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References

Cochran, W. G., and Cox, G. M. Experimental Designs, 2nd ed. New York: Wiley, 1957.
Conover, W. J. Practical Nonparametric Statistics, 2nd ed. New York: Wiley, 1980.
Daniel, W. W. Applied Nonparametric Statistics, 2nd ed. Boston: PWS-Kent, 1990.
Dunn, O. J. “Multiple comparisons using rank sums.” Technometrics, Vol. 6, 1964.
Freedman, D., Pisani, R., and Purves, R. Statistics. New York: W. W. Norton and Co., 1978.
Friedman, M. “The use of ranks to avoid the assumption of normality implicit in the analysis of variance.” Journal of the American Statistical Association, Vol. 32, 1937.
Gibbons, J. D. Nonparametric Statistical Inference, 2nd ed. New York: McGraw-Hill, 1985.
Hollander, M., and Wolfe, D. A. Nonparametric Statistical Methods. New York: Wiley, 1973.
Hsu, J. C. Multiple Comparisons: Theory and Methods. London: Chapman & Hall, 1996.
Kramer, C. Y. “Extension of multiple range tests to group means with un-equal number of replications.” Biometrics, Vol. 12, 1956, pp. 307–310.
Kruskal, W. H., and Wallis, W. A. “Use of ranks in one-criterion variance analysis.” Journal of the American Statistical Association, Vol. 47, 1952.
Kutner, M., Nachtsheim, C., Neter, J., and Li, W. Applied Linear Statistical Models, 5th ed. New York: McGraw-Hill/Irwin, 2005.
Lehmann, E. L. Nonparametrics: Statistical Methods Based on Ranks. (revised). New York: Springer, 2006.
Marascuilo, L. A., and McSweeney, M. Nonparametric and Distribution-Free Methods for the Social Sciences. Monterey, CA: Brooks/Cole, 1977.
Mason, R. L., Gunst, R. F., and Hess, J. L. Statistical Design and Analysis of Experiments. New York: Wiley, 1989.
Mendenhall, W. Introduction to Linear Models and the Design and Analysis of Experiments. Belmont, CA: Wadsworth, 1968.
Miller, R. G., Jr. Simultaneous Statistical Inference. New York: Springer-Verlag, 1981.
Rekab, K. & Shaikh, M. Statistical Design of Experiments with Engineering Applications, CRC publication, 2005.
Satterthwaite, F. W. “An approximate distribution of estimates of variance components.” Biometrics Bulletin, Vol. 2, 1946, pp. 110–114.
Scheffé, H. “A method for judging all contrasts in the Analysis of Variance,” Biometrica, Vol. 40, 1953, pp. 87–104.
Scheffé, H. The Analysis of Variance. New York: Wiley, 1959.
Snedecor, G. W., and Cochran, W. Statistical Methods, 7th ed. Ames, IA: Iowa State University Press, 1980.
Steel, R. G. D., and Torrie, J. H. Principles and Procedures of Statistics, 2nd ed. New York: McGraw-Hill, 1980.
Tukey, J. “Comparing individual means in the Analysis of Variance,” Biometrics, Vol. 5, 1949, pp. 99–114.
Wilcoxon, F., and Wilcox, R. A. “Some rapid approximate statistical procedures.” The American Cyanamid Co., 1964.
Winer, B. J. Statistical Principles in Experimental Design, 2nd ed. New York: McGraw-Hill, 1971.

Using Technology MINITAB: Comparing Means

MINITAB can be used to make both parametric and nonparametric inferences about $μ_{1} - μ_{2}$ $μ_{1} - μ_{2}$ for independent samples, $μ_{d}$ $μ_{d}$ for paired samples, and $μ_{1}, μ_{2} \dots, μ_{k}$ $μ_{1}, μ_{2} \dots, μ_{k}$ in an analysis of variance.

Comparing Two Means with Independent Samples

Step 1 Access the MINITAB worksheet that contains the sample data.
Step 2 Click on the “Stat” button on the MINITAB menu bar and then click on “Basic Statistics” and “2-Sample t,” as shown in Figure 7.M.1. The resulting dialog box appears as shown in Figure 7.M.2.

Figure 7.M.1

MINITAB menu options for comparing two means
Step 3a If the worksheet contains data for one quantitative variable (which the means will be computed on) and one qualitative variable (which represents the two groups or populations), select “Samples in one column” and then specify the quantitative variable in the “Samples” area and the qualitative variable in the “Subscripts” area. (See Figure 7.M.2.)

