Transcript Section 7.3 Sample Means

Section 7.3
Sample Means
A look back at proportions…
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You may have noticed that proportions
ALWAYS deal with categorical data.
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We’ve looked at the proportion of first-year
college students who applied to more than
one school.
We’ve looked at the proportion of people who
are left handed.
Quantitative Variables
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When we are interested in quantitative
variables, like the income of a household,
how long a car lasts, or the blood pressure
of a patient, we look at other statistics.
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One of the most popular statistics for
quantitative variables is the sample mean.
Why Be Mean?
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Averages are less variable than individual
observations.
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Which is more likely: finding one person
whose IQ is at least 130 or finding a random
sample of individuals whose AVERAGE IQ is at
least 130?
In fact, averages are more NORMAL than
individual observations.
The Mean and Standard Deviation
of x-bar
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Since x-bar is an unbiased estimator of μ,
what should the mean of x-bar be?
x  

x 
n
What happens
to the standard
deviation as n
increases?
Some facts about x-bar
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X-bar is an unbiased estimator of μ.
The larger the sample, the smaller the
variation of x-bar values.
BONUS!!!!!
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Thank you for checking the PowerPoints.
If you write down the word “Joetro” at the
top of your test, I will give you 4 bonus
points.
Example
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A grinding machine in an auto parts plant
prepares axles with a target diameter µ =
40.125 mm. The standard deviation is σ
= 0.002 mm. The machine operator
inspects a random sample of 4 axles each
hour and records the sample mean
diameter. What are the mean and
standard deviation of the sampling
distribution of x-bar?
Example
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A grinding machine in an auto parts plant
prepares axles with a target diameter µ =
40.125 mm. The standard deviation is σ
= 0.002 mm. How many axles would you
need to sample if you wanted the
standard deviation of the sampling
distribution of x-bar to be within 0.0005
mm?
PCFS for Means
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Parameter
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Conditions
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This time our parameter is µ.
Normality: today, the problem will tell you
that the population is distributed normally.
Independence: population ≥ 10n
Formula
Sentence
Example
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A grinding machine in an auto parts plant
prepares axles with a target diameter µ =
40.125 mm. The standard deviation is σ
= 0.002 mm. Assume the population of
all axles made is distributed normally.
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What is the probability that a random sample
of 4 axles has a mean diameter of at least
40.1265 mm?
What is the probability that an individual axle
is at least 40.1265 mm in diameter?
Don’t you have any other
examples?
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Women’s heights are distributed normally
with a mean of 64.5 inches and a standard
deviation of 2.5 inches.
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What is the probability that a randomly
selected young woman is taller than 66.5
inches?
What is the probability that the mean height
of a sample of 10 young women is greater
than 66.5 inches?