Economic Statistics:
Homework 9
Confidence Interval Estimation of the Population Variance:
1.
The weights of 20 oranges (in ounces) are shown below.
a.
Construct a 95% CI for the population mean.
The mean of the sample is 7.775.
The population variance is unknown, but
the sample variance is 1.775 – so we will use the t distribution.
SMPER =
n
s
t
/
2
/
α
= 2.093*(1.332/4.472) = 0.6234; so the CI is from
7.1516 to 8.3984
b.
Construct a 95% CI for the population variance.
Use the Chisquared distribution.
P(
2
2
,
1
2
)
1
(
α
χ


n
s
n
<σ
2
<
2
2
1
,
1
2
)
1
(
α
χ



n
s
n
) = 1α
P(
85
.
32
775
.
7
)
19
(
<σ
2
<
91
.
8
775
.
7
)
19
(
) = 0.95; so the Lower Bound = 4.48 and the
Upper bound = 16.53
c.
What must be assumed about the population distribution to calculate the
above estimates?
The population must be normal in order to use the Chi Squared, and also to
use the t for relatively small samples.
Hypothesis Testing:
2.
When an Olympic athlete is tested for performance enhancing drugs like steroids,
the presumption is that the athlete is in compliance with the rules.
Samples are
taken as evidence.
a.
State the null and alternative hypothesis.
H
0
:
No Illegal steroid use
:
1
H
Illegal steroid use
b.
Describe what the Type I and Type II error would be for this type of test.
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Type I error is unfairly disqualifying an athlete who is “clean”.
Type II error is letting
the drug user get away with it and have an unfair competitive advantage.
c.
Which error is more to be feared?
The cost of Type I error is hard feelings and unnecessary embarrassment.
The cost of
Type II error is tarnishing the Olympic image and rewarding those who break the rules.
You decide.
3.
The width of a sheet of standard size copier paper should be 216mm (8.5 inches).
There is some variation in paper width due to the nature of the production process
and the population variance is 0.84mm.
However, only a certain number of
nonconforming sheets per million are allowed by industry standards.
Samples are
taken and the mean is used to test a hypothesis.
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 Spring '08
 Martin
 Normal Distribution, Null hypothesis, Statistical hypothesis testing, Type I and type II errors, rejection region

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