●
The sampling distribution of the difference in
sample
means is
normal
.
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Hypothesis Testing
●
We often use the null hypothesis that there is no difference in population means.
●
There are three possible alternative hypotheses to this:
○
Population mean 1 is greater than Population mean 2
○
Population mean 1 is smaller than Population mean 2
○
The Population means are not equal.
●
What we chose depends on what we are trying to test.
○
E.g. if we want to test that a new drug is
better
, we may use a one-
sided hypothesis.
If the standard deviation of the populations are known, use a Z test statistic.
●
If the null hypothesis is true, then the statistic will have the standard
normal distribution.
If the standard deviation of the populations are
not
known, we have two options.
Pooled-variance two-sample t procedure
●
Assumptions
○
Assumes that the population standard deviations are equal to each other.
■
Ratio of differences is what is important.
○
Assumes normally distributed populations.
○
Assumes independent random samples.
○
Advantage
: Test statistic is exactly a t-distribution
●
Pooled sample variance
○
Combine sample variances together to get the best estimation of
the common population variance.
●
Standard error of the difference in sample means
○
The square root of its estimated variance.
○
Estimated standard deviation of the sampling distribution of the difference
in the
sample
means.
●
p-values
○
Alternative hypothesis: population mean 1 is greater than
population mean 2
■
p-value = area to the right of the test statistic
○
Alternative hypothesis: population mean 2 is greater than
population mean 1
■
p-value = are to the left of the test statistic
○
Alternative hypothesis: population means are not equal
■
p-value = Double the area to the left or right (whichever is small)
●
Degrees of Freedom = the degrees of freedom for the variance estimator
○
Lose two degrees of freedom when we use the two sample means
to estimate the two population means.
○
n1 + n2 - 2
●
When population sizes differ, it is a problem for this procedure.
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○
Worst when the small population has the large sample variance.
Welch procedure
●
Assumes normally distributed populations.
●
Used when population variances are not equal.
●
The test statistic is similar to that of the pooled-variance t test.
○
The only difference is the form of the standard error in the denominator
○
Standard error of the difference in sample means estimates the true
standard deviation of the sampling distribution of the difference
between the two sample means.
●
Used in the same kind of situations as pooled-variance t test.