Figure 7.M.2

MINITAB 2-sample t dialog box
Step 3b If the worksheet contains the data for the first sample in one column and the data for the second sample in another column, select “Samples in different columns” and then specify the “First” and “Second” variables. Alternatively, if you have only summarized data (i.e., sample sizes, sample means, and sample standard deviations), select “Summarized data” and enter these summarized values in the appropriate boxes.
Step 4 Click the “Options” button on the MINITAB “2-Sample T” dialog box. Specify the confidence level for a confidence interval, the null-hypothesized value of the difference, $μ_{1} - μ_{2},$ $μ_{1} - μ_{2},$ and the form of the alternative hypothesis (lower tailed, two tailed, or upper tailed) in the resulting dialog box, as shown in Figure 7.M.3.

Figure 7.M.3

MINITAB options dialog box
Step 5 Click “OK” to return to the “2-Sample T” dialog box and then click “OK” again to generate the MINITAB printout.

Important Note: The MINITAB two-sample t-procedure uses the t-statistic to conduct the test of hypothesis. When the sample sizes are small, check the “Assume equal variances” box in Figure 7.M.2. When the sample sizes are large, leave the “Assume equal variances” box unchecked; the t-value will be approximately equal to the large-sample z-value, and the resulting test will still be valid.

Comparing Two Means with Paired Samples

Step 1 Access the MINITAB worksheet that contains the sample data. The data file should contain two quantitative variables—one with the data values for the first group (or population) and one with the data values for the second group. (Note: The sample size should be the same for each group.)
Step 2 Click on the “Stat” button on the MINITAB menu bar and then click on “Basic Statistics” and “Paired t” (see Figure 7.M.1).
Step 3 On the resulting dialog box, select the “Samples in columns” option and specify the two quantitative variables of interest in the “First sample” and “Second sample” boxes, as shown in Figure 7.M.4. [Alternatively, if you have only summarized data of the paired differences, select the “Summarized data (differences)” option and enter the sample size, sample mean, and sample standard deviation in the appropriate boxes.]

Figure 7.M.4

MINITAB paired-samples t dialog box
Step 4 Click the “Options” button and specify the confidence level for a confidence interval, the null-hypothesized value of the difference $μ_{d},$ $μ_{d},$ and the form of the alternative hypothesis (lower tailed, two tailed, or upper tailed) in the resulting dialog box. (See Figure 7.M.3.)
Step 5 Click “OK” to return to the “Paired t” dialog box and then click “OK” again to generate the MINITAB printout.

Rank Sum Test

Step 1 Access the MINITAB worksheet file with the sample data. It should contain two quantitative variables, one for each of the two samples being compared.
Step 2 Click on the “Stat” button on the MINITAB menu bar, then click on “Nonparametrics” and “Mann-Whitney” (see Figure 7.M.5).

Figure 7.M.5

MINITAB nonparametric menu options
Step 3 On the resulting dialog box (see Figure 7.M.6), specify the variable for the first sample in the “First Sample” box and the variable for the second sample in the “Second Sample” box.

Figure 7.M.6

MINITAB Mann-Whitney (rank sum) test dialog box
Step 4 Specify the form of the alternative hypothesis (“not equal,” “less than,” or “greater than”).
Step 5 Click “OK” to generate the MINITAB printout.

Signed Rank Test

Step 1 Access the MINITAB worksheet file with the matched-pairs data. It should contain two quantitative variables, one for each of the two groups being compared.
Step 2 Compute the difference between these two variables and save it in a column on the worksheet. (Use the “Calc” button on the MINITAB menu bar.)
Step 3 Click on the “Stat” button on the MINITAB menu bar, then click on “Nonparametrics” and “1-Sample Wilcoxon” (see Figure 7.M.5).
Step 4 On the resulting dialog box (see Figure 7.M.7), enter the variable representing the paired differences in the “Variables” box.

Figure 7.M.7

MINITAB 1-Sample Wilcoxon (signed rank) test dialog box
Step 5 Select the “Test median” option and specify the hypothesized value of the median as “0.” Select the form of the alternative hypothesis (“not equal,” “less than,” or “greater than”).
Step 6 Click “OK” to generate the MINITAB printout.

