6.1.1. t and FTests
All t and FTests can be accessed under this menu item and the results presented in a single page of output.
If you wish to perform a One Sample tTest, you can select only one variable. If you select two or more variables, then for each pair, two separate one sample ttests will be performed on each variable, alongside the two sample tests between them. A paired ttest will be performed only when the two selected variables have the same size. Output Options Dialogue will allow you to choose which tests to appear in the output.
The ttest is used to determine whether the difference between two means is significant. The null hypothesis tested in all four types of ttest is that “the difference between two population means is zero”. When the alternative hypothesis is “the difference is not equal to zero”, the twotailed probability should be compared against the given significance level α (usually 0.05). If the calculated probability is greater than α, then the null hypothesis cannot be rejected. Otherwise, we can conclude that the two means are significantly different. In this case, the confidence interval for the difference will not enclose 0. When the alternative hypothesis is a difference in one direction (i.e. one mean is greater or less than the other), then the onetailed probability is compared against α. UNISTAT reports both one and twotailed probabilities, where the former is the half of the latter.
The data for this test can be in one of the three types supported for Two Sample Tests.
After the variable selection is complete, you will be able to select which tests you wish to have displayed in the output. The output consists of the tvalue, its degrees of freedom, one and twotailed probabilities and the confidence interval for the specified confidence level. When the Report summary statistics box is checked, summary information (number of valid cases, missing observations or pairs, mean and standard deviation) about the selected variables is also displayed.
6.1.1.1. One Sample tTest
If only one variable is selected, the program will perform only a onesample ttest against the given mean. By default, the given mean is 0, testing whether the mean of the sample is different from zero. If two or more variables have been selected, then the program will perform two separate onesample ttests on each pair of variables, using the same given mean specified in the Output Options Dialogue. Missing cases are omitted by case and the degrees of freedom is adjusted accordingly.
The null hypothesis that “the population mean is equal to the given mean” (a scalar) is tested. The population variance is assumed to be unknown. The tstatistic is computed as:
where M is the given mean.
Example
Example 4.2 on p. 101 from Armitage & Berry (2002). The null hypothesis “the population mean is not significantly different from 24” is tested at 95% and 99% levels.
Open PARTESTS, select Statistics 1 → Parametric Tests → t and FTests, select the first column Weight (C1) as [Variable] and click [Next]. Type 24 into the Given Mean box, select all output options (including the Report summary statistics box) and click [Next] to obtain the following results:
t and FTests
For Weight

Valid Cases 
Missing 
Mean 
Standard Deviation 
Difference 
Standard Error 
Mean(Weight) – 24 
20 
0 
21.0000 
5.9116 
3.0000 
1.3219 

tStatistic 
Degrees of Freedom 
1Tail Probability 
2Tail Probability 
Lower 95% 
Upper 95% 
Mean(Weight) – 24 
2.2695 
19.0000 
0.0175 
0.0351 
5.7667 
0.2333 
Since the twotailed probability is less than 5% we reject the null hypothesis and conclude that the population mean is significantly different from 24 at a 95% level. The same result can also be obtained from the reported confidence interval for the difference between means (5.7667 to 0.2333), since it does not include zero.
If, however, the confidence level is increased to 99%, the null hypothesis should not be rejected, as the twotailed probability is greater than 1%. This can also be observed from the confidence interval by repeating the test with a 99% confidence level. Select the test again and edit the Confidence Level box to 0.99 in Variable Selection Dialogue. This time, the confidence interval includes zero.

Lower 99% 
Upper 99% 
Mean(Weight) – 24 
6.7818 
0.7818 
6.1.1.2. Pooled Variance tTest
The null hypothesis “two population means are equal” is tested. It is assumed that the two populations are independent and their standard deviations are the same. The assumption of equal variances can be tested by using the Ftest or Levene’s FTest, which is part of the standard output of this procedure. If the twotailed probability for the Fvalue is greater than the specified α (such as 0.01 or 0.05), then the hypothesis of equal variances is not rejected and the ttest can use the pooledvariance estimate (equal variances). Otherwise the ttest should be based on separate variance estimates (unequal variances).
The tstatistic for equal population variances is calculated as follows:
where:
is the pooled estimate of the population variance.
Missing values are omitted by case and the degrees of freedom is adjusted accordingly.
Example 1
Table 87 on p. 231 from Cohen, L. & M. Holliday (1983). The null hypothesis “empathy scores of social and nonsocial work students have the same mean” is tested at a 95% confidence level.
Open PARTESTS and select Statistics 1 → Parametric Tests → t and FTests. Select the data option 1 and Social and Nonsocial (C2 and C3) as [Variable]s. Enter 0 for the Given Mean and select all output options. The following results are obtained:
t and FTests
For Social and Nonsocial

