CHAPTER 14 – STATISTICAL INFERENCE: OTHER TWOSAMPLE TEST STATISTICS
14.1 Introduction
Learning about the F statistic and F sampling distribution that are used to test hypotheses about 2
variances and how to use a z statistic to test hypotheses about 2 population proportions
14.2 TwoSample F Test and Confidence Interval for Variances Using Independent Samples
F Test for 2 Variances (Independent Samples)
Researcher might want to see if 2 populations differ in dispersion or they might want to test one of the
assumptions of the t test for independent samples – that the 2 unknown population variances are equal
F statistic for testing the hypotheses
:
=
:
≥
:
≤
H0
σ12
σ22
H0
σ12
σ22
H0
σ12
σ22
:
≠
:
<
:
>
H1
σ12
σ22
H1
σ12 σ22
H1
σ12 σ22
is
=
F
σlarger2σsmaller2
where
σlarger2
and
σsmaller2
denote, respectively, the larger and
smaller sample variance and each sample variance is computed using
=


σ 2
Xi X2n 1
. The degrees of
freedom for the numerator and denominator are, respectively,
=

ν1
nlarger σ2 1
and
=
ν2
nsmaller

σ2 1
.
Sampling distribution of the F derived by Ronald A. Fisher and G.W. Snedecor named it after him.
Like the t distr., is a family of distributions whose shape depends on its d.f.
Unlike the z and t distributions that are symmetrical, the F distribution is positively skewed
Shape of the F distribution approaches the normal for large values of
ν1
and
ν2
.
F is a ratio of nonnegative numbers so it can take on values from 0 to ∞. Values around 1 are expected if
the null hypothesis that
=
σ12
σ22
is true
Assumptions for the F to test a null hypothesis:
1. Independent samples
2. Populations are normally distributed
3. Participants are random samples from the populations of interest or the participants have been
randomly assigned to the conditions in the experiment.
F is not robust to violation of the normality assumption like the t is regardless of how large your sample
is. So, unless the normality assumption is fulfilled, the probability of making a Type 1 error will not equal
the preselected value of α. Do NOT use the F unless you have good reason to believe the 2 variables
X1
and
X2
are normal.
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;
,
Fα
ν1 ν2
is the critical value of F that cuts off the upper α region of the sampling distribution for
ν1
and ν2
degrees of freedom. The first
ν
denotes the df for the numerator of the F and the second
ν
denotes the df for the denominator of the F.
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 Spring '08
 kirk
 Statistics, Normal Distribution, Variance, independent samples

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