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8.1 Sampling Distributions
LEARNING GOAL
Understand the fundamental ideas of sampling distributions
and how the distribution of sample means and the
distribution of sample proportions are formed. Also learn the
notation used to represent sample means and proportions.
Copyright © 2009 Pearson Education, Inc.
Sample Means: The Basic Idea
Table 8.1 lists the weights of the five
starting players (labeled A through E
for convenience) on a professional
basketball team. We regard these five
players as the entire population (with
a mean of 242.4 pounds).
Samples drawn from this population
of five players can range in size from
n = 1 (one player out of the five) to n = 5 (all five players).
With a sample size of n = 1, there are 5 different samples that
could be selected: Each player is a sample. The mean of each
sample of size n = 1 is simply the weight of the player in
the sample.
Copyright © 2009 Pearson Education, Inc.
Slide 8.1- 2
Figure 8.1 shows a
histogram of the means of
the 5 samples; it is called a
distribution of sample
means, because it shows
the means of all 5 samples
of size n = 1.
The distribution of sample
Figure 8.1 Sampling distribution
means created by this process
for sample size n 1.
is an example of a sampling
distribution. This term simply refers to a distribution of a
sample statistic, such as a mean, taken from all possible
samples of a particular size.
Copyright © 2009 Pearson Education, Inc.
Slide 8.1- 3
Notice that the mean of the 5
sample means is the mean of the
entire population:
215 + 242 + 225 + 215 + 315
5
= 242.4 pounds
This demonstrates a general rule: The mean of a
distribution of sample means is the population mean.
Copyright © 2009 Pearson Education, Inc.
Slide 8.1- 4
Let’s move on to samples of size n = 2, in which each sample
consists of two different players. With five players, there are 10
different samples of size n = 2. Each sample has its own mean.
Table 8.2 lists the 10 samples with their
means.
Figure 8.2 shows
the distribution
of all 10 sample
means.
Again, notice
that the mean of
the distribution
of sample
means is equal to the population mean,
242.4 pounds.
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Slide 8.1- 5
Ten different samples of size n = 3 are possible in a population
of five players.
Table 8.3 shows these samples and their means, and Figure
8.3 shows the distribution of these sample means.
Again, the
mean of the
distribution
of sample
means is
equal to the
population mean, 242.4 pounds.
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Slide 8.1- 6
With a sample size of n = 4, only 5 different samples are
possible.
Table 8.4 shows these samples and their means, and Figure 8.4
shows the distribution of these sample means.
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Slide 8.1- 7
Finally, for a population of five
players, there is only 1 possible
sample of size n = 5: the entire
population. In this case, the
distribution of sample means is
just a single bar (Figure 8.5).
Again the mean of the distribution
of sample means is the population
mean, 242.4 pounds.
Figure 8.5 Sampling distribution
for sample size n = 5.
To summarize, when we work
with all possible samples of a
population of a given size, the mean of the distribution of
sample means is always the population mean.
Copyright © 2009 Pearson Education, Inc.
Slide 8.1- 8
Sample Means with Larger Populations
In typical statistical applications, populations are huge and
it is impractical or expensive to survey every individual in
the population; consequently, we rarely know the true
population mean, μ.
Therefore, it makes sense to consider using the mean of a
sample to estimate the mean of the entire population.
Although a sample is easier to work with, it cannot
possibly represent the entire population exactly. Therefore,
we should not expect an estimate of the population mean
obtained from a sample to be perfect.
The error that we introduce by working with a sample is
called the sampling error.
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Slide 8.1- 9
Sampling Error
The sampling error is the error introduced because a
random sample is used to estimate a population
parameter. It does not include other sources of error,
such as those due to biased sampling, bad survey
questions, or recording mistakes.
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Slide 8.1- 10
Results from a survey of students who were asked how
many hours they spend per week using a search engine
on the Internet.
n = 400
μ = 3.88
σ = 2.40
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Slide 8.1- 11
A sample of 32 students selected from the 400 on
the previous slide.