Valid Cases 
Missing 
Mean 
Standard Deviation 
Mean(Social) – 0 
10 
0 
75.5000 
4.5031 
Mean(Nonsocial) – 0 
10 
0 
63.1000 
5.9712 
Pooled Variance 



5.2884 
Separate Variance 




Paired 
10 
0 

5.2957 

Difference 
Standard Error 
tStatistic 
Degrees of Freedom 
Mean(Social) – 0 
75.5000 
1.4240 
53.0196 
9.0000 
Mean(Nonsocial) – 0 
63.1000 
1.8883 
33.4169 
9.0000 
Pooled Variance 
12.4000 
2.3650 
5.2431 
18.0000 
Separate Variance 
12.4000 
2.3650 
5.2431 
16.7351 
Paired 
12.4000 
1.6746 
7.4045 
9.0000 

1Tail Probability 
2Tail Probability 
Lower 95% 
Upper 95% 
Mean(Social) – 0 
0.0000 
0.0000 
72.2787 
78.7213 
Mean(Nonsocial) – 0 
0.0000 
0.0000 
58.8284 
67.3716 
Pooled Variance 
0.0000 
0.0001 
7.4313 
17.3687 
Separate Variance 
0.0000 
0.0001 
7.4042 
17.3958 
Paired 
0.0000 
0.0000 
8.6117 
16.1883 

Variance 1 
Variance 2 
FStatistic 
d.f. Numerator 
d.f. Denominator 
FTest 
20.2778 
35.6556 
1.7584 
9 
9 
Levene’s F Test 


0.6515 
1 
18 

RightTail Probability 
2Tail Probability 
Lower 95% 
Upper 95% 
FTest 
0.2066 
0.4132 
0.4368 
7.0791 
Levene’s F Test 

0.4301 


This result shows that there is a significant difference at the 0.1% level, between the empathy scores of social work students and nonsocial work students.
Example 2
Example on p. 30, Gardner & Altman (2000). Blood pressure level data for diabetics and non diabetics are not available but all necessary parameters to perform a ttest are given.
Size of Group 1 
100 
Size of Group 2 
100 
Mean 1 
146.4 
Mean 2 
140.4 
Standard Deviation 1 
18.5 
Standard Deviation 2 
16.8 
Select Statistics 1 → Parametric Tests → t and FTests, select the data option 3 Test Statistics are Given and enter the above data. Leave the default value of the confidence level unchanged at 0.95. Check all output options except the Report summary statistics box. The following results are obtained:
t and FTests
Test Statistics are Given

Difference 
Standard Error 
tStatistic 
Degrees of Freedom 
Mean(Group 1) – 0 
146.4000 
1.8500 
79.1351 
99.0000 
Mean(Group 2) – 0 
140.4000 
1.6800 
83.5714 
99.0000 
Pooled Variance 
6.0000 
2.4990 
2.4010 
198.0000 
Separate Variance 
6.0000 
2.4990 
2.4010 
196.1884 

1Tail Probability 
2Tail Probability 
Lower 95% 
Upper 95% 
Mean(Group 1) – 0 
0.0000 
0.0000 
142.7292 
150.0708 
Mean(Group 2) – 0 
0.0000 
0.0000 
137.0665 
143.7335 
Pooled Variance 
0.0086 
0.0173 
1.0720 
10.9280 
Separate Variance 
0.0086 
0.0173 
1.0717 
10.9283 

FStatistic 
d.f. Numerator 
d.f. Denominator 
FTest 
1.2126 
99 
99 

RightTail Probability 
2Tail Probability 
Lower 95% 
Upper 95% 
FTest 
0.1696 
0.3391 
0.8159 
1.8022 
Note that paired ttest and Levene’s FTest cannot be computed when the data option Test Statistics are Given is selected.
Next, click on the [Last Procedure Dialogue] button to redisplay the Variable Selection Dialogue. Edit the value of the confidence level to 0.99 and click [Finish]. All results will be as above except for the confidence intervals. The interval for pooled variance ttest will be:

Lower 99% 
Upper 99% 
Pooled Variance 
0.4996 
12.4996 
And finally, edit the confidence level to 0.9 and repeat the procedure to obtain:

