Inference (hypothesis tests and estimation)
Variance of one population, comparing two variances
Tests and estimation on means and proportions focus on “typical value” or “where is the center of the values”.
The other huge aspect of distributions (that is still simpler than “shape”) is variability. In dealing with (for example)
the stock market and/or particular stocks, the interest is not just in “expected value” but in “risk” - the amount by
which the value fluctuates. The most common measurement used is standard deviation and (its theoretically-nicer-but
conceptually-wierder-brother) variance.
We would like to be able to estimate a population standard deviation, to perform tests on a standard deviation, and
to test for equality (or inequality) of standard deviations from two populations - but the work is carried out in terms of
variance. Our best estimator for σ2is s2, if we look at all samples of size n, the mean of all the s2values will be exactly
σ2. [Because of the behavior of square roots, the values of swould not average out to the value of σ].
Distribution: The distribution of s2values is not normal — we can’t convert to a Zor t. If the variable (X) being
observed is approximately normal, then the distribution of values of s2does fit the χ2(chi-square) family. We tend to
get a χ2distribution if we add the squares of a lot of normal variables (as we do, in calculating s2, if X is normal). As
with the tdistributions, there is a different version of the χ2for each value of “‘degrees of freedom”; the distribution is
not symmetric, and the mean gets larger with larger degrees of freedom.
Table: We will work with a table of critical values for χ2— it’s a little more complicated than the ttable because all
values are positive (no ±tαtricks can be used) and the distribution is not symmetric. The value which cuts off area αon
the right (larger) side is called χ2
α(same as with Zor t) but the value that cuts off area αon the left can’t be −χ2
α- we
notice that it must cut off area 1 −αto the em right and call it χ2
(1−α)
Test Statistic: To work with s2we use a normalization — but instead of subtracting the mean of the distribution,
we divide by the mean (of the s2values — that is σ2of the X values) and multiply by degrees of freedom - so our test
statistic will be sample χ2=(n−1)s2
σ2.
Estimation
We do not get an estimate of the form (statistic) ±E - we will be looking at multiplication/division:
To estimate σ2, with confidence 1 −αwe use the interval
(n−1)s2
χ2
α/2
to (n−1)s2
χ2
(1−α/2)
with df =n−1
Tests
For tests, we have the usual six steps, but the hypotheses are about σ2, the test statistic is sample χ2=(n−1)s2
σ2and we
have to look for χ2
(1−α)or χ2
(1−α/2) where we would have used −tαor −tα/2for tests on a mean.
“Greater”
H0:σ2=σ2
0
Ha:σ2> σ2
0
“Less”
H0:σ2=σ2
0
Ha:σ2< σ2
0
“not equal”
H0:σ2=σ2
0
Ha:σ26=σ2
0
sample χ2=(n−1)s2
σ2
0
df =n−1
Reject H0if sample χ2> χ2
αReject H0if sample χ2< χ2
(1−α)Reject H0if sample χ2< χ2
(1−α
2)
or sample χ2> χ2
(1−α
2)
Tests on difference of two variances
To decide whether the variances of two populations are equal, we will consider the ratio of sample variances s2
1
s2
2
. The
distribution of this ratio will follow yet another distribution - this one for a ratio of χ2distributions. It’s called the F
distribution and changes based on not one but two values - degrees of freedom for the numerator and degrees of freedom
for the denominator. In this note I will use the notation F(1−α)a
bfor “the value of F, with degrees of freedom afor the
numerator, vfor the denominator, that cuts off area αto the right”. In your text (pp. 925-927) there is a table of critical
values for the right-side only (that is Fαa
bfor small α). The conversion for the left-side value is not pretty but not terribly
hard: The left-side critical value (for “ <00 and “ 6=00 tests) is given by F(1−α)a
b=1
Fαb
a.
For the tests, we have the usual six steps, but the hypotheses are about σ2
1and σ2
2, the test statistic is sample F=s2
1
s2
2
and we have to look for F(1−α)abor F(1−α/2)abwhere we would have used −tαor −tα/2for a test on means.
“Greater”
H0:σ2
1=σ2
2
Ha:σ2
1> σ2
2
“Less”
H0:σ2
1=σ2
2
Ha:σ2
1< σ2
2
“not equal”
H0:σ2
1=σ2
2
Ha:σ2
16=σ2
2
sample F=s2
1
s2
2
numerator df =a=n1−1,
denominator df =b=n2−1
Reject H0if sample F > Fαa
bReject H0if sample F < F(1−α)a
bReject H0if sample F < F(1−α
2)a
b
or sample F > F(1−α
2)a
b
1