1.1
3.8
1.7
7.8
5.7
2.1
6.8
6.5
1.2
4.9
2.7
0.3
3.0
2.6
0.9
6.5
1.4
2.4
5.2
7.1
2.5
2.2
5.5
7.8
5.1
3.1
3.4
5.0
4.7
6.8
7.0
6.5
Sample 1
The mean of this sample is x̄x = 4.17; we use the standard
notation x̄x to denote this mean.
We say that x̄x is a sample statistic because it comes from a
sample of the entire population. Thus, x̄x is called a sample
mean.
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Slide 8.1- 12
Notation for Population and Sample Means
n = sample size
m = population mean
x¯ = sample mean
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Slide 8.1- 13
A different sample of 32 students selected from the 400.
1.8
5.2
0.5
0.4
5.7
3.9
4.0
6.5
3.1
2.4
1.2
5.8
0.8
5.4
2.9
6.2
5.7
7.2
0.8
7.2
0.9
6.6
5.1
4.0
5.7
3.2
7.9
3.1
2.5
5.0
3.6
3.1
Sample 2
For this sample x̄x is = 3.98.
Now you have two sample means that don’t agree with each
other, and neither one agrees with the true population mean.
x̄x1 = 4.17 (slide 13)
x̄x2 = 3.98
m = 3.88 (slide 10)
Copyright © 2009 Pearson Education, Inc.
Slide 8.1- 14
Figure 8.6 shows a histogram that results from 100 different
samples, each with 32 students. Notice that this histogram is
very close to a normal distribution and its mean is very close
to the population mean, μ = 3.88.
Figure 8.6 A distribution of 100 sample means, with a sample size of n = 32,
appears close to a normal distribution with a mean of 3.88.
Copyright © 2009 Pearson Education, Inc.
Slide 8.1- 15
The Distribution of Sample Means
The distribution of sample means is the distribution
that results when we find the means of all possible
samples of a given size.
The larger the sample size, the more closely this
distribution approximates a normal distribution.
In all cases, the mean of the distribution of sample
means equals the population mean.
If only one sample is available, its sample mean, x,
x̄ is
the best estimate for the population mean, m.
Copyright © 2009 Pearson Education, Inc.
Slide 8.1- 16
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Slide 8.1- 17
If we were to include all possible samples of size n = 32,
this distribution would have these characteristics:
• The distribution of sample means is approximately a
normal distribution.
• The mean of the distribution of sample means is 3.88 (the
mean of the population).
• The standard deviation of the distribution of sample
means depends on the population standard deviation and
the sample size. The population standard deviation is σ =
2.40 and the sample size is n = 32, so the standard
deviation of sample means is
σ
2.40
=
= 0.42
n
32
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Slide 8.1- 18
Suppose we select the following random sample of 32
responses from the 400 responses given earlier:
5.8 7.5 5.8 5.2 3.9 3.4 7.3 4.1 0.5 7.9 7.7 7.7 5.0
2.3 7.8 2.3 5.0 6.8 6.5 1.7 2.1 7.3 4.0 2.2 5.6 4.7
5.3 3.5 6.5 3.4 6.6 5.0
Sample 3
The mean of this sample is x̄x = 5.01.
Given that the mean of the distribution of sample means is
3.88 and the standard deviation is 0.42, the sample mean
of xx̄ = 5.01 has a standard score of
z=
sample mean – pop. mean
5.01 – 3.88
=
= 2.7
standard deviation
0.42
Copyright © 2009 Pearson Education, Inc.
Slide 8.1- 19
The sample (from the previous slide) has a standard score
of z = 2.7, indicating that it is 2.7 standard deviations
above the mean of the sampling distribution.
From Table 5.1, this standard score corresponds to the
99.65th percentile, so the probability of selecting another
sample with a mean less than 5.01 is about 0.9965.
It follows that the probability of selecting another sample
with a mean greater than 5.01 is about 1 – 0.9965 =
0.0035.
Apparently, the sample we selected is rather extreme
within this distribution.
Copyright © 2009 Pearson Education, Inc.
Slide 8.1- 20
EXAMPLE 1 Sampling Farms
Texas has roughly 225,000 farms, more than any other state in
the United States. The actual mean farm size is μ = 582 acres
and the standard deviation is σ = 150 acres. For random samples
of n = 100 farms, find the mean and standard deviation of the
distribution of sample means. What is the probability of
selecting a random sample of 100 farms with a mean greater
than 600 acres?