Lower 90% 
Upper 90% 
Pooled Variance 
1.8702 
10.1298 
6.1.1.3. Separate Variance tTest
The null hypothesis “the means of two populations are equal” is tested. It is assumed that their standard deviations may be different. The resulting tstatistic is based on a number of degrees of freedom which is reduced by a factor depending on the extent of the differences in variances.
where:
and where:
The reported degrees of freedom (Satterthwaite’s approximation) may not be an integer but the nearest integer is used to calculate the tail probabilities.
Missing values are omitted by case and the degrees of freedom is adjusted accordingly.
Example
Table 89 on p. 233 from Cohen, L. & M. Holliday (1983). The raw data on social perceptiveness scores of nursery school and nonnursery school children are not available, but all parameters necessary to perform a ttest are given.
Size of Group 1 
71 
Size of Group 2 
64 
Mean 1 
19.5 
Mean 2 
15.3 
Standard Deviation 1 
3.4 
Standard Deviation 2 
4.6 
Select Statistics 1 → Parametric Tests → t and FTests. Select the data option 3 Test Statistics are Given and enter the above values. Check only the Separate Variance ttest output option to obtain the following results:
t and FTests
Test Statistics are Given

Difference 
Standard Error 
tStatistic 
Degrees of Freedom 
Separate Variance 
4.2000 
0.7025 
5.9790 
115.1866 

1Tail Probability 
2Tail Probability 
Lower 95% 
Upper 95% 
Separate Variance 
0.0000 
0.0000 
2.8086 
5.5914 
This result shows that there is a significant difference at the 0.1% level, of the social perceptiveness of nursery school and nonnursery school children.
6.1.1.4. Paired tTest
This test will be available only when the following conditions are met:
Ā· The data option 1 is selected
Ā· At least two variables are selected
Ā· The selected pairs have the same length.
Two or more columns can be selected by clicking on [Variable]. The test will be performed between all possible pairs, as long as the two columns have the same size. For each test, any pair of cases with one or more missing values is omitted and the degrees of freedom adjusted. It is also possible to perform ttests between subgroups of two variables defined by one or more factor columns. In this case, the Run a separate analysis for each option selected box must be unchecked.
A paired ttest is performed between two variables, such as values of a sample before and after a certain treatment. The null hypothesis tested is “the difference between pairs is zero” against the alternative hypothesis that “the difference between pairs is not equal to zero”.
The tstatistic is calculated as follows:
where MD and sD are the mean and standard error of D and:
Example 1
Example 8.3.4 on pp. 45454, Larson, H. J. (1982). The null hypothesis “reaction times before consumption of beverage x and after consumption y on individuals are equal” is tested.
Open PARTESTS and select Statistics 1 → Parametric Tests → t and FTests. Select x and y (C4 and C5) as [Variable]s and check only the Onesample ttest and Paired ttest boxes to obtain the following results:
t and FTests
For x and y

Difference 
Standard Error 
tStatistic 
Degrees of Freedom 
Mean(x) – 0 
602.4000 
29.3342 
20.5358 
9.0000 
Mean(y) – 0 
803.7000 
19.6413 
40.9190 
9.0000 
Paired 
201.3000 
15.1056 
13.3262 
9.0000 

1Tail Probability 
2Tail Probability 
Lower 95% 
Upper 95% 
Mean(x) – 0 
0.0000 
0.0000 
536.0415 
668.7585 
Mean(y) – 0 
0.0000 
0.0000 
759.2684 
848.1316 
Paired 
0.0000 
0.0000 
235.4712 
167.1288 
This result shows that there is a significant difference at the 5% level, of the reaction time of individuals before and after consumption of beverage.
Example 2
Example on p. 31, Gardner & Altman (2000). Data on testing the difference between the systolic blood pressure levels for 16 middle aged men before and after a standard exercise are given. The difference between the two columns should be in the order of After – Before.
Open PARTESTS and select Statistics 1 → Parametric Tests → t and FTests and select Before and After (C6 and C7) as [Variable]s and check all output options to obtain the following results:
t and FTests
For After and Before

Valid Cases 
Missing 
Mean 
Standard Deviation 
Mean(After) – 0 
16 
0 
147.7500 
12.3477 
Mean(Before) – 0 
16 
0 
141.1250 
13.6229 
Pooled Variance 



13.0010 
Separate Variance 




Paired 
16 
0 

5.9652 

Difference 
Standard Error 
tStatistic 
Degrees of Freedom 
Mean(After) – 0 
147.7500 
3.0869 
47.8630 
15.0000 
Mean(Before) – 0 
141.1250 
3.4057 
41.4376 
15.0000 
Pooled Variance 
6.6250 
4.5965 
1.4413 
30.0000 
Separate Variance 
6.6250 
4.5965 
1.4413 
29.7148 
Paired 
6.6250 
1.4913 
4.4425 
15.0000 

