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Chapter 10: t-Test for Two Independent Samples - Coggle Diagram
Chapter 10: t-Test for Two Independent Samples
Most research means comparing 2+ sets of sample data. That data can come from either:
Different groups of participants
Repeated Measures Design
Within-Subjects
The same group of participants
Independent Measures Design
Between-Subjects
Identifying which sample you're talking about when working with multiple samples
Subscripting a 1 when referring to a sample designated Sample 1, a 2 for Sample 2, etc, on number of participants (n), means (m) or sums of squares (SS) identifies which sample that information belongs to
t-Testing
Hypotheses
Unlike before, our hypothesis now involves how two samples will compare to each other.
Alternative hypothesis: u1 - u2 != 0, or there is a difference between the population means
Null Hypothesis: u1 - u2 = 0, or there is no difference between the population means
In the case of a one-tailed test, the greater than or less than signs can be present in the alternative, while the less than or equal to or greater than or equal to signs can be present in the null hypothesis, just as with other designs
The actual testing is much the same: After computing your statistic, compare it to the critical region and either reject or do not reject the null hypothesis
Formula: t=sample mean difference minus the population mean difference, divided by the estimated standard error of the sample mean difference
Similar to regular t formula, but incorporates two sample means
Usually the null hypothesis indicates there is no change, so the population mean difference is 0
Estimated Standard Error of the Mean Differences: A value that measures how far, on average, the sample mean difference is from zero.
If the null hypothesis is true, this will also be the average mean difference
The standard error formula for each mean is the square root of variance divided by number of scores
The Estimated Standard Error of the Mean Differences adds the variance divided by number of scores values together,
then
finds the square root of that new value
Finding Variance
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Degrees of freedom for the t statistic are equal to the combined degrees of freedom of the samples
Assumptions
Observations must be independent
Two populations sampled from must be normal
Homogeneity of Variance: Two populations sampled from must have equal variances
Determining Homogeneity of Variance
Hartley's F-Max Test
Formula: Divide the larger variance by the smaller variance.
F-Max values near 1.0 indicate more homogeneity, meaning the assumption is reasonable
High values indicate homogeneity is likely not present
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Measuring Effect Size
Recall Cohen's d and r squared
Cohen's d can be estimated using the estimated mean difference and the estimated standard deviation (which is equal to the square root of the pooled variance)
r squared is calculated exactly the same as before, using the t statistic found using the estimated standard error of the mean and the sample mean difference