Download x-bar - San Jose State University

Chapter 10
Sampling Distributions
BPS - 3rd Ed.
Chapter 10
1
Parameters and Statistics

Parameter
– Is a fixed number that describes the location or spread
of a population
– Its value is NOT known in statistical practice

Statistic
– A calculated number from data in the sample
– Its value IS known in statistical practice
– Some statistics are used to estimate parameters

Sampling variability
– Different samples or experiments from the same
population yield different values of the statistic
BPS - 3rd Ed.
Chapter 10
2
Parameters and statistics
mean of a population is called µ  this
is a parameter
 The mean of a sample is called “x-bar”  this
is a statistic
 Illustration:
 The
– Average age of all SJSU students (µ) is 26.5
– A SRS of 10 San Jose State students yields a
mean age (x-bar) of 22.3
– x-bar and µ are related but are not the same thing!
BPS - 3rd Ed.
Chapter 10
3
The Law of Large Numbers
BPS - 3rd Ed.
Chapter 10
4
Example 10.2 (p. 251)
Does This Wine Smell Bad?




Dimethyl sulfide (DMS) is
sometimes present in wine
causing off-odors
Different people have
different thresholds for
smelling DMS
Winemakers want to know
the odor threshold that the
human nose can detect.
Population values:
– Mean threshold of all adults
 is 25 µg/L wine
– Standard deviation  of all
adults is 7 µg/L
– Distribution is Normal
BPS - 3rd Ed.
Chapter
10
5
Simulation: Example 10.2 (cont.)
 Suppose
you take a simple random sample
(SRS) from the population and the first
individual in sample has a value of 28
 The second individual has a value of 40
 The average of the first two individuals =
(28+40)/2 = 34
 Continue sampling individuals at random and
calculating means
 Plot the means
BPS - 3rd Ed.
Chapter 10
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Simulation: law of large numbers
Fig 10.1
The sample mean
gets close to
population mean 
as we take more
and more samples
BPS - 3rd Ed.
Chapter 10
7
Sampling distribution of xbar
Key questions:
What would happen if we took many samples or did the
experiment many times?
How would the statistics from these repeated samples vary?
BPS - 3rd Ed.
Chapter 10
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Case Study
Does This Wine Smell Bad?
• Recall  = 25 µg / L,  = 7 µg / L and
the distribution is Normal
• Suppose you take 1,000 repetitions of
samples, each of n =10 from this
population
• You calculate x-bar in each sample
• You plot the x-bars as a histogram
• You study the histogram (next slide)
BPS - 3rd Ed.
Chapter 10
9
Simulation: 1000 sample means
Example 10.4
BPS - 3rd Ed.
Chapter 10
10
Mean and Standard Deviation of
“x-bar”
BPS - 3rd Ed.
Chapter 10
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Mean and Standard Deviation of
Sample Means
the mean of X is , we say that X is
an unbiased estimator of 
Since
Individual
observations have standard
deviation , but sample means X from
samples of size n have standard deviation

n . Averages are less variable than
individual observations.
BPS - 3rd Ed.
Chapter 10
12
Case Study
Does This Wine Smell Bad?
(Population distribution)
BPS - 3rd Ed.
Chapter 10
13
Illustration Exercise 10.8 (p. 258)
Suppose blood cholesterol in a population of men is
Normal with  = 188 and  = 41. You select 100 men
at random
(a) What is mean and standard deviation of the 100 x-bars from
these samples?
– x-barx-bar = 188 (same as µ)
– sx-bar = 41 / sqrt(100) = 4.1 (one-tenth of )
(b) What is probability a given x-bar is less than 180?
Pr(x-bar < 180)  standardize  z = (180 – 188) / 4.1 = -1.95
= Pr(Z < –1.95)  TABLE A  = .0256
BPS - 3rd Ed.
Chapter 10
14
Central Limit Theorem
No matter what the shape of the population, the sampling
distribution of xbar, will follow a Normal distribution when the
sample size is large.
BPS - 3rd Ed.
Chapter 10
15
Central Limit Theorem in Action
Example 10.6 (p. 259)

Data = time to perform an
activity
– NOT Normal (Fig a)
– µ = 1 hour
– σ = 1 hour
Fig (a) is for single observations
Fig (b) is for x-bars based on n = 2
Fig (c) is for x-bars based on n =
10
Fig (d) is for x-bars based on n =
25

Notice how distribution
becomes increasingly Normal
as n increases!
BPS - 3rd Ed.
Chapter 10
16
Example 10.7 (cont.)


Sample n = 70 activities
What is the distribution of x-bars?
– x-barx-bar = 1 (same as µ)
– sx-bar = 1 / sqrt(70) = 0.12 (by formula)
– Normal (central limit theorem)

What % of x-bars will be less than
0.83 hours?
Pr(x-bar < 0.83)  standardize 
z = (0.83 – 1) / 0.12 = -1.42
Pr(Z < - 1.42)  TABLE A  = 0.0778
Notice that if Pr(x-bar < 0.83 = 0.0778, then Pr(Z
> 0.83) = 1 – 0.0778 = 0.9222
BPS - 3rd Ed.
Chapter 10
17
BPS - 3rd Ed.
Chapter 10
18
Statistical process control*
Skip pp. 262 – 269
BPS - 3rd Ed.
Chapter 10
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