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Chp. 10 - The t Test for Two Independent Samples - Coggle Diagram
Chp. 10 - The t Test for Two Independent Samples
Intro to the Independent-Measures Design
Most research studies require the comparison of 2 or more sets of sample data or 2 or more treatment conditions
The research designs that are used to obtain the two sets of sample data can be classified in two general categories:
The two sets of data could come from two completely separate groups of participants.
For example, the study could involve a sample of young adults from the Millennial generation compared with a sample of Gen X adults. Or the study could compare grades for one group of freshmen who are given laptop computers with the grades for a second group who are not given computers.
The two sets of data could come from the same group of participants.
For example, the researcher could obtain one set of scores by measuring depression for a sample of patients before they begin therapy and then obtain a second set of data by measuring the same individuals after six weeks of therapy. Or a developmental psychologist who is interested in the development of morality in children might measure moral judgment when they are 5 years old, and then again in the same sample of children when they are 10 years old.
The second research strategy, in which the two sets of data are obtained from the same group of participants, is called a
repeated-measures research design or a within-subjects design
.
A research design that uses a separate group of participants for each treatment condition (or for each population) is called an
independent-measures research design
or a
between-subjects design
.
The Hypotheses and the Independent-Measures t Statistic
The Hypotheses 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).
null hypothesis can be stated as
μ1 = μ2
Alternative hypothesis can be stated as H1: μ1 not equal to μ2
The Formulas for an Independent-Measures Hypothesis Test
The basic structure of the t statistic is the same for both the independent-measures and the single-sample hypothesis tests.
The independent-measures t is basically a two-sample t that doubles all the elements of the single-sample t formulas.
The Overall t formula
The independent-measures t uses the difference between two sample means to evaluate a hypothesis about the difference between two population means.
t = (sample mean difference - population mean difference) / estimated standard error of sample mean difference
The Estimated Standard Error
In each of the t-score formulas, the standard error in the denominator measures how much error is expected between the sample statistic and the population parameter.
It measures the standard distance between and μ1 - μ2 .
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
Each of the two sample means represents its own population mean, but in each case there is some 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.
Pooled Variance
For situations in which the two sample sizes are different, the formula is biased and, therefore, inappropriate. The bias comes from the fact that Equation treats the two sample variances equally. However, when the sample sizes are different, the two sample variances are not equally good estimates for error and should not be treated equally.
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.
The pooled variance is obtained by averaging or “pooling” the two sample variances using a procedure that allows the bigger sample to carry more weight in determining the final value.
Estimated 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 formula combines the error for the first sample mean with the error for the second sample mean. Also note that the pooled variance from the two samples is used to compute the standard error for the two samples.
The Final Formula & Degrees of Freedom
t = (sample mean difference - population mean difference) / estimated standard error
Hypothesis Tests with the Independent-Measures t Statistic
The independent-measures t statistic uses the data from two separate samples to help decide whether there is a significant mean difference between two populations or between two treatment conditions.
Directional Hypotheses & One-Tailed Tests
One-tailed tests can lead to rejecting H0 when the mean difference is relatively small compared to the magnitude required by a two-tailed test.
As a result, one-tailed tests should be used when clearly justified by theory or previous findings.
Assumptions Underlying the Independent-Measures t Formula
The observations within each sample must be independent
The two populations from which the samples are selected must be normal.
The two populations from which the samples are selected must have equal variances.
Referred to as
homogeneity of variance
and states that the two populations being compared must have the same variance.
Homogeneity of variance is most important when there is a large discrepancy between the sample sizes.
Hartley's F-Max Test
A more objective procedure involves a statistical test to evaluate the homogeneity assumption.
An additional advantage is that this test can also be used to check homogeneity of variance with more than two independent samples.
Effect Size & Confidence Intervals for the Independent-Measures t
In general, increasing the size of the sample increases the likelihood of rejecting the null hypothesis. As a result, even a very small treatment effect can be significant if the sample is large enough. Therefore, a hypothesis test is usually accompanied by a report of effect size to provide an indication of the absolute magnitude of the treatment effect independent of the size of the sample.
Cohen's Estimated d
One technique for measuring effect size is Cohen’s d, which produces a standardized measure of mean difference.
estimated d = estimated mean difference / estimated standard deviation
Cohen's d is typically reported as a positive value
Explained Variance and r2
The independent-measures t hypothesis test also allows for measuring effect size by computing the percentage of variance accounted for, r squared
Confidence Intervals for Estimating M1 - M2
It is possible to compute a confidence interval as an alternative method for measuring and describing the size of the treatment effect.
In this case, the confidence interval literally estimates the size of the population mean difference between the two populations or treatment conditions.
In particular, the width of the interval depends on the percentage of confidence used so that a larger percentage produces a wider interval. Also, the width of the interval depends on the sample size, so that a larger sample produces a narrower interval. Because the interval width is related to sample size, the confidence interval is not a pure measure of effect size like Cohen’s d or .
Confidence Intervals & Hypothesis Tests
In addition to describing the size of a treatment effect, estimation can be used to get an indication of the significance of the effect.
The Role of Sample Variance & Sample Size in the Independent-Measures t Test
Error and the Role of Individual Differences
The differences that exist among a sample of participants within each treatment group will have an influence on the variance (and standard deviation) of that treatment group. Thus, individual differences are one factor that have an effect on standard error.
Random assignment helps ensure that there is no bias between the groups in terms of individual differences.
It is possible, however, that assignment of participants to groups will be unintentionally biased and that the groups differ at the beginning of the experiment prior to administration of the treatments. Pretesting participants prior to treatment for differences in certain variables (such as attitudes or motivation) is one way to ensure the groups are equivalent at the start.
Although variance and sample size both influence the hypothesis test, only variance has a large influence on measures of effect size such as Cohen’s d and r squared ; larger variance produces smaller measures of effect size. Sample size, on the other hand, has no effect on the value of Cohen’s d and only a small influence on r squared .