Transcript Samples The means of these samples

Samples
1 and 3
1 and 5
1 and 7
1 and 9
3 and 5
3 and 7
3 and 9
5 and 7
5 and 9
7 and 9
The means of these samples
2
3
4
5
4
5
6
6
7
8
Example
Random samples of size n =2 are drawn from a finite
population that consists of the numbers 2, 4, 6 and 8
without replacement.
a-) Calculate the mean and the standard deviation of this
population
b-) List six possible random samples of size n=2 that can be
drawn from this population and calculate their means.
c-) Use the results in b-) to construct the sampling
distribution of the mean.
d-) Calculate the standard deviation of the sampling
distribution.
Example for Correction Factor:
What is the value of the finite population correction
factor when
a-) n= 20 and N=200 ?
b-) n= 20 and N= 2000 ?
Example 2: Tuition Cost
The mