Purpose:
Test if two population means are equal
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The two-sample t-test
(Snedecor and
Cochran, 1989) is used to determine if two population means
are equal. A common application is to test if a
new process or treatment is superior to a current process
or treatment.
There are several variations on this test.
- The data may either be paired or not paired. By
paired, we mean that there is a one-to-one
correspondence between the values in the two samples.
That is, if X1, X2,
..., Xn and Y1,
Y2, ... , Yn are
the two samples, then Xi corresponds to
Yi. For paired samples, the difference
Xi - Yi is usually
calculated. For unpaired samples, the sample sizes for
the two samples may or may not be equal. The formulas
for paired data are somewhat simpler than the formulas
for unpaired data.
- The variances of the two samples may be assumed
to be equal or unequal. Equal variances yields
somewhat simpler formulas, although with computers
this is no longer a significant issue.
- In some applications, you may want to adopt a new
process or treatment only if it exceeds the current
treatment by some threshold. In this case, we can
state the null hypothesis in the form that the
difference between the two populations means is
equal to some constant
\(\mu_{1} - \mu_{2} = d_{0}\)
where the constant is the desired threshold.
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Definition
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The two-sample t-test for unpaired data is defined as:
H0:
\( \mu_{1} = \mu_{2} \)
Ha:
\( \mu_{1} \neq \mu_{2} \)
Test Statistic:
\( T = \frac{\bar{Y_{1}} - \bar{Y_{2}}}
{\sqrt{{s^{2}_{1}}/N_{1} + {s^{2}_{2}}/N_{2}}} \)
where N1 and N2 are the
sample sizes,
\( \bar{Y_{1}} \) and \( \bar{Y_{2}} \) are
the sample means, and
\( {s^{2}_{1}} \) and \( {s^{2}_{2}} \) are the
sample variances.
If equal variances are assumed, then the formula reduces
to:
\( T = \frac{\bar{Y_{1}} - \bar{Y_{2}}} {s_{p}\sqrt{1/N_{1} + 1/N_{2}}} \)
where
\( s_{p}^{2} = \frac{(N_{1}-1){s^{2}_{1}} + (N_{2}-1){s^{2}_{2}}} {N_{1} + N_{2} - 2} \)
Significance Level:
α.
Critical Region:
Reject the null hypothesis that the two means are equal if
where t1-α/2,ν
is the
critical value of the
t distribution with ν
degrees of freedom where
\( \upsilon = \frac{(s^{2}_{1}/N_{1} + s^{2}_{2}/N_{2})^{2}} {(s^{2}_{1}/N_{1})^{2}/(N_{1}-1) + (s^{2}_{2}/N_{2})^{2}/(N_{2}-1) } \)
The above degrees of freedom are the
Welch-Satterthwaite
formulation. The Welch-Satterthwaite degrees of freedom are
robust for unequal sample sizes and/or unequal variances. If
equal variances are assumed and the sample sizes are similar,
then the following simpler formula for degrees of freedom can
be used: ν = N1 + N2 - 2.
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Two-Sample t-Test Example
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The following two-sample t-test was generated for the
AUTO83B.DAT data set. The data set
contains miles per gallon for U.S. cars (sample 1) and for
Japanese cars (sample 2); the summary statistics for each
sample are shown below.
SAMPLE 1:
NUMBER OF OBSERVATIONS = 249
MEAN = 20.14458
STANDARD DEVIATION = 6.41470
STANDARD ERROR OF THE MEAN = 0.40652
SAMPLE 2:
NUMBER OF OBSERVATIONS = 79
MEAN = 30.48101
STANDARD DEVIATION = 6.10771
STANDARD ERROR OF THE MEAN = 0.68717
We are testing the hypothesis that the population means are equal
for the two samples. We assume that the variances for the two
samples are equal.
H0: μ1 = μ2
Ha: μ1 ≠ μ2
Test statistic: T = -12.62059
Pooled standard deviation: sp = 6.34260
Degrees of freedom: ν = 326
Significance level: α = 0.05
Critical value (upper tail): t1-α/2,ν = 1.9673
Critical region: Reject H0 if |T|> 1.9673
The absolute value of the test statistic for our example, 12.62059,
is greater than the critical value of 1.9673, so we reject
the null hypothesis and conclude that the two population means
are different at the 0.05 significance level.
In general, there are three possible alternative hypotheses and
rejection regions for the one-sample t-test:
| Alternative Hypothesis |
Rejection Region |
| Ha: μ1 ≠ μ2 |
|T|> t1-α/2,ν |
| Ha: μ1> μ2 |
T> t1-α,ν |
| Ha: μ1 < μ2 |
T < tα,ν |
For our two-tailed t-test, the critical value is
t1-α/2,ν = 1.9673, where α = 0.05
and ν = 326. If we were to perform an upper, one-tailed test,
the critical value would be t1-α,ν = 1.6495.
The rejection regions for three posssible alternative hypotheses using
our example data are shown below.
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