CHAPTER 13 – STATISTICAL INFERENCE: TWO SAMPLES
13.1 Introduction to Hypothesis Tests for 2 Samples
Examples at the beginning of the chapter involve a comparison of 2 population
distributions (example: are men able to withstand weightlessness better than
women?, etc.)
Population distributions can differ in central tendency, dispersion, skewness, and
kurtosis.
Our inferences in this chapter are just extensions of the 1 sample case you learned
in chapters 1012
13.2 TwoSample t Test and Confidence Interval for
Using
Independent Samples
A t test statistic is used to test a hypothesis about the means,
and
, of 2
populations
The statistic can be used to test any of the following null hypotheses:
is the hypothesized difference between the population means
Usually we are interested in testing whether the 2 population means are equal, in
which case
.
The t test statistic is given by:
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The denominator of the t statistic,
, is an estimator of the
standard error of
the difference between 2 population means.
Degrees of freedom for the t statistic:
Sample 1 contributes
degrees of freedom, the number of d.f. associated with
Sample 2 contributes
degrees of freedom, the number of d.f. associated with
The null hypothesis is rejected if the observed t statistic exceeds or equals the
critical value of t given in Appendix Table D.3.
1 and 2 tailed critical values for t are
t
α,ν
and t
α/2,ν
To use the t statistic, assume that 2 random samples of sizes n
1
and n
2
have been
obtained from the populations of interest or that participants have been randomly
assigned to 2 groups often called experimental and control groups. These sampling
procedures produce
independent samples
in which the selection of elements in 1
sample is not affected by the selection of elements in the other.
The use of random sampling or random assignment helps to ensure that the
samples are statistically independent. Random assignment helps to distribute the
unique characteristics of the participants equally to the 2 groups.
Assume that the 2 populations are normally distributed and that the variances of
the populations
and
are unknown but assumed to be equal.
The pooled variance in the t formula is used since we are assuming the unknown
population variances,
and
, are equal. If the equality assumption is tenable,
the sample variances,
and
, are both estimators of the same population
variance,
. Whenever 2 estimators of
are available, a pooled estimator is likely
to provide a better estimate of the unknown population variance than either of the
sample estimators taken alone.
The pooled variance is just a weighted mean of
and
where the weights are
the respective degrees of freedom.
Homogeneity of variance assumption –
the assumption that the variances of
populations 1 and 2 are equal
When we expose 1 group to a treatment and use the other group as the control, we
often expect the treatment to raise or lower the scores of those exposed to it by a
constant amount. Remember that adding or subtracting a constant (4.2 exercise 8 &
4.7 exercise 11) does not affect the standard deviation (or variance) of the scores.
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 Spring '08
 kirk
 Normal Distribution, Standard Deviation, independent samples, Dependent Samples

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