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Background for Extended Application III
Confidence Intervals & Hypothesis Tests for a Single Variance and Standard Deviation
Math 15, Moorpark College, Fall 2017
Just like we can use confidence intervals to estimate p, p1 – p2 , µ, µd , and µ1 – µ2 (i.e., the Big Five
Parameters), we can also use confidence intervals to estimate the unknown population variance, s 2 , and
unknown population standard deviation, s. Likewise, we can perform hypothesis tests on variances and
standard deviations as well.
Confidence Intervals for a single variance or standard deviation
For these confidence intervals we use the ?2 -distribution. This is the same distribution that we used
when comparing categorical data, and as mentioned in class, the ?2 -distribution comes from a sampling
distribution of the variances. Since the standard deviation is the square root of the variance, it follows that
for the confidence interval formula for the standard deviation is merely the square root of the formula for
the variance.
Before we go through examples of these confidence intervals, we should note the following:
(1) Since the ?2 -distribution is not symmetric, the confidence interval is not point estimate ± margin
of error since the bounds of the left and right will be different distances from the point estimate of
s2 (for variances) or s (for standard deviations).
(2) In order for this confidence interval to work well, the underlying data must be normally distributed
with no outliers. In other words, this test is not robust since it is very sensitive to bad data.
Confidence Interval for a Population Variance. To estimate s 2 :
(n – 1) s2 (n – 1) s2
,
?2a/2
?2(1-a/2)
!
n = sample size
s2 = sample variance
?2a/2 = the critical value where there is a/2 area to the right of the critical value
?2(1-a/2) = the critical value where there is (1 – a/2) area to the right of the critical value
1
2
Confidence Interval for a Population Standard Deviation. To estimate s:
?
? p
p
s (n – 1) s (n – 1) ?
(n –
(n –
, q
,
=? q
?2a/2
?2(1-a/2)
?2a/2
?2(1-a/2)
p
Note that the square on ?2 is part of the symbol, so ?2 6= ?, in other words do not square root the
s
1) s2
s
1) s2
!
?2 values in the standard deviation formula.
Unfortunately, the TI Calculator does not have a way to perform these confidence intervals unless we
program them ourselves. So, I will give you the appropriate ?2a/2 and ?2(1-a/2) , whereas usually you have to
look up the values on a table.
Hypothesis Tests for a single variance or standard deviation
The test statistic for these hypothesis tests (both variance and standard deviation) is
?2 =
(n – 1) s2
s2
where s 2 is from the null value. There are left tail, right tail, and two-tailed tests, of course, but lets just
pick one of them for the exercise. Note, the ?2 distribution only has values such that ?2 = 0, so we reject
the null differently than the Big 5 Parameters, viz.
Reject the null (?2 for one population variance/standard deviation)
(1) Left-tailed test: test statistic < critical value (2) Right-tailed test: test statistic > critical value
(3) Two-tailed: left test statistic < critical value OR right test statistic > critical value

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