Download Sampling Distribution of Sample Mean

Sampling Distribution of Sample Mean


The sampling distribution of the Sample
Mean, X, is the distribution of the sample mean
values for all possible samples of the same size
from the same population.
If the parent population IS N(μ, σ),
then for any sample size (small or large),
the distribution:
Central Limit Theorem (CLT)


The sampling distribution of the Sample Mean,
, is the distribution
of the sample mean values for
X
all possible samples of the same size from the
same population.
If the parent population IS NOT N(μ, σ),
then for n>30 the distribution is
approximately:
Example: Research on
eating disorder

Bulimia is an illness in which a person
binges on food or has regular episodes
of significant overeating and feels a
loss of control. Usually is observed in
young females.
Research on eating disorder

During the American Statistician (May 2001) study of
female students who suffer from bulimia, each student
completed a questionnaire from which a “fear of negative
evaluation” (FNE) score was produced. Suppose the FNE
scores of bulimic students have a distribution with mean
18 and standard deviation of 5.

What is the probability that a randomly selected bulimic
female has a greater than 15?
Research on eating disorder

During the American Statistician (May 2001) study of
female students who suffer from bulimia, each student
completed a questionnaire from which a “fear of negative
evaluation” (FNE) score was produced. Suppose the FNE
scores of bulimic students have a distribution with mean
18 and standard deviation of 5.

What is the probability that the sample mean FNE score
(of 100) is greater than 15?
Research on eating disorder

During the American Statistician (May 2001) study of
female students who suffer from bulimia, each student
completed a questionnaire from which a “fear of negative
evaluation” (FNE) score was produced. Suppose the FNE
scores of bulimic students have a distribution with mean
18 and standard deviation of 5.

What is the probability that the sample mean FNE score
(of 25) is greater than 15?
Example: Body fat of American
Men

The percentage of fat (not the same
as body mass index) in the bodies of
American men is an approx. normal
with mean of 15% and std of 5%. If
these values were used to describe the
body fat of U.S. Army men, then a
measure of 20% or more body fat
would characterize a soldier as obese.
Body fat of American Men

The percentage of fat of American men is an
approx. normal (mean=15, std =5) Measure of 20%
or more body fat would characterize a U.S. Army
soldier as obese.

What is the probability that a randomly chosen
soldier would be considered obese, i.e., the
percentage of body fat of a randomly chosen soldier
in the U.S. Army is 20% or higher?
Body fat of American Men

The percentage of fat of American men is an
approx. normal (mean=15, std =5) Measure of 20%
or more body fat would characterize a U.S. Army
soldier as obese.

What is the probability that an average percentage
of body fat of a randomly chosen 100 soldiers
would be 20% or higher?
Body fat of American Men

The percentage of fat of American men is an
approx. normal (mean=15, std =5) Measure of 20%
or more body fat would characterize a U.S. Army
soldier as obese.

Given that the distribution of the percentage of the
body fat of U.S. Army soldiers has the mean of
13% and the standard deviation of 3% (not
necessarily normal), what is the probability that an
average percentage of body fat of a randomly
chosen 100 soldiers would be 20% or higher?
Confidence Interval Estimation

Confidence Interval estimates a range of values
for the population parameter with a pre-defined
level of confidence (e.g., 95% confidence
interval).


A value not in a CI can be rejected as possible value of
the population parameter
A value in a CI is an “acceptable” or “reasonable”
possibility for the value of a population parameter
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is
Normal(μ=35 minutes, σ=9.5 minutes).

Using the Empirical Rule, define the lower and the upper
bounds of the interval within which falls 95% of the most
typical data values.
Empirical Rule
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is
Normal(μ=35 minutes, σ=9.5 minutes).

The lower and the upper bounds of the interval within
which falls 95% of the most typical data values.

So we expect that 95% of the times patients will have to
wait at ER during weekends 16 to 54 minutes.
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is
Normal(μ=35 minutes, σ=9.5 minutes).

Describe the sampling distribution of a sample
average waiting time based on a random sample of
n=100 patients who came during the last weekend.
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is a ~
Normal(μ=35 minutes, σ=9.5 minutes).

Describe the sampling distribution of a sample average
waiting time based on a random sample of n=100 patients
who came during the last weekend.
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is a ~
Normal(μ=35 minutes, σ=9.5 minutes).

Describe the sampling distribution of a sample average
waiting time based on a random sample of n=100 patients
who came during the last weekend.
Using CLT (or the Sampling distribution rule) for
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is a ~
Normal(μ=35 minutes, σ=9.5 minutes).

Using the Empirical Rule, identify the lower and the upper
bounds of the interval within which falls 95% of the most
typical values for a sample mean (in a random sample of 100
patients).
Example: Estimating Mean Waiting Time
in the Emergency Room.


Waiting time in the ER during weekends is a ~
Normal(μ=35 minutes, σ=9.5 minutes).
, we expect that 95% of the
times patients will have to wait
The loweratand
upperweekends
bounds of the
interval
within which
ERthe
during
33.1
to
on average.
falls 95%36.9
of theminutes,
most typical
values for a sample mean (in a
random sample of 100 patients).
We expect that 95% of the times patients will have to wait at ER
during weekends 33.1 to 36.9 minutes, on average.
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is a ~
Normal(μ=35 minutes, σ=9.5 minutes).

Based on two independent random samples of 100 patients
we computed two values for sample mean 37.85 minutes and
33.75 minutes. Which one is better? Why?
Example: Estimating Mean Waiting Time
in the Emergency Room.


Waiting time in the ER during weekends is a ~
Normal(μ=35 minutes, σ=9.5 minutes).
Now assume that true population μ is unknown. How can we
estimate μ?
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is a ~
Normal(μ=35 minutes, σ=9.5 minutes).

But if the true population mean is unknown, then there is no
way to say which estimate is better (more accurate). In terms
of the problem, people would like to believe that the average
waiting time in ER on weekends is actually closer to 33.75
minutes than 37.85 minutes.
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is
Normal(μ=35 minutes, σ=9.5 minutes).

Assume that the true population mean is unknown,
but the std is known σ=9.5.

With a random sample of 100 patients, we estimated
the sample mean 33.75 minutes.

What would a rough guess for the lower and the
upper bounds of the interval within which the true
population mean falls with the 95% of confidence?
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is
Normal(μ=35 minutes, σ=9.5 minutes).

In the part c) we found that 95% of the times patients
have to wait at ER during weekends 33.1 to 36.9 minutes,
on average, or

or
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is
Normal(μ=35 minutes, σ=9.5 minutes).

We can show that
Is the same as
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is
Normal(μ=35 minutes, σ=9.5 minutes).

Then our approximate 95% confidence interval for
the true population mean μ is:

Does the 95% Confidence Interval contain the true
population mean of 35?
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is
Normal(μ=35 minutes, σ=9.5 minutes).

The approximate 95% confidence interval for the true
population mean μ (based on the sample mean of
37.85):

Does the 95% Confidence Interval contain the true
population mean of 35?