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Dataplot Vol 1 Vol 2

T TEST

Name:
    T TEST
Type:
    Analysis Command
Purpose:
    Perform either a one sample t-test, an unpaired two sample t-test, or a paired two sample t-test. Both one-tailed and two-tailed tests are supported.
Description:
    There are three distinct tests supported by this command.

    1. The test that the mean for a sample is equal to a specified value can be formulated as follows:
        H0: \( \mu \) = <value>
        Ha: \( \mu \) ≠ <value>
        Test Statistic: \( T = \frac{\bar{x} - \mu} {s/\sqrt{n}} \)

        with \( \bar{x} \), s, n, and \( \mu \) denoting the sample mean, sample standard deviation, sample size and the hypothesized value, respectively.

        Significance Level: \( \alpha \) (typically set to .05)
        Critical Region: For a two-tailed test

          \( T < t_{\alpha/2,n-1} \hspace{0.2in} \mbox{or} \hspace{0.2in} T > t_{1 - \alpha/2,n-1} \)

        where t is the percent point function of the t distribution.

        For a lower-tailed test

          \( T < t_{\alpha,n-1} \)

        For an upper-tailed test

          \( T > t_{1 - \alpha,n-1} \)
        Conclusion: Reject null hypothesis if T in critical region

    2. The test that the means for two independent samples (i.e., unpaired) are equal can be formulated as follows:
        H0: \( \mu_{1} = \mu_{2} \)
        Ha: \( \mu_{1} \ne \mu_{2} \)
        Test Statistic (assuming equal population variances): \( T = \frac{\bar{x_1} - \bar{x_2}} {s_{p} \sqrt{\frac{1}{n_1} + \frac{1}{n_{2}}}} \)

        where \( \bar{x}_{1} \), n1, \( \bar{x}_{2} \), and n2 are the sample one mean and sample size and the sample two mean and sample size, respectively, and where Sp is the pooled standard deviation:

        \( S_{p} = \sqrt{\frac{(n_1 - 1)s_{1}^2 + (n_2 - 1)s_{2}^2}{n_1 + n_2 -2}} \)

        where \( s_{1}^{2} \) and \( s_{2}^{2} \) denote the variances of sample 1 and sample 2, respectively.

        The degrees of freedom equals n1 + n2 - 2.

        Test Statistic (assuming unequal population variances): \( T = \frac{\bar{x_1} - \bar{x_2}} {\sqrt{\frac{s_{1}^{2}}{n_1} + \frac{s_{2}^{2}}{n_2}}} \)

        where \( \bar{x}_{1} \), \( s_{1}^{2} \) and n1 are the sample one mean, variance, and sample size and \( \bar{x}_{2} \), \( s_{2}^{2} \), and n2 are the sample two mean, variance, and sample size, respectively.

        The degrees of freedom equals:

        \( \frac{(\frac{s_1^{2}}{n_1} + \frac{s_2^{2}}{n_2})^2} {\frac{(s_1^{2}/n_1)^2}{n_1 - 1} + \frac{(s_2^{2}/n_2)^2}{n_2 - 1}} \)

        Significance Level: \( \alpha \) (typically set to .05)
        Critical Region: For a two-tailed test

          \( T < t_{\alpha/2,df} \hspace{0.2in} \mbox{or} \hspace{0.2in} T > t_{1 - \alpha/2,df} \)

        where t is the percent point function of the t distribution and df is as defined above.

        For a lower-tailed test

          \( T < t_{\alpha,df} \)

        For an upper-tailed test

          \( T > t_{1 - \alpha,df} \)
        Conclusion: Reject null hypothesis if T in critical region

    3. The test that the means for two independent samples are equal can be formulated as follows:
        H0: \( \mu_{1} = \mu_{2} \)
        Ha: \( \mu_{1} \ne \mu_{2} \)
        Test Statistic: \( T = \frac{\bar{x_1} - \bar{x_2}} {s_{xdel}/\sqrt{n}} \)

        where \( \bar{x}_{1} \) and \( \bar{x}_{2} \) are the sample one and sample two means, respectively, \( s_{xdel} \) is the standard deviation of the differences of the paired samples, and n is the number of paired samples.

