The Hypothesis Test for an Independent Measures Test
The goal of an independent-measures research study is to evaluate the mean difference between two populations (or between two treatment conditions).
As always, the null hypothesis states that there is no change, no effect, or, in this case, no difference. Thus, in symbols, the null hypothesis for the independent-measures test is
Ho: mu1 - mu2 = 0
The alternative hypothesis states that there is a mean difference between the two populations,
H1: mu1 - mu2 does not = 0
Formulas for an Independent Measures Hypothesis Test
1. The basic structure of the t statistic is the same for both the independent-measures and the single-sample hypothesis tests. In both cases,
t = actual difference btween sample data and the hypothesis / expected difference between sample data and hypothesis with no treatment effect
2. The independent-measures t is basically a two-sample t that doubles all the elements of the single-sample t formulas.
t = (M1 - M2) - (mu1 - mu2) /
s(M1-M2)
In a hypothesis test, the null hypothesis sets the population mean difference equal to zero, so the independent measures t formula can be simplified further,
t - sample mean difference / estimated standard error
The overall structure of the t statistic can be reduced to the following:
t = (data - hypothesis)/error
Estimated Standard ErrorWhen the null hypothesis is true, however, the population mean difference is zero. In this case, the standard error is measuring how far, on average, the sample mean difference is from zero. However, measuring how far it is from zero is the same as measuring how big it is. Thus, there are two ways to interpret the estimated standard error of (M1 - M2)
- It measures the standard distance between (M1 - M2) and (mu1 - mu2)
- When the null hypothesis is true, it measures the standard, or average size of (M1 - M2). That is, it measures how much difference is reasonable to expect between the two sample means.
Calculating the Estimated Standard Error
For the independent-measures t statistic, we want to know the total amount of error involved in using two sample means to approximate two population means. To do this, we will find the error from each sample separately and then add the two errors together. The resulting formula for standard error is
s(M1 - M2) = square root of (s1squared/n1) + (s2squared/n2)
Estmated Standard Error
Using the pooled variance in place of the individual sample variances, we can now obtain an unbiased measure of the standard error for a sample mean difference. The resulting formula for the independent-measures estimated standard error is
s(M1 - M2) = square root of (spsquared/n1) +(spsquared/n2)
Pooled Variance
When the sample sizes are different, the two sample variances are not equally good estimates for error and should not be treated equally.
The variance obtained from a large sample is a more accurate estimate of sigma squared than the variance obtained from a small sample.
One method for correcting the bias in the standard error is to combine the two sample variances into a single value called the pooled variance.
pooled variance = ssquaredp = (SS1 + SS2)/(df1 + df2)
When computing the pooled variance, the weight for each of the individual sample variances is determined by its degrees of freedom. Because the larger sample has a larger df value, it carries more weight when averaging the two variances. This produces an alternative formula for computing pooled variance:
ssquaredp = (df1s1squared + df2s2squared)/(df1 + df2)
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