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¾Section 6.5
How Close Are Sample Means to
Population Means?
Agresti/Franklin Statistics, 1e, 1 of 19
The Sampling Distribution of the
Sample Mean
z
z
z
The sample mean, X, is a random
variable.
The sample mean varies from sample
to sample.
By contrast, the population mean, µ,
is a single fixed number.
Agresti/Franklin Statistics, 1e, 2 of 19
Mean and Standard Error of the
Sampling Distribution of the Sample
Mean
z
For a random sample of size n from a population
having mean µ and standard deviation σ, the
sampling distribution of the sample mean X has:
•
Center described by the mean µ (the same as the
mean of the population).
•
Spread described by the standard error, which
equals the population standard deviation divided by
the square root of the sample size: σ n
Agresti/Franklin Statistics, 1e, 3 of 19
Example: How Much Do Mean
Sales Vary From Week to Week?
z
Daily sales at a pizza restaurant vary
from day to day.
z
The sales figures fluctuate around a
mean µ = $900 with a standard
deviation σ = $300.
Agresti/Franklin Statistics, 1e, 4 of 19
Example: How Much Do Mean
Sales Vary From Week to Week?
z
z
z
The mean sales for the seven days in a
week are computed each week.
The weekly means are plotted over time.
These weekly means form a sampling
distribution.
Agresti/Franklin Statistics, 1e, 5 of 19
Example: How Much Do Mean
Sales Vary From Week to Week?
z
What are the center and spread (standard
error) of the sampling distribution?
μ = $900
300
σ=
= 113
7
Agresti/Franklin Statistics, 1e, 6 of 19
Sampling Distribution vs.
Population Distribution
Agresti/Franklin Statistics, 1e, 7 of 19
Standard Error
z
Knowing how to find a standard error
gives us a mechanism for
understanding how much variability
to expect in sample statistics “just by
chance.”
Agresti/Franklin Statistics, 1e, 8 of 19
Standard Error
z
The standard error of the sample mean:
σ
n
z
z
z
As the sample size n increases, the denominator
increase, so the standard error decreases.
With larger samples, the sample mean is more
likely to fall close to the population mean.
This property applies to the sampling
distribution of the sample proportion too.
Agresti/Franklin Statistics, 1e, 9 of 19
Central Limit Theorem
z
Question: How does the sampling
distribution of the sample mean relate
with respect to shape, center, and
spread to the probability distribution
from which the samples were taken?
Agresti/Franklin Statistics, 1e, 10 of 19
Central Limit Theorem
z
z
For random sampling with a large
sample size n (at least 30), the
sampling distribution of the sample
mean is approximately
a
normal
⎛ σ2 ⎞
distribution: X ~ N ⎜⎜ μ , ⎟⎟, if n ≥ 30.
n ⎠
⎝
This result applies no matter what the
shape of the probability distribution
from which the samples are taken.
Agresti/Franklin Statistics, 1e, 11 of 19
Central Limit Theorem:
How Large a Sample?
z
The sampling distribution of the sample mean takes
more of a bell shape as the random sample size n
increases.
•
•
z
z
For large n, the sampling distribution is approximately normal
even if the population distribution is not.
This enables us to make inferences about population means
regardless of the shape of the population distribution.
The more skewed the population distribution, the
larger n must be before the shape of the sampling
distribution is close to normal.
In practice, the sampling distribution is usually close
to normal when the sample size n is at least about 30.
Agresti/Franklin Statistics, 1e, 12 of 19
A Normal Population Distribution
and the Sampling Distribution
z
If the population distribution is
approximately normal, then the
sampling distribution is
approximately normal for all sample
sizes.
Agresti/Franklin Statistics, 1e, 13 of 19
Notation
z
the mean of the sample means
µx = µ
z
the standard deviation of sample mean
σ
σx = n
(often called standard error of the mean)
Agresti/Franklin Statistics, 1e, 14 of 19
Example: Given the population of men has normally distributed weights (X) with
a mean of 172 lb and a standard deviation of 29 lb,
a) if one man is randomly selected, find the probability that his weight is greater
than 167 lb. P(X>167)=?
b) if 12 different men are randomly selected, find the probability that their mean
weight is greater than 167 lb.
P(X>167)=?
Agresti/Franklin Statistics, 1e, 15 of 19
Example: Note μ=172, σ=29.
a) if one man is randomly selected, find the probability that his weight is greater
than 167 lb. P(X>167)=?
Step 1: Compute z-score of 167
z = 167 – 172 = –0.17
29
Step 2: From table A-2.
We get
P(X>167)=P(Z>-.17)
=1-.4325
=.5675
Agresti/Franklin Statistics, 1e, 16 of 19
Example:
Note μ=172, σ=29.
x -μ x
x -μ
z=
=
.
σx
σ/ n
(b) Find P(X>167)
Step 1: Note that 167 is the mean
of 12 men. So z-score of 167 is
z = 167 – 172 = –0.60
29
12
z − score for x is
Step 2: From table A-2,
P(X>167)=P(Z>-.60)
=1-.2743
=.7257
Agresti/Franklin Statistics, 1e, 17 of 19
Example: Summary:
a) if one man is randomly selected, find the probability
that his weight is greater than 167 lb.
P(x > 167) = 0.5675
b) if 12 different men are randomly selected, their mean
weight is greater than 167 lb.
P(x > 167) = 0.7257
It is much easier for an individual to deviate from the
mean than it is for a group of 12 to deviate from the mean.
Agresti/Franklin Statistics, 1e, 18 of 19
Sampling Without Replacement
If n > 0.05 N
σx =
σ
n
N–n
N–1
finite population
correction factor
Agresti/Franklin Statistics, 1e, 19 of 19