independent measures
A research design that uses a separate sample for each treatment condition or each population being compared.
The goal of an independent-measures research study is to evaluate the mean difference between two populations (or between two treatment conditions).
between-subjects
An alternative term for an independent-measures design.
repeated measures
A research design in which the different groups of scores are all obtained from the same group of participants. Also known as repeated-measures design.
within subjects
A research design in which the different groups of scores are all obtained from the same group of participants. Also known as repeated-measures design.
in symbols, the null hypothesis and alternative hypothesis for the independent-measures test is
H0 = mu1 - mu 2 = 0
H1 = mu1 - mu2 does not equal to 0
independent measures t-statistic
In a between-subjects design, a hypothesis test that evaluates the statistical significance of the mean difference between two separate groups of participants.
- The basic structure of the t statistic is the same for both the independent-measures and the single-sample hypothesis tests. In both cases,
- The independent-measures t is basically a two-sample t that doubles all the elements of the single-sample t formulas.
the independent-measures t formula is
(M1 - M2) - (mu1 - mu2 (if null hypothesis, it is 0) / S (M1-M2) (which is the standard error of the mean)
estimated standard error of M1-M2
The estimated standard error (sM) is used as an estimate of the real standard error σ when the value of
σ is unknown. It is computed from the sample variance or sample standard deviation and provides an estimate of the standard distance between a sample mean M and the population mean μ.
there are two ways to interpret the estimated standard error of (M1-M2).
- It measures the standard distance between and μ1 and μ2
- It measures the standard, or average size of (M1-M2) if the null hypothesis is true. That is, it measures how much difference is reasonable to expect between the two sample means.
To develop the formula for s(mu1 - mu2) we consider the following three points
- Each of the two sample means represents it own population mean, but in each case there is some error.
- The amount of error associated with each sample mean is measured by the estimated standard error of M.
- 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
square root of s^2 (first) / n (first) + s^2 (second) / n (second)
pooled variance
A single measure of sample variance that is obtained by averaging two sample variances. It is a weighted mean of the two variances.
When the pooled variance has equal samples, then the pooled variance is exactly half way between the two sample variances
when the pooled variance has unequal samples, then the pooled variance is between the two sample variances
but closer to the variance for the larger sample
pooled variance equation
s^2 (p) = SS1 + SS2 / df1 + df2
the df value for the independent-measures t statistic can be expressed as
df = n1 + n2 - 2
There are three assumptions that should be satisfied before you use the independent-measures t formula for hypothesis testing:
- 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.
homogeneity of variance
An assumption that the two populations from which the samples were obtained have equal variances.
F-Max test procedure
- in order to test the homogeneity of variance
- Compute the sample variance, for each of the separate samples.
- Select the largest and the smallest of these sample variances and compute
F max = s^2 (largest) / s^2 (smallest)
- The F-max value computed for the sample data is compared with the critical value. If the sample value is larger than the table value, you conclude that the variances are different and that the homogeneity assumption is not valid.
confidence interval equation
mu 1 - mu 2 = M1 - M2 +- t*s(M1-M2)
