Download The t-Test for the Difference Between Means

Section 8.3:
Testing the Difference Between Means (Dependent Samples)
Today we will study
• How to decide whether two samples are independent or dependent
• How to perform a two-sample t-test to test the mean of the differences for a population of paired data
The t-Test for the Difference Between Means
Paraphrased from our text - section 8.3
DEFINITION
Two samples are independent if the sample values selected from one population are not related to or somehow
paired with the sample values selected from the other population.
Two samples are dependent if each member of one sample corresponds to a member of the other sample.
Dependent samples are also called paired samples or matched pairs.
Three conditions must be satisfied to perform this t-test.
• The two samples must be dependent (matched pairs).
• Both populations must have normal distributions.
( if n ≥ 30 we can drop the normal distribution requirement)
• The samples must be randomly selected.
Remarks
In prior sections we typically used the random variable x, in this section we will work with the variable d, defined to be
the difference between entries for a data pair, d = x1 − x2 .
¯
Therefore our test statistic will be the mean of this variable, d.
Symbol
Description
n
number of pairs of data
d
d¯
individual difference between the two values in a single matched pair, d = x1 − x2
sd
the standard deviation of the differences d for the paired data entries in the dependent samples
µd
the hypothesized mean of the differences of paired data in the population
the mean of the differences d for the paired data entries in the dependent samples
t-Test for the Difference Between Means
A t-test can be used to test the difference of two population means when a sample is randomly selected from
each population. The requirements for performing the test are that each population must be normal and each
¯ and
member of the first sample must be paired with a member of the second sample. The test statistic is d,
the standardized test statistic is
d¯ − µd
t = sd .
√
n
The degrees of freedom are
d.f. = n − 1.
GUIDELINES
Using the t-Test for the Difference Between Means
(Dependent Samples)
1. Write the null hypothesis H0 and alternative hypothesis Ha ; then identify the claim.
2. Specify the level of significance α.
3. Identify the degrees of freedom and sketch the sampling distribution, add the test statistic, critical value(s)
and rejection region(s).
4. Determine the critical value(s), t0 .
5. Determine the rejection region(s).
6. Calculate the test statistic d¯ and std dev sd .
7. Calculate the standardized test statistic t.
8. Make a decision to reject H0 or fail to reject H0 .
(a) If t is in the rejection region - reject H0 .
(b) If t is not in the rejection region - fail to reject H0 .
9. Interpret the decision in the context of the original claim.
z < z0
Fail to reject H 0
Fail to reject H 0
Reject H 0
z
z0
0
Left−Tailed Test
Fail to reject H 0
Reject H 0
z0
0
Right−Tailed Test
z > z0
Reject H 0
z
z < z0
Reject H 0
z0
z0 z > z0
0
Two−Tailed Test
Table from our text on page 353
α
Tail
0.10
Left
Right
Two
-1.28
1.28
±1.645
0.05
Left
Right
Two
-1.645
1.645
±1.96
0.01
Left
Right
Two
-2.33
2.33
±2.575
z0
Claim
Decision
Claim is H0 .
Claim is Ha
Reject H0 .
There is enough evidence to reject the claim
There is enough evidence to
support the claim
Fail to Reject H0 .
There is Not enough evidence
to reject the claim
There is Not enough evidence
to support the claim
STATISTICS(REG) - M109
z