1Tail Probability 
2Tail Probability 
Lower 95% 
Upper 95% 
Mean(After) – 0 
0.0000 
0.0000 
141.1704 
154.3296 
Mean(Before) – 0 
0.0000 
0.0000 
133.8659 
148.3841 
Pooled Variance 
0.0799 
0.1599 
2.7624 
16.0124 
Separate Variance 
0.0800 
0.1600 
2.7662 
16.0162 
Paired 
0.0002 
0.0005 
3.4464 
9.8036 

Variance 1 
Variance 2 
FStatistic 
d.f. Numerator 
d.f. Denominator 
FTest 
152.4667 
185.5833 
1.2172 
15 
15 
Levene’s F Test 


0.1228 
1 
30 

RightTail Probability 
2Tail Probability 
Lower 95% 
Upper 95% 
FTest 
0.3542 
0.7084 
0.4253 
3.4838 
Levene’s F Test 

0.7284 


6.1.1.5. FTest
The FTest is used to compare variances or standard deviations of two samples. The null hypothesis tested is “the two populations have equal variances”. Columns selected for this test need not be equal in size. Output displays the Fvalue, degrees of freedom, the right and twotailed probabilities from the Fdistribution and the confidence interval for the specified confidence level. When the alternative hypothesis is “the two population variances are not equal”, use the twotailed probability. When the Fvalue is calculated as follows:
If then the Fvalue is inverted and the two degrees of freedom are interchanged. In other words, the Fvalue is always the larger variance divided by the smaller variance.
Example 1
Example 5.1 on p. 151 from Armitage & Berry (2002). The null hypothesis “the two population variances are not significantly different” is tested at 95% level. The raw data are not available, but it is sufficient to know the number of cases in each group and their variances to perform an Ftest .
Size of Group 1 
10 
Size of Group 2 
20 
Variance 1 
1.232 
Variance 2 
0.304 
Select Statistics 1 → Parametric Tests → t and FTests, the data option 3 Test Statistics are Given. As this dialogue asks for standard deviations rather than variances, enter Sqr(1.232) and Sqr(0.304) for the two standard deviations. The mean values are not used for Ftest. In the Output Options Dialogue, check only the Ftest and Levene’s FTest boxes. The following results are obtained:
t and FTests
Test Statistics are Given

FStatistic 
d.f. Numerator 
d.f. Denominator 
FTest 
4.0526 
9 
19 

RightTail Probability 
2Tail Probability 
Lower 95% 
Upper 95% 
FTest 
0.0049 
0.0099 
1.4071 
14.9272 
Since the null hypothesis suggests a twotailed test (equal vs. not equal) then the twotailed probability should be compared with α. This result shows there is no significant difference between the two population variances at 5% level for the twotailed test.
When the data option 3 Test Statistics are Given is selected the Levene’s test cannot be computed.
Example 2
Example 8.3.2 on pp. 45051, Larson, H. J. (1982). The null hypothesis “the population variances for hours of services given by 60 watt light bulbs of brand G and brand W are the same” is tested.
Open PARTESTS and select Statistics 1 → Parametric Tests → t and FTests. Select Brand G and Brand W (C8 and C9) as [Variable]s and check only the Ftest, Levene’s FTest and Report summary statistics boxes to obtain the following results:
t and FTests
For Brand G and Brand W

Variance 1 
Variance 2 
FStatistic 
d.f. Numerator 
d.f. Denominator 
FTest 
2222.2143 
653.8778 
3.3985 
7 
9 
Levene’s F Test 


1.4112 
1 
16 

RightTail Probability 
2Tail Probability 
Lower 95% 
Upper 95% 
FTest 
0.0459 
0.0917 
0.8097 
16.3918 
Levene’s F Test 

0.2522 


Since the null hypothesis suggests a twotailed test (equal vs. not equal) then we should look at the 2tail probability. This result shows that there is no significant difference between the two population variances at 5% level.
6.1.1.6. Levene’s FTest
Levene’s FTest has the advantage of being less sensitive to deviations from normality and is considered to be more powerful than the classical Ftest. The alternative hypothesis for Levene’s test is “the two population variances are not equal” and the probability reported is comparable to the twotailed probability for the Ftest. The test statistic, which has an Fdistribution, is computed as follows:
where:
Missing values are omitted by case. If the data option 3 Test Statistics are Given is selected then the Levene’s test will not be available.