Solution
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Slide 8.1- 21
EXAMPLE 1 Sampling Farms
Solution: (cont.)
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Slide 8.1- 22
Sample Proportions
In a survey where 400 students were asked if they own a
car, 240 replied that they did.
The exact proportion of car owners is
240
p=
= 0.6
400
This population proportion, p = 0.6, is another example of a
population parameter.
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Slide 8.1- 23
A sample of 32 was selected from the 400 students and 21
were car owners.
21
pp̂ =
= 0.656
32
This proportion is another example of a sample statistic.
In this case, it is a sample proportion because it is the
p̂ symbol p
proportion of car owners within a sample; we use the
(read “p-hat”) to distinguish this sample proportion from the
population proportion, p.
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Slide 8.1- 24
Notation for Population and Sample Proportions
n = sample size
p = population proportion
p
ˆ = sample proportion
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Slide 8.1- 25
Figure 8.7 shows such a histogram of sample proportions
from 100 samples of size n = 32. As we found for sample
means, this distribution of sample proportions is very close
to a normal distribution. Furthermore, the mean of this
distribution is very close to the population proportion of 0.6.
Figure 8.7 The distribution of 100 sample proportions, with a sample size
of 32, appears to be close to a normal distribution.
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Slide 8.1- 26
Suppose it were possible to select all possible samples
of size n = 32. The resulting distribution would be
called a distribution of sample proportions.
The mean of this distribution equals the population
proportion exactly.
This distribution approaches a normal distribution as
the sample size increases.
In practice, we often have only one sample to work
with. In that case, the best estimate for the population
p̂
proportion, p, is the sample proportion, p.
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Slide 8.1- 27
The Distribution of Sample Proportions
The distribution of sample proportions is the
distribution that results when we find the proportions (p̂ )
in all possible samples of a given size.
The larger the sample size, the more closely this
distribution approximates a normal distribution.
In all cases, the mean of the distribution of sample
proportions equals the population proportion.
If only one sample is available, its sample proportion, p̂ ,
is the best estimate for the population proportion, p.
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Slide 8.1- 28
EXAMPLE 2 Analyzing a Sample Proportion
Consider the distribution of sample proportions shown in Figure
8.7 (slide 30). Assume that its mean is p = 0.6 and its standard
deviation is 0.1. Suppose you randomly select the following
sample of 32 responses:
YYNYYYYNYYYYYYNYYNYYYNYYNYYNYNYY
p̂ for this sample. How far does
Compute the sample proportion, p,
it lie from the mean of the distribution? What is the probability of
selecting another sample with a proportion greater than the one
you selected?
Solution:
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Slide 8.1- 29
EXAMPLE 2 Analyzing a Sample Proportion
Solution: (cont.)
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Slide 8.1- 30
8.2 Estimating Population Means
LEARNING GOAL
Learn to estimate population means and compute the
associated margins of error and confidence intervals.
Copyright © 2009 Pearson Education, Inc.
Estimating a Population Mean: The Basics
When we have only a single sample, the sample
mean is the best estimate of the population mean, μ.
However, we do not expect the sample mean to be
equal to the population mean, because there is likely
to be some sampling error. Therefore, in order to
make an inference about the population mean, we
need some way to describe how well we expect it to
be represented by the sample mean.
The most common method for doing this is by way
of confidence intervals.
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Slide 8.2- 32
A precise calculation shows that if the distribution of
sample means is normal with a mean of μ, then 95%
of all sample means lie within 1.96 standard
deviations of the population mean; for our purposes
in this book, we will approximate this as 2 standard
deviations.
A confidence interval is a range of values likely to
contain the true value of the population mean.
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Slide 8.2- 33
95% Confidence Interval for a Population Mean
The margin of error for the 95% confidence interval is
2s
margin of error = E ≈
n
where s is the standard deviation of the sample.
We find the 95% confidence interval by adding and
subtracting the margin of error from the sample mean.
That is, the 95% confidence interval ranges
from (xx̄ – margin of error)
to
x̄ + margin of error)
(x
We can write this confidence interval more formally as
x̄x – E < μ < x̄x + E
or more briefly as
x̄x ± E
Copyright © 2009 Pearson Education, Inc.
