It is commonly
believed that anyone who tabulates numbers is a statistician.
This is like believing that anyone who owns a scalpel is a surgeon.
Comparing Two Means:
Comparative studies are more convincing than single one-sample investigations, so one-sample inference is NOT as common as comparative (two-sample) inference.
In a comparative study, we may want to compare two treatments, or we may want to compare two populations. In either case, the samples must be chosen randomly and independently in order to perform statistical inference. (These are the assumptions, conditions, for comparing two means.
Logically, because matched pairs are NOT chosen independently, we will NOT use two-sample inference for a matched pairs design. (For a matched pairs design, apply the one-sample t procedures to the observed differences.)
Otherwise, we may use two-sample inference to compare two treatments or two populations.
The null hypothesis is that there is no difference between the two parameters.
The alternative hypothesis could be that
or (one-sided)(left tail)
or (one-sided)(right tail)
Here we go….
Before you begin, check your assumptions! For comparing two means, both samples must be an SRS and must be chosen independently. Also, both populations must be normally distributed. (Check the data for outliers or skewedness.)
IF these assumptions hold, then the difference in sample means is an unbiased estimator of the difference in population means, so is equal to .
Also, the variance of is the sum of the variances of and ,
which is .
If both populations are normally distributed, then is also normally distributed.
In order to standardize , subtract the mean and divide by the standard deviation:
If we do not know and , we will substitute the standard error and for the standard deviation. This gives the standardized t value:
Not to worry...
Since the math is cumbersome…
We will use the TI-83 to perform two-sample T tests.
You should still be familiar with the formulas being used.
2-SampTTest for a hypothesis test
2-SampTInt for a confidence interval