Download Hypothesis Test on mu X

Stat211/Labpack/Hypothesis Test on mx - my
Hypothesis Test for mX - mY:
Goals of the lab:
To give students an understanding of hypothesis tests for the difference in means of two independent
populations.
Background:
In Statistics 211 we are often interested in testing whether or not the means of two different populations are
equal. We can classify the types of populations that we observe into two distinct settings - paired
populations and independent populations. For the case of two paired populations, there is some "link"
between the two populations that "pairs" observations from one population with those from the other
population. One example of this is a "before and after" experiment, where data is collected from subjects
before the experiment is performed, and again after the experiment is performed. Consequently for each
measurement from the first population (the "before" population) there is a corresponding measurement
from the second population (the "after" population). Another way to think of this situation is that we are
selecting a sample from a single population and making "pairs" of measurements on each subject in the
sample.
The second case of the "two populations" setting is "independent" populations setting. In this situation there
are actually two populations that are distinct and separate. Thus we obtain two samples - one sample from
each of the two distinct populations. The observations from the two samples are not "paired".
For both of these types of experiments we are interested in whether or not the difference in the population
means is zero. A value of zero for the difference indicates that the two populations have the same mean.
For the "independent populations" scenario, we must compute a sample mean and sample standard
deviation for each of our samples. The computation of the test statistic will use the sample statistics
computed from each sample. Otherwise the structure of the hypothesis test for mX - mY looks very similar
to the hypothesis test for m, the mean of a single population.
Relevant Formulas:
H 0 : m X - mY = 0
HA: mX - mY ≠ 0 or HA: mX - mY > 0
or HA: mX - mY < 0
Significance level: a
Test Statistic:
1) Small Samples Case (n < 20 and m < 20)
x-y
t=
1 1
sp
+
n m
where
sp =
(n - 1)sX 2 + (m - 1)sY 2
n+ m-2
Critical Value (table value): t(n+m-2, 1-a/2) for two-sided alternate hypothesis HA: mX - mY ≠ 0
t(n+m-2, 1-a) for one-sided alternate hypothesis HA: mX - mY > 0
-t(n+m-2, 1-a) for one-sided alternate hypothesis HA: mX - mY < 0
Assumptions:
The observations are independent.
The data in each sample are (approximately) normally distributed.
The population standard deviations are equal, sx = sy .
Stat211/Labpack/Hypothesis Test on mx - my
2) Large Samples Case (n ≥ 20 and m ≥ 20)
Z=
x -y
sx 2 sy 2
+
n
m
Critical Value (table value): z(1-a/2) for two-sided alternate hypothesis HA: mX - mY ≠ 0
z(1-a) for one-sided alternate hypothesis HA: mX - mY > 0
-z(1-a) for one-sided alternate hypothesis HA: mX - mY < 0
Assumptions:
The observations are independent.
Decision Rule: There are two methods that can be used to make the decision about whether to reject the
null hypothesis. One is called the "Classical" method and the other is called the "p-value" method. The
classic approach was developed before the advent of computers and makes use of t-tables. The "p-value"
method was developed in order to eliminate the need for t-tables when using statistical computing software
packages, and is the method used in the IDEAL system.
Classical Method:
For a two-sided hypothesis test (i.e., HA: mX - mY ≠ 0):
We reject the null hypothesis if | Test Statistic | ≥ Critical Value.
We fail to reject the null hypothesis if | Test Statistic | < Critical Value.
For a one-sided hypothesis test (i.e., HA: mX - mY > 0):
We reject the null hypothesis if Test Statistic ≥ Critical Value.
We fail to reject the null hypothesis if Test Statistic < Critical Value.
For a one-sided hypothesis test of (i.e., HA: mX - mY < 0):
We reject the null hypothesis if Test Statistic ≤ Critical Value.
We fail to reject the null hypothesis if Test Statistic > Critical Value.
p-value Method:
Most statistical computer packages will print three different p-values. The user must determine which of
these three p-values is appropriate for the test of hypothesis being performed:
For a two-sided hypothesis test (i.e., HA: mX - mY ≠ 0), the p-value is 2*P( T > | test stat | ).
For a one-sided hypothesis test (i.e., HA: mX - mY > 0), the p-value is P( T > test stat ).
For a one-sided hypothesis test (i.e., HA: mX - mY < 0), the p-value is P( T < test stat ).
Once the appropriate p-value has been determined, we make our decision by comparing the pvalue of the test statistic to the significance a:
We reject the null hypothesis if the (appropriate) p-value ≤ a.
We fail to reject the null hypothesis if the (appropriate) p-value > a.
For examples, go to http://www.stat.wvu.edu/~abilling/stat211notes.html
And download the lecture notes from Sec. 9.5.