        Significance Level: \( \alpha \) (typically set to .05)
        Critical Region: For a two-tailed test

          \( T < t_{\alpha/2,n-1} \hspace{0.2in} \mbox{or} \hspace{0.2in} T > t_{1 - \alpha/2,n-1} \)

        where t is the percent point function of the t distribution.

        For a lower-tailed test

          \( T < t_{\alpha,n-1} \)

        For an upper-tailed test

          \( T > t_{1 - \alpha,n-1} \)
        Conclusion: Reject null hypothesis if T in critical region

Syntax 1:
    <LOWER TAILED/UPPER TAILED> T TEST <y> <mu>
                            <SUBSET/EXCEPT/FOR qualification>
    where <LOWER TAILED/UPPER TAILED> is an optional keyword that specifies either a lower tailed or an upper tailed test;
                <y> is a response variable;
                <mu> is a parameter or number;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs a one sample t-test that the mean is equal to <mu> If <mu> is omitted, it is assumed to be zero.

    The response variable can be a matrix.

    If neither LOWER TAILED or UPPER TAILED is specified, the t test will return the results for the two-tailed case, the lower tailed case, and the upper tailed case. If LOWER TAILED is specified, then only the results for the lower tailed case will be printed. If UPPER TAILED is specified, then only the results for the upper tailed case will be printed.

Syntax 2:
    MULTIPLE <LOWER TAILED/UPPER TAILED> ONE SAMPLE T TEST <y1> ... <yk> <mu>
                <SUBSET/EXCEPT/FOR qualification>
    where <LOWER TAILED/UPPER TAILED> is an optional keyword that specifies either a lower tailed or an upper tailed test;
                <y1> ... <yk> is a list of one to 30 response variables;
                <mu> is a parameter or number;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs a one sample t-test that the mean is equal to <mu> for each of the response variables. If <mu> is omitted, it is assumed to be zero.

    The response variables can be matrices.

    If neither LOWER TAILED or UPPER TAILED is specified, the t test will return the results for the two-tailed case, the lower tailed case, and the upper tailed case. If LOWER TAILED is specified, then only the results for the lower tailed case will be printed. If UPPER TAILED is specified, then only the results for the upper tailed case will be printed.

Syntax 3:
    <LOWER TAILED/UPPER TAILED> <PAIRED> T TEST <y1> <y2>
                            <SUBSET/EXCEPT/FOR qualification>
    where <LOWER TAILED/UPPER TAILED> is an optional keyword that specifies either a lower tailed or an upper tailed test;
                <PAIRED> is an optional keyword that specifies whether a paired or unpaired t test is performed;
                <y1> is the first response variable;
                <y2> is the second response variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs a two sample t-test. If the keyword PAIRED is given, a paired t test is performed. Otherwise an unpaired t test is performed.

    The response variables can be matrices.

    If neither LOWER TAILED or UPPER TAILED is specified, the t test will return the results for the two-tailed case, the lower tailed case, and the upper tailed case. If LOWER TAILED is specified, then only the results for the lower tailed case will be printed. If UPPER will be printed.

Syntax 4:
    MULTIPLE <LOWER TAILED/UPPER TAILED> <PAIRED> T TEST <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <LOWER TAILED/UPPER TAILED> is an optional keyword that specifies either a lower tailed or an upper tailed test;
                <PAIRED> is an optional keyword that specifies whether a paired or unpaired t test is performed;
                <y1> ... <yk> is a list of 1 to 30 response variables;
                <y2> is the second response variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs all the pairwise t tests for the listed response variables.

    If the keyword PAIRED is given, a paired t test is performed. Otherwise an unpaired t test is performed.

    The response variables can be matrices.

    If neither LOWER TAILED or UPPER TAILED is specified, the t test will return the results for the two-tailed case, the lower tailed case, and the upper tailed case. If LOWER TAILED is specified, then only the results for the lower tailed case will be printed. If UPPER TAILED is specified, then only the results for the upper tailed case will be printed.