Slide 8.2- 34
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Slide 8.2- 35
EXAMPLE 1 Computing the Margin of Error
Compute the margin of error and find the 95% confidence
interval for the protein intake sample of n = 267 men, which
has a sample mean of xx̄ = 77.0 grams and a sample standard
deviation of s = 58.6 grams.
Solution:
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Slide 8.2- 36
EXAMPLE 1 Computing the Margin of Error
Solution: (Cont.)
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Slide 8.2- 37
Interpreting the Confidence Interval
Figure 8.10 This figure
illustrates the idea behind
confidence intervals. The
central vertical line
represents the true
population mean, μ. Each of
the 20 horizontal lines
represents the 95%
confidence interval for a
particular sample, with the sample mean marked by the dot in the
center of the confidence interval. With a 95% confidence interval, we
expect that 95% of all samples will give a confidence interval that
contains the population mean, as is the case in this figure, for 19 of the
20 confidence intervals do indeed contain the population mean. We
expect that the population mean will not be within the confidence
interval in 5% of the cases; here, 1 of the 20 confidence intervals (the
sixth from the top) does not contain the population mean.
Copyright © 2009 Pearson Education, Inc.
Slide 8.2- 38
EXAMPLE 2 Constructing a Confidence Interval
A study finds that the average time spent by eighth-graders
watching television is 6.7 hours per week, with a margin of error
of 0.4 hour (for 95% confidence). Construct and interpret the
95% confidence interval.
Solution:
x̄
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Slide 8.2- 39
Choosing Sample Size
Solve the margin of error formula E ≈ 2s /
 2s 
n   
E 
n for n.
2
Choosing the Correct Sample Size
In order to estimate the population mean with a specified
margin of error of at most E, the size of the sample should
2
be at least
 2 

n  
E 
where σ is the population standard deviation (often
estimated by the sample standard deviation s).
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Slide 8.2- 40
EXAMPLE 6 Constructing a Confidence Interval
You want to study housing costs in the country by sampling
recent house sales in various (representative) regions. Your goal
is to provide a 95% confidence interval estimate of the housing
cost. Previous studies suggest that the population standard
deviation is about $7,200. What sample size (at a minimum)
should be used to ensure that the sample mean is within
a. $500 of the true population mean?
b. $100 of the true population mean?
Solution:
a.
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Slide 8.2- 41
EXAMPLE 6 Constructing a Confidence Interval
Solution:
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Slide 8.2- 42
8.3 Estimating Population
Proportions
LEARNING GOAL
Learn to estimate population proportions and compute the
associated margins of error and confidence intervals.
Copyright © 2009 Pearson Education, Inc.
The Basics of Estimating a Population
Proportion
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Slide 8.3- 44
95% Confidence Interval for a Population
Proportion
For a population proportion, the margin of error for the
95% confidence interval is
pˆ (1  pˆ )
E 2
n
where p̂ is the sample proportion.
The 95% confidence interval ranges
from p̂ – margin of error
to p̂ + margin of error
We can write this confidence interval more formally as
pˆ – E  p  pˆ  E
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Slide 8.3- 45
EXAMPLE 2 TV Nielsen Ratings
The Nielsen ratings for television use a random sample of
households. A Nielsen survey results in an estimate that a
women’s World Cup soccer game had 72.3% of the entire
viewing audience. Assuming that the sample consists of n =
5,000 randomly selected households, find the margin of error
and the 95% confidence interval for this estimate.
Solution:
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Slide 8.3- 46
EXAMPLE 2 TV Nielsen Ratings
Solution: (cont.)
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Slide 8.3- 47
Choosing Sample Size
Choosing the Correct Sample Size
In order to estimate a population proportion with a 95%
degree of confidence and a specified margin of error of E,
the size of the sample should be at least
n=
1
E2
Copyright © 2009 Pearson Education, Inc.
Slide 8.3- 48
EXAMPLE 4 Minimum Sample Size for Survey
You plan a survey to estimate the proportion of students on your
campus who carry a cell phone regularly. How many students
should be in the sample if you want (with 95% confidence) a
margin of error of no more than 4 percentage points?
Solution:
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Slide 8.3- 49