T TEST

T TEST

Analysis Command

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.

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 dedependent (i.e., paired) 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

<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.

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.

<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.

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.

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

Sample sizes do not need to be equal unless paired t tests are specified.

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.

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

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.

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.

TTEST is a synonym for T TEST.
1 SAMPLE is a synonym for ONE SAMPLE.

CONFIDENCE LIMITS	= Compute the confidence limits for the mean of a sample.
BIHISTOGRAM	= Generates a bihistogram.
QUANTILE QUANTILE PLOT	= Generates a quantile qauntile plot.
BOX PLOT	= Generates a box plot.

T tests are discussed in most introductory statistics books.

Confirmatory Data Analysis

Pre-1987
2011/3: Added support for paired t test
2011/3: Reformatted output for greater clarity
2011/3: Support for one-tailed tests

 
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
    

 
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
    

 
.  Example of paired t-test from p. 178 of Bowker and Lieberman
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