Download distribution of sample means

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.
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8.1-1
Sample Means: The Basic Idea
Consider the weights of the five
starting players on a professional
basketball team. We regard these five
players as the entire population (with
a mean of 242.4 pounds).
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.
Q:What is the mean of the
distribution of sample means?
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With a sample size of 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.
Q: What is the mean of the
distribution of sample means with
sample size n=2?
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The mean of the distribution of sample
means is equal to the population mean,
242.4 pounds.
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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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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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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.
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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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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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Notation for Population and Sample Means
n = sample size
m = population mean
x¯ = sample mean
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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.
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EXAMPLE 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?
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Notation for Population and Sample Proportions
n = sample size
p = population proportion
p
ˆ = sample proportion
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EXAMPLE: Sample and population Proportions.
The college of Los Angeles had 2,444 students and 269 of
them are left-handed. You conduct a survey of 50 students
and find that 8 of them are left-handed.
a . What is the population proportion of left-handed
students?
b. What is the sample proportion of left-handed students?
c. Does your sample appear to be representative of the
college?
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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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