratio Degrees of Freedom for Numerator α Degrees of Freedom Denominator

27 (Continued) Degrees of Freedom Denominator 461

28 TABLE B.11. (Continued) 462 Degrees of Freedom for Numerator α Degrees of Freedom Denominator

29 Degrees of Freedom Denominator 463