How do we go about establishing whether the differences in the two means are statistically significant or due to chance? You begin by formulating a hypothesis about the difference. This hypothesis states that the two means are equal or the difference between the two means is zero and is called the null hypothesis.

Using the null hypothesis, you begin testing the significance by saying:
  • "There is no difference in the score obtained in science between subjects in the Demonstration group and the Lecture group" 
  • More commonly the null hypothesis may be stated as follows:

                  a)    Ho :  U=  U2                which translates into         43.0 = 38.0
                 b)    Ho :   U- U2   =  0     which translate into          43.0 -  38.0 = 0

  • If you reject the null hypothesis, it means that the difference between the two means have statistical significance
  • If you do not reject the null hypothesis, it means that the difference between the two means are NOT statistically significant and the difference is due to chance.

Note: For a null hypothesis to be accepted, the difference between the two means need not be equal to zero since sampling may account for the departure from zero. Thus, you can accept the null hypothesis even if the difference between the two means is not zero provided the difference is likely to be due to chance. However, if the difference between the two means appears too large to have been brought about by chance, you reject the null hypothesis and conclude that a real difference exists.
At the end of this Module, you should be able to:
  • explain what is the t-Test and its use in hypothesis testing
  • demonstrate using the t-Test for  INDEPENDENT MEANS
  • identify the assumptions for using the t-test
  • demonstrate the use of the t-Test for DEPENDENT MEANS

  • State TWO null hypothesis in your area of interest that can be tested using the t-test.
  • What do you mean when you reject or do not reject the null hypothesis?

The t-test was developed by a statistician, W.S. Gossett (1878-1937) who worked in a brewery in Dublin, Ireland. His pen name was ‘student’ and hence the term ‘student’s t-test’ which was published in the scientific journal, Biometrika in 1908. The t-test is a statistical tool used to infer differences between small samples based on the mean and standard deviation.
In many educational studies, the researcher is interested in testing the differences between means on some variable. The researcher is keen to determine whether the differences observed between two samples represents a real difference between the populations from which the samples were drawn. In other words, did the observed difference just happen by chance when, in reality, the two populations do not differ at all on the variable studies.
For example, a teacher wants to find out whether the Demontration method of teaching science to primary school children was more effective than the Lecture method. She conducts an experiment among 70 primary school children of which 35 pupils were taught using the Demonstration method and 35 children were taught using the Lecture method.
The results of the study showed that subjects in the Demonstration group score 43.0 marks while subjects in the Lecture method group score 38.0 marks on a the science test.
Yes, the Demonstration group did better than the Lecture group. Does the difference between the two groups represent a real difference or was it due to chance? To answer this question, the t-test is often used by researchers.
copyright 2008
W.S. Gossett
Watch this 4 min video clip by Dr. A.G. Picciano explaining the t-test.
Do not worry if you do not understand as you will be learning all the concepts mentioned in this chapter.
Jot down some of the terms and concepts mentioned and it will clearer as you read the rest of the chapter.
Science in primary school
5. Homogeneity of Variance.
It has often been suggested by some researchers that homogeneity of variance or equality of variance is actually more important than the assumption of normality. In other words, are the standard deviations of the two groups pretty close to equal? Most statistical software packages provide a "test of equality of variances" along with the results of the t-test and the most common being Levene's test of homogeneity of variance.
Begin by putting forward the null hypothesis that:
   "There are no significant differences between the variances of the two groups" and you set the significant level at .05.

If the Levene statistic is significant, i.e LESS than .05 level (p < .05), then the null hypothesis is
  • REJECTED and one accepts the alternative hypothesis and conclude that the VARIANCES ARE UNEQUAL. [The unequal variances in the SPSS output is used]
  • If the Levene statistic is not significant, i.e. MORE than .05 level (p > .05), then you DO NOT REJECT (or Accept) the null hypothesis and conclude that the VARIANCES ARE EQUAL. [The equal variances in the SPSS output is used]    

The Levene test is robust in the face of departures from normality. The Levene's test is based on deviations from the group mean.
  • SPSS provides two options'; i.e. "homogenity of variance assumed" and "homogeneity of varance not assumed" (see Table below).
  • The Levene test is more robust in the face of non-normality than more traditional tests like Bartlett's test.
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   F                     Sig.           t             df          Sig. (2-tailed)    Mean          Std. Error     95% Confidence Interval
                                                                                                  Difference  Difference     Lower          Upper  
Levene's Test of Equality of Variances
  3.39               .080          .848        20           .047                  1.00               1.18            -1.46           3.46
                                        .848     16.704       .049                  1.00               1.18            -1.49           3.49 
Equal variances assumed
Equal variances not assumed
Independent Samples Test

While the t-test has been described as a a robust statistical tool, it is based on a model that makes several assumptions about the data that must be met prior to analysis. Unfortunately, students conducting research tend not to report whether their data meet the assumptions of the t-test. These assumptions need ro be evaluated, because the accuracy of your interpretation of the data depends on wether assumptions are violated. The following are three main assumptions that are generic to all t-tests.

1. Instrumentation (Scale of Measurement)
The data that you collect for the dependent variable should be based on an instrument or scale that is continuous or ordinal. For example, scores that you obtain from a 5-point Likert scale; 1,2,3,4,5 or marks obtained in a mathematics test, the score obtained on an IQ test or the score obtained on an aptitude test.

2. Random Sampling
The sample of subjects should be randomly sampled from the population of interest.

3. Normality
The data come from a distribution that has one of those nice bell-shaped curves known as a normal distribution. Refer to Chapter 3: The Normal Distribution which provides both graphical and statistical methods for assessing normality of a sample or samples.

4. Sample Size
Fortunately, it has been shown that if the sample size is reasonably large, quite severe   epartures from normality do not seem to affect the conclusions reached. Then again what is a reasonable sample size? It has been argued that as long as you have enough people in each group (typically greater or equal to 30 cases) and the groups are close to equal in size, you can be confident that the t-test will be a good, strong tool for getting the correct conclusions. Statisticians say that the t-test is a "robust" test. Departure from normality is most serious when sample sizes are small. As sample sizes increase, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the original population.
a) Explain how the Levene's Test is used to establish homogeneity  
   of variance for a distribution.
b) What will you do if the Levene'sTest reported a significant level of      0.045?
c) What do you think is an appropriate sample? Explain.