Ch. 24 - Comparing Two Population Means
Assumptions:
1. The two samples are random and independent.
2. At least one of the following is also true:
i. Both samples are large (n ≥130 and n ≥ 32)
ii. If either one or both sample sizes are small, then both populations from which
the samples are drawn are normally distributed.
3. The standard deviations σ a1d σ of 2he two populations are unknown and unequal to
each other; that is, σ1≠ σ 2
Checking the Assumptions:
The first assumption can be “checked” by analyzing the experimental design. The 2 nd
assumption can be “checked” just like in Ch. 23. The third should require a formal test
s
that is highly sensitive, but for now, check if max ≥ 2 .
s min
Hypotheses:
As in Ch. 22, there are two population means (a.k.a. parameters) in our data structure and
we consider them together as ONE parameter: µ – µ . 1hus,2we have
H 0 µ 1 µ =20 H A µ 1 µ ≠20
Note that we could use any value to compare to, but zero has a ‘special’ interpretation.
Also, tests can be one-sided, too.
Test statistic:
If the assumptions hold, then we may use the t-distribution.
Thus, the standard error of y − y is
1 2
s2 s2
SE(y −1y ) 2 1 + 2
n1 n 2
and the test statistic 0 is
y − y − ( µ − µ )
t0= 1 2 1 2
SE(y −1y ) 2
and t follows a t-distribution with a complicated df (see footnote on p. 657). Thus, we
will instead use a conservative lower bound: df ≥ min{n – 1,1n – 1}.2
P-value: No different than how we calculated it in Ch. 23.
Conclusion: Reject/do not reject as in one-sample test; answer hypotheses/question
posed. Confidence Interval
The (1 – α)100% CI for µ – 1 is2
y1− y 2± tα/2, dfSE(y 1 y )2
Assumptions: as per hypothesis test.
Notes: - CI tends to be more informative than a test.
- check if zero falls within the interval; check sign and magnitude.
The Pooled t-Test rd
Recall the three assumptions from the previous test. The 3 assumption now changes to
3. The standard deviations σ 1nd σ o2 the two populations are unknown and equal to
each other; that is, 1 = σ2. (Checking the assumption reverses as well.)
The consequence of this change is that the standard error of y1− y 2 now uses the pooled
sample standard deviation, or s p
n −1) s2 + (n −1)s2
s = 1 1 2 2

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