Examples:
    T TEST Y
    T TEST Y MU
    LOWER TAILED T TEST Y MU
    MULTIPLE ONE SAMPLE T TEST Y1 TO Y5 MU
    T TEST Y1 Y2
    T TEST Y1 Y2 SUBSET TAG > 2
    UPPER TAILED PAIRED T TEST Y1 Y2
    MULTIPLE T TEST Y1 TO Y10
Note:
    Sample sizes do not need to be equal unless paired t tests are specified.
Note:
    By default, Dataplot prints the test statistic for both the equal and unequal population variances assumptions for the two sample test. The following command can be used to specify this:

      SET T TEST VARIANCE EQUAL
      SET T TEST VARIANCE UNEQUAL
      SET T TEST VARIANCE BOTH

    The EQUAL keyword specifies that only the equal variance case will be printed, UNEQUAL specifies that only the unequal variances case will be printed, and BOTH resets the default that both the equal and unequal variance cases will be printed.

Note:
    Dataplot saves the following internal parameters after a t test:

      STATVAL: the value of the test statistic
      STATCDF: the CDF of the test statistic
      PVALUE: the p-value for the two-sided test
      PVALUELT: the p-value for the lower tailed test
      PVALUEUT: the p-value for the upper tailed test
      CUTLOW50: the 50% lower tailed critical value
      CUTUPP50: the 50% upper tailed critical value
      CUTLOW80: the 80% lower tailed critical value
      CUTUPP80: the 80% upper tailed critical value
      CUTLOW90: the 90% lower tailed critical value
      CUTUPP90: the 90% upper tailed critical value
      CUTLOW95: the 95% lower tailed critical value
      CUTUPP95: the 95% upper tailed critical value
      CUTLOW99: the 99% lower tailed critical value
      CUTUPP99: the 99% upper tailed critical value
      CUTLO999: the 99.9% lower tailed critical value
      CUTUP999: the 99.9% upper tailed critical value
Note:
    The following statistics are also supported:

      LET A = ONE SAMPLE T TEST Y MU
      LET A = ONE SAMPLE T TEST CDF Y MU
      LET A = ONE SAMPLE T TEST PVALUE Y MU
      LET A = ONE SAMPLE T TEST LOWER TAIL PVALUE Y MU
      LET A = ONE SAMPLE T TEST UPPER TAIL PVALUE Y MU

    If MU is omitted in the above commands, it is assumed to be zero.

      LET A = TWO SAMPLE T TEST Y1 Y2
      LET A = TWO SAMPLE T TEST CDF Y1 Y2
      LET A = TWO SAMPLE T TEST PVALUE Y1 Y2
      LET A = TWO SAMPLE T TEST LOWER TAIL PVALUE Y1 Y2
      LET A = TWO SAMPLE T TEST UPPER TAIL PVALUE Y1 Y2

      LET A = TWO SAMPLE PAIRED T TEST Y1 Y2
      LET A = TWO SAMPLE PAIRED T TEST CDF Y1 Y2
      LET A = TWO SAMPLE PAIRED T TEST PVALUE Y1 Y2
      LET A = TWO SAMPLE PAIRED T TEST LOWER TAIL PVALUE Y1 Y2
      LET A = TWO SAMPLE PAIRED T TEST UPPER TAIL PVALUE Y1 Y2

    Enter HELP STATISTICS to see what commands can use these statistics.

Default:
    Two-tailed tests will be performed unless the LOWER TAILED or UPPER TAILED keywords are specified. For the two sample test, an unpaired test will be assumed unless the PAIRED keyword is specified.
Synonyms:
    TTEST is a synonym for T TEST.
    1 SAMPLE is a synonym for ONE SAMPLE.
Related Commands: Reference:
    T tests are discussed in most introductory statistics books.
Applications:
    Confirmatory Data Analysis
Implementation Date:
    Pre-1987
    2011/3: Added support for paired t test
    2011/3: Reformatted output for greater clarity
    2011/3: Support for one-tailed tests
Program 1:
     
    SKIP 25
    READ AUTO83B.DAT Y1 Y2
    T TEST Y1 Y2
        
    The following output is generated.
                Two Sample t-Test for Equal Means
     
    First Response Variable:  Y1
    Second Response Variable: Y2
     
    H0: Population Means Are Equal
    Ha: Population Means Are Not Equal
     
    Sample One Summary Statistics:
    Number of Observations:                             249
    Sample Mean:                                   20.14457
    Sample Standard Deviation:                      6.41469
    Sample Standard Deviation of the Mean:          0.40651
     
