I will attach a file called Extended Application 3 and you will need to read that before you answer the four questions, it tells you how to answer it. After you read that, please answer the questions and send the answers back as such:Answer 1a:Answer 1b:Answer 2a:Answer 2b:Answer 3a:Answer 4:Just make it clear which answer is for which question, this is only 4 questions and will take YOU 30 MINUTES OR LESS if you know statistics. Please make you answers clear and make sure they are 100% RIGHT! I CAN not get any of them wrong so please make sure. Make sure you read extended application 3, the PDF attached FIRST, because it will tell you how the professor wants it done.

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extended_applicaton_3.pdf

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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 lets 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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