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CHAPTER 12:
INFERENCE ABOUT A POPULATION
Stats & Prob. for Bus. Mgmt (Stat1100)
Jochem
Chap 12: Inference about a Pop.
b
Goals of Chapter 12
Be able to do inference* about …
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1. a Population Mean when the Std Dev is
unknown
.
. a Population Variance.
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b
2. a Population Variance.
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3. a Population Proportion.
s
Selecting the right sample size to estimate a population
proportion with a given margin of error.
* i.e. computing an interval estimate and testing a hypothesis
b
1. Interval Estimator for Population Mean
b
In Chap 10, we introduced the interval estimator:
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Chap 12: Inference about a Pop.
s
This estimator assumes that we do know the standard
deviation of the population,
σ.
In reality, however, we
usually don’t know
σ.
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1. Interval Estimator for Population Mean
b
With a slight change we can modify the interval
estimator to work also when we do not know the
standard deviation:
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Chap 12: Inference about a Pop.
s
where
s
is now the
sample
standard deviation, and
s
t
is the
t
statistic (the number of sample std dev. from mean)
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1. Interval Estimator for Population Mean
b
The tstatistic is (or ttest statistic) is just like the z(test)
statistic the number of std. dev. away from the mean.
s
Before: # of population std dev; now: # of sample std dev.
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Chap 12: Inference about a Pop.
s
1. we replaced
σ
with the sample std deviation “
s
”
s
2. tstatistic:
s
zstatistic : std. normally distr. ; we use Table 3 to look up values.
s
tstatistic:
Student t distributed, i.e. we now need to look up
values in Table 4.
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Chap 12: Inference about a Pop.
b
1. Interval Estimator for Population Mean
b
Recall: the Student
t
distribution is bellshaped and
has only one parameter, the “degrees of freedom
ν
”
(
pronounced: nu), which equals the sample size minus 1.
s
When
ν
goes to infinity, the Student t distribution becomes the
Normal distribution. In general however, the Student t
distribution is flatter (i.e. has a greater spread) than the
normal distribution.
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Chap 12: Inference about a Pop.
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1. Interval Estimator for Population Mean
b
How to look up values in Table 4?
s
Example: look up the tstatistic for a sample size of 30
when the significance level is 5% and it’s a righttail test.
. Sample size = n = 30
s
1. Sample size = n = 30
B
degrees of freedom
ν = n1=29.
s
2. Locate “29” as the degree of
freedom (df) in Table 4
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3. Go to the column where it says t
0.050
.
B
t
ν
=29,0.050
=1.699.
Example
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Chap 12: Inference about a Pop.
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1. Interval Estimator for Population Mean
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Example1: Q12.24 (p393)
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A federal agency responsible for enforcing laws governing
weights and measure routinely inspects packages to determine
whether the weight of the contents is at least as great as that
advertised on the package. A random sample of 18 containers
whose packaging states that the contents weigh 8 ounces was
drawn. Its sample mean is 7.91 ounces and the sample
standard deviation is 0.085.
s
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 Spring '08
 Chiappetta
 Variance

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