Assume John is conducting a one-way ANOVA with three groups. Naturally he specif

Question

Assume John is conducting a one-way ANOVA with three groups. Naturally he specifies the null hypothesis as Mean1=Mean2=Mean3. After learned about the t-test, he claims that the ANOVA can also be accomplished by three separate t-tests on the three pairs of means. In other words, the null hypotheses in the three t-tests would be: Mean1=Mean2; Mean2=Mean3; Mean1=Mean3. He further argues that the t-tests are prefered since they help to identify which mean(s) is(are) significantly different from other(s) while the one-way ANOVA could not without further tests. Do you agree with John? Why or why not?

Explanation / Answer

Yes, one should agree with John.

T-test is a hypothesis test that is used to compare the means of two populations.

In t-test null hypothesis takes the form of H0: µ(x) = µ(y) against alternative hypothesis H1: µ(x) µ(y), wherein µ(x) and µ(y) represents the population means. The degree of freedom of t-test is n1 + n2 – 2

ANOVA is a statistical technique that is used to compare the means of more than two populations.

Null hypothesis (H0) where in all population means are the same, or alternative hypothesis (H1) where in at least one population mean is different.

Like in the example,
From the above three t tests, one may find that Mean 2 is significantly different from Mean 1 and Mean 3, where from Anova it is not possible to check to identify which mean(s) is(are) significantly different from other.

But,

It can be said that t-test is a special type of ANOVA that can be used when we have only two populations to compare their means. Although the chances of errors might increase if t-test is used when we have to compare more than two means of the populations concurrently, that is why ANOVA is used.

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