ANOVA `F`-Test

Step 1 Access the MINITAB worksheet file that contains the sample data. The data file should contain one quantitative variable (the response, or dependent, variable) and one factor variable with at least two levels.
Step 2 Click on the “Stat” button on the MINITAB menu bar and then click on “ANOVA” and “One-Way,” as shown in Figure 7.M.8.

Figure 7.M.8

MINITAB menu options for one-way ANOVA
Step 3 On the resulting dialog screen Figure 7.M.9, specify the response variable in the “Response” box and the factor variable in the “Factor” box.
Step 4 Click “OK” to generate the MINITAB printout.

Figure 7.M.9

MINITAB one-way ANOVA dialog box

TI-83/TI-84 Plus Graphing Calculator: Comparing Means

The TI-83/ TI-84 plus graphing calculator can be used to conduct tests and form confidence intervals for the difference between two means with independent samples, the difference between two means with matched pairs, and an analysis of variance.

Confidence Interval for $μ_{1} - μ_{2}$ $μ_{1} - μ_{2}$

Step 1Enter the data (Skip to Step 2 if you have summary statistics, not raw data)
- Press STAT and select 1:Edit
  
  Note: If the lists already contain data, clear the old data. Use the up ARROW to highlight “L1.”
- Press CLEAR ENTER
- Use the up ARROW to highlight “L2”
- Press CLEAR ENTER
- Use the ARROW and ENTER keys to enter the first data set into L1
- Use the ARROW and ENTER keys to enter the second data set into L2
Step 2 Access the statistical tests menu
- Press STAT
- Arrow right to TESTS
- Arrow down to 2-SampTInt
- Press ENTER
Step 3Choose “Data” or “Stats” (“Data” is selected when you have entered the raw data into the Lists. “Stats” is selected when you are given only the means, standard deviations, and sample sizes)
- Press ENTER
- If you selected “Data,” set List1 to L1 and List2 to L2
- Set Freq1 to 1 and set Freq2 to 1
- Set C-Level to the confidence level
- If you are assuming that the two populations have equal variances, select Yes for Pooled
- If you are not assuming equal variances, select No
- Press ENTER
- Arrow down to “Calculate”
- Press ENTER
- If you selected “Stats,” enter the means, standard deviations, and sample sizes
- Set C-Level to the confidence level
- If you are assuming that the two populations have equal variances, select Yes for Pooled
- If you are not assuming equal variances, select No
- Press ENTER
- Arrow down to “Calculate”
- Press ENTER
(The accompanying screen is set up for an example with a mean of 100, a standard deviation of 10, and a sample size of 15 for the first data set and a mean of 105, a standard deviation of 12, and a sample size of 18 for the second data set.)

The confidence interval will be displayed with the number of degrees of freedom, the sample statistics, and the pooled standard deviation (when appropriate).

Hypothesis Test for $μ_{1} - μ_{2}$ $μ_{1} - μ_{2}$

Step 1 Enter the data (Skip to Step 2 if you have summary statistics, not raw data)
- Press STAT and select 1:Edit
  
  Note: If the lists already contain data, clear the old data. Use the up ARROW to highlight “L1.”
- Press CLEAR ENTER
- Use the up ARROW to highlight “L2”
- Press CLEAR ENTER
- Use the ARROW and ENTER keys to enter the first data set into L1
- Use the ARROW and ENTER keys to enter the second data set into L2
Step 2 Access the statistical tests menu
- Press STAT
- Arrow right to TESTS
- Arrow down to 2-SampTTest
- Press ENTER
Step 3 Choose “Data” or “Stats” (“Data” is selected when you have entered the raw data into the Lists. “Stats” is selected when you are given only the means, standard deviations, and sample sizes)
- Press ENTER
- If you selected “Data,” set List1 to L1 and List2 to L2
- Set Freq1 to 1 and set Freq2 to 1
- Use the ARROW to highlight the appropriate alternative hypothesis
- Press ENTER
- If you are assuming that the two populations have equal variances, select Yes for Pooled
- If you are not assuming equal variances, select No
- Press ENTER
- Arrow down to “Calculate”
- Press ENTER
- If you selected “Stats,” enter the means, standard deviations, and sample sizes
- Use the ARROW to highlight the appropriate alternative hypothesis
- Press ENTER
- If you are assuming that the two populations have equal variances, select Yes for Pooled
- If you are not assuming equal variances, select No
- Press ENTER
- Arrow down to “Calculate”
- Press ENTER
(The screen that follows is set up for an example with a mean of 100, a standard deviation of 10, and a sample size of 15 for the first data set and a mean of 120, a standard deviation of 12, and a sample size of 18 for the second data set.)