    Sample Two Summary Statistics:
    Number of Observations:                             249
    Sample Mean:                                 -672.37751
    Sample Standard Deviation:                    480.11125
    Sample Standard Deviation of the Mean:         30.42581
     
    Test When Assume Equal Variances:
    Pooled Standard Deviation:                    339.52022
    Difference (Delta) in Means:                  692.52208
    Standard Deviation of Delta:                   30.42853
    t-Test Statistic Value:                        22.75897
    Degrees of Freedom:                                 496
    CDF Value:                                      0.99999
    P-Value (2-tailed test):                        0.00000
    P-Value (lower-tailed test):                    0.99999
    P-Value (upper-tailed test):                    0.00000
     
    Test When Assume Unequal Variances:
    Bartlett CDF Value:                             1.00000
    Difference (Delta) in Means:                  692.52208
    Standard Deviation of Delta:                   30.42853
    t-Test Statistic Value:                        22.75897
    Degrees of Freedom:                                 248
    CDF Value:                                      1.00000
    P-Value (2-tailed test):                        0.00000
    P-Value (lower-tailed test):                    1.00000
    P-Value (upper-tailed test):                    0.00000
     
     
                Two-Tailed Test (Assume Equal Variances)
     
    H0: u1 = u2; Ha: u1 <> u2
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic    Value (+/-)     Conclusion
    ------------------------------------------------------------
              50.0%       22.75897        0.67498         REJECT
              80.0%       22.75897        1.28326         REJECT
              90.0%       22.75897        1.64793         REJECT
              95.0%       22.75897        1.96475         REJECT
              99.0%       22.75897        2.58577         REJECT
              99.9%       22.75897        3.31024         REJECT
     
     
                Lower One-Tailed Test (Assume Equal Variances)
     
    H0: u1 = u2; Ha: u1 < u2
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (<)     Conclusion
    ------------------------------------------------------------
              50.0%       22.75897        0.00000         ACCEPT
              80.0%       22.75897       -0.84234         ACCEPT
              90.0%       22.75897       -1.28326         ACCEPT
              95.0%       22.75897       -1.64793         ACCEPT
              99.0%       22.75897       -2.33388         ACCEPT
              99.9%       22.75897       -3.10674         ACCEPT
     
     
                Upper One-Tailed Test (Assume Equal Variances)
     
    H0: u1 = u2; Ha: u1 > u2
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (>)     Conclusion
    ------------------------------------------------------------
              50.0%       22.75897        0.00000         REJECT
              80.0%       22.75897        0.84234         REJECT
              90.0%       22.75897        1.28326         REJECT
              95.0%       22.75897        1.64793         REJECT
              99.0%       22.75897        2.33388         REJECT
              99.9%       22.75897        3.10674         REJECT
     
     
                Two-Tailed Test (Assume Unequal Variances)
     
    H0: u1 = u2; Ha: u1 <> u2
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic    Value (+/-)     Conclusion
    ------------------------------------------------------------
              50.0%       22.75897        0.67547         REJECT
              80.0%       22.75897        1.28497         REJECT
              90.0%       22.75897        1.65101         REJECT
              95.0%       22.75897        1.96957         REJECT
              99.0%       22.75897        2.59579         REJECT
              99.9%       22.75897        3.33017         REJECT
     
     
                Lower One-Tailed Test (Assume Unequal Variances)
     
    H0: u1 = u2; Ha: u1 < u2
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (<)     Conclusion
    ------------------------------------------------------------
              50.0%       22.75897        0.00000         ACCEPT
              80.0%       22.75897       -0.84307         ACCEPT
              90.0%       22.75897       -1.28497         ACCEPT
              95.0%       22.75897       -1.65101         ACCEPT
              99.0%       22.75897       -2.34147         ACCEPT
              99.9%       22.75897       -3.12340         ACCEPT
     
     
                Upper One-Tailed Test (Assume Unequal Variances)
     
