Analyzing Data From A Simple Experiment
I. Describing results:
A. Average score for each group, if have ratio or interval data, average is usually the mean
B. Measure that tells you how well the mean characterizes the group. This measure of spread (variability) is typically the standard deviation
C. If the sample distributions are normally distributed, you can describe those distributions very accurately with just the mean and the standard deviation.
II. Problem: If you subtract the group means from one another, you will probably find a difference. However, you won't
know whether this difference is due to chance (random
error) or whether the difference is due, at least in part, to the treatment. (For example, if we'd started in a
different place on the random numbers table, we'd have
drawn two different samples and may have obtained
different pattern of results). In other words, the difference between the
sample means is only an estimate of what we really want
If everyone had been in the experimental group, would their scores be different than if everyone had been in control group?
III. Factors that determine whether we can believe that difference between the two sample means reflects that there really is a difference between the two population means
A. Size of the difference between the means
B. How closely the difference between sample means reflects a difference between population means. That, in turn, depends on how well each sample mean reflects its population mean (control group mean to control group's population mean, experimental group mean to experimental group population mean)
IV. Factors that determine how closely a sample mean will reflect population mean
A. Variability in population (size of sd): If all the population's scores are close to the mean, it's hard to get a sample that would have a mean far away from the population mean.
B. Sample size: The larger the independent random sample, the more accurate it will tend to be.
Note: Low variability in the population and a large sample size will result in a small standard error of the mean.C. If you were to draw numerous independent random samples from a population and plot the mean for each sample, you'd end up with a normal curve.
The middle of this curve would be the population mean,
and 2/3 of the sample means would be within 1 standard error of the population mean,
95% within 2 standard errors,
and 99% within 3 standard errors.
V. The degree to which the difference between sample means reflects random error is a function of:
A. Failure of the control group mean to accurately reflect its population mean PLUS
B. Failure of the experimental group mean to accurately reflect its population mean
C. Hence, one formula for the standard error of the difference (an index of the extent to which the difference between sample means may be due to random error) is:
square root of [ (s1) squared/N1 + (s2) squared/N2]
Note: Although this formula helps you understand the general logic of the t test, this formula does not hold under all conditions. So, if you are actually computing t, use a different formula, such as formula on page 521 of your text.
D. If you were to draw numerous independent random samples from two identical populations, compute the sample means, subtract sample mean 1 from sample mean 2, and plot this difference for each pair of samples, you'd end up with a normal curve.
The middle of this curve would be zero
and 2/3 of the sample means would be within 1 standard error of the difference,
95% within 2 standard errors of the difference,
and 99% within 3 standard errors of the difference.
VI. To go back to our original question (Is the difference between our groups due to more than random error?) is
to subtract Mean 1 from Mean 2 and then divide by
the standard error of the difference.
This is our t ratio.
A. Calculate df (N-2).
B. Look in the t table under .05 level of significance.
C. If the absolute value of the calculated "t "is greater than tabled value, the result is statistically significant.
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