The results of the hypothesis test will be displayed with the p-value, the number of degrees of freedom, the sample statistics, and the pooled standard deviation (when appropriate).

Confidence Interval for a Paired Difference Mean

Note: There is no paired difference option on the calculator. These instructions demonstrate how to calculate the differences and then use the 1-sample t-interval.

Step 1 Enter the data and calculate the differences
- Press STAT and select 1:Edit
  
  Note: If the lists already contain data, clear the old data. Use the up ARROW to highlight “L1.”
- Press CLEAR ENTER
- Use the up ARROW to highlight “L2”
- Press CLEAR ENTER
- Use the ARROW and ENTER keys to enter the first data set into L1
- Use the ARROW and ENTER keys to enter the second data set into L2
- The differences will be calculated in L3
- Use the up ARROW to highlight “L3”
- Press CLEAR—This will clear any old data, but L3 will remain highlighted
- To enter the equation $L 3 = L 1 - L 2,$ $L 3 = L 1 - L 2,$ use the following keystrokes:
- Press $2^{N D}$ $2^{N D}$ “1” (this will enter L1)
- Press the MINUS button
- Press $2^{N D}$ $2^{N D}$ “2” (this will enter L2)
  
  (Notice the equation at the bottom of the screen.)
- Press ENTER (the differences should be calculated in L3)
Step 2 Access the statistical tests menu
- Press STAT
- Arrow right to TESTS
- Arrow down to TInterval (even for large sample case)
- Press ENTER
Step 3 Choose “Data”
- Press ENTER
- Set List to L3
- Set Freq to 1
- Set C-Level to the confidence level
- Arrow down to “Calculate”
- Press ENTER
The confidence interval will be displayed with the mean, standard deviation, and sample size of the differences.

Hypothesis Test for a Paired Difference Mean

Note: There is no paired difference option on the calculator. These instructions demonstrate how to calculate the differences and then use the 1-sample t-test.

Step 1 Enter the data and calculate the differences
- Press STAT and select 1:Edit
  
  Note: If the lists already contain data, clear the old data. Use the up ARROW to highlight “L1.”
- Press CLEAR ENTER
- Use the up ARROW to highlight “L2”
- Press CLEAR ENTER
- Use the ARROW and ENTER keys to enter the first data set into L1
- Use the ARROW and ENTER keys to enter the second data set into L2
- The differences will be calculated in L3
- Use the up ARROW to highlight “L3”
- Press CLEAR—This will clear any old data, but L3 will remain highlighted
- To enter the equation $L 3 = L 1 - L 2,$ $L 3 = L 1 - L 2,$ use the following keystrokes:
- Press $2^{N D}$ $2^{N D}$ “1” (this will enter L1)
- Press the MINUS button
- Press $2^{N D}$ $2^{N D}$ “2” (this will enter L2)
(Notice the equation at the bottom of the screen.)
- Press ENTER (the differences should be calculated in L3)
Step 2 Access the statistical tests menu
- Press STAT
- Arrow right to TESTS
- Arrow down to T-Test (even for a large-sample case)
- Press ENTER
Step 3 Choose “Data”
- Press ENTER
- Enter the values for the hypothesis test, where $μ_{0} = t h e$ $μ_{0} = t h e$ value for $μ_{d}$ $μ_{d}$ in the null hypothesis
- Set List to L3
- Set Freq to 1
- Use the ARROW to highlight the appropriate alternative hypothesis
- Press ENTER
- Arrow down to “Calculate”
- Press ENTER
The test statistic and the p-value will be displayed, as will the sample mean, standard deviation, and sample size of the differences.

ANOVA `F`-Test

Step 1 Enter each data set into its own list (i.e., sample 1 into L1, sample 2 into L2, sample 3 into L3, etc.)
Step 2 Access the statistical test menu
- Press STAT
- Arrow right to TESTS
- Arrow down to ANOVA
- Press ENTER
- Type in each List name, separated by commas (e.g., L1, L2, L3, L4)
- Press ENTER
Step 3 View display

The calculator will display the F-test statistic, as well as the p-value, the factor degrees of freedom, sum of squares, mean square, and by arrowing down, the Error degrees of freedom, sum of squares, mean square, and the pooled standard deviation.

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