    H0: u1 = u2; Ha: u1 > u2
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (>)     Conclusion
    ------------------------------------------------------------
              50.0%       22.75897        0.00000         REJECT
              80.0%       22.75897        0.84307         REJECT
              90.0%       22.75897        1.28497         REJECT
              95.0%       22.75897        1.65101         REJECT
              99.0%       22.75897        2.34147         REJECT
              99.9%       22.75897        3.12340         REJECT
        
Program 2:
     
    let z = normal rand numb for i = 1 1 50
    let mu0 = 0.3
    set write decimals 5
    .
    t test z mu0
    lower tailed t test z mu0
    upper tailed t test z mu0
        
    The following output is generated.
                One Sample t-Test for the Mean
     
    Response Variable: Z
     
    H0: Mean Equal                                  0.30000
    Ha: Mean Not Equal                              0.30000
     
    Summary Statistics:
    Number of Observations:                              50
    Sample Mean:                                   -0.00822
    Sample Standard Deviation:                      0.71196
    Sample Standard Deviation of the Mean:          0.10068
     
    Test:
    Mean - Mu0:                                    -0.30822
    t-Test Statistic Value:                        -3.06121
    Degrees of Freedom:                                  49
    CDF Value:                                      0.00178
    P-Value (2-tailed test):                        0.00357
    P-Value (lower-tailed test):                    0.00178
    P-Value (upper-tailed test):                    0.99821
     
     
                Two-Tailed Test
     
    H0: u = m0; Ha: u <> m0
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic    Value (+/-)     Conclusion
    ------------------------------------------------------------
              50.0%       -3.06121        0.67952         REJECT
              80.0%       -3.06121        1.29906         REJECT
              90.0%       -3.06121        1.67655         REJECT
              95.0%       -3.06121        2.00957         REJECT
              99.0%       -3.06121        2.67995         REJECT
              99.9%       -3.06121        3.50044         ACCEPT
     
     
                Lower One-Tailed Test
     
    H0: u = m0; Ha: u < m0
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (<)     Conclusion
    ------------------------------------------------------------
              50.0%       -3.06121        0.00000         REJECT
              80.0%       -3.06121       -0.84901         REJECT
              90.0%       -3.06121       -1.29906         REJECT
              95.0%       -3.06121       -1.67655         REJECT
              99.0%       -3.06121       -2.40489         REJECT
              99.9%       -3.06121       -3.26507         ACCEPT
     
     
                Upper One-Tailed Test
     
    H0: u = m0; Ha: u > m0
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (>)     Conclusion
    ------------------------------------------------------------
              50.0%       -3.06121        0.00000         ACCEPT
              80.0%       -3.06121        0.84901         ACCEPT
              90.0%       -3.06121        1.29906         ACCEPT
              95.0%       -3.06121        1.67655         ACCEPT
              99.0%       -3.06121        2.40489         ACCEPT
              99.9%       -3.06121        3.26507         ACCEPT
     
     
                One Sample t-Test for the Mean
     
    Response Variable: Z
     
    H0: Mean Equal                                  0.30000
    Ha: Mean Not Equal                              0.30000
     
    Summary Statistics:
    Number of Observations:                              50
    Sample Mean:                                   -0.00822
    Sample Standard Deviation:                      0.71196
    Sample Standard Deviation of the Mean:          0.10068
     
    Test:
    Mean - Mu0:                                    -0.30822
    t-Test Statistic Value:                        -3.06121
    Degrees of Freedom:                                  49
    CDF Value:                                      0.00178
    P-Value (2-tailed test):                        0.00357
    P-Value (lower-tailed test):                    0.00178
    P-Value (upper-tailed test):                    0.99821
     
     
                Lower One-Tailed Test
     
    H0: u = m0; Ha: u < m0
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (<)     Conclusion
    ------------------------------------------------------------
              50.0%       -3.06121        0.00000         REJECT
              80.0%       -3.06121       -0.84901         REJECT
              90.0%       -3.06121       -1.29906         REJECT
              95.0%       -3.06121       -1.67655         REJECT
              99.0%       -3.06121       -2.40489         REJECT
              99.9%       -3.06121       -3.26507         ACCEPT
     
     
                One Sample t-Test for the Mean
     
    Response Variable: Z
     
    H0: Mean Equal                                  0.30000
    Ha: Mean Not Equal                              0.30000
     
    Summary Statistics:
    Number of Observations:                              50
    Sample Mean:                                   -0.00822
    Sample Standard Deviation:                      0.71196
    Sample Standard Deviation of the Mean:          0.10068
     
    Test:
    Mean - Mu0:                                    -0.30822
    t-Test Statistic Value:                        -3.06121
    Degrees of Freedom:                                  49
    CDF Value:                                      0.00178
    P-Value (2-tailed test):                        0.00357
    P-Value (lower-tailed test):                    0.00178
    P-Value (upper-tailed test):                    0.99821
     
     
                Upper One-Tailed Test
     
    H0: u = m0; Ha: u > m0
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (>)     Conclusion
    ------------------------------------------------------------
              50.0%       -3.06121        0.00000         ACCEPT
              80.0%       -3.06121        0.84901         ACCEPT
              90.0%       -3.06121        1.29906         ACCEPT
              95.0%       -3.06121        1.67655         ACCEPT
              99.0%       -3.06121        2.40489         ACCEPT
              99.9%       -3.06121        3.26507         ACCEPT
        
Program 3:
     
    .  Example of paired t-test from p. 178 of Bowker and Lieberman
    skip 25
    read w1 w2
    73  51
    43  41
    47  43
    53  41
    58  47
    47  32
    52  24
    38  43
    61  53
    56  52
    56  57
    34  44
    55  57
    65  40
    75  68
    end of data
    set write decimals 5
    paired t test w1 w2
        
    The following output is generated.
                Two Sample Paired t-Test for Equal Means
     
    First Response Variable:  W1
    Second Response Variable: W2
     
    H0: Population Means Are Equal
    Ha: Population Means Are Not Equal
     
    Sample One Summary Statistics:
    Number of Observations:                              15
    Sample Mean:                                   54.20000
    Sample Standard Deviation:                     11.57707
     
    Sample Two Summary Statistics:
    Number of Observations:                              15
    Sample Mean:                                   46.20000
    Sample Standard Deviation:                     10.77165
     
    Summary Statistics of Paired Data:
    Number of Observations:                              15
    Sample Mean:                                    8.00000
    Sample Standard Deviation:                     11.02594
    Sample Standard Deviation of the Mean:          2.84688
     
    Test:
    Difference (Delta) in Means:                    8.00000
    t-Test Statistic Value:                         2.81008
    Degrees of Freedom:                                  14
    CDF Value:                                      0.99304
    P-Value (2-tailed test):                        0.01390
    P-Value (lower-tailed test):                    0.99304
    P-Value (upper-tailed test):                    0.00695
     
     
                Two-Tailed Test
     
    H0: u1 = u2; Ha: u1 <> u2
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic    Value (+/-)     Conclusion
    ------------------------------------------------------------
              50.0%        2.81008        0.69241         REJECT
              80.0%        2.81008        1.34503         REJECT
              90.0%        2.81008        1.76130         REJECT
              95.0%        2.81008        2.14478         REJECT
              99.0%        2.81008        2.97682         ACCEPT
              99.9%        2.81008        4.14028         ACCEPT
     
     
                Lower One-Tailed Test
     
    H0: u1 = u2; Ha: u1 < u2
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (<)     Conclusion
    ------------------------------------------------------------
              50.0%        2.81008        0.00000         ACCEPT
              80.0%        2.81008       -0.86805         ACCEPT
              90.0%        2.81008       -1.34503         ACCEPT
              95.0%        2.81008       -1.76130         ACCEPT
              99.0%        2.81008       -2.62448         ACCEPT
              99.9%        2.81008       -3.78729         ACCEPT
     
     
                Upper One-Tailed Test
     
    H0: u1 = u2; Ha: u1 > u2
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (>)     Conclusion
    ------------------------------------------------------------
              50.0%        2.81008        0.00000         REJECT
              80.0%        2.81008        0.86805         REJECT
              90.0%        2.81008        1.34503         REJECT
              95.0%        2.81008        1.76130         REJECT
              99.0%        2.81008        2.62448         REJECT
              99.9%        2.81008        3.78729         ACCEPT
        

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Date created: 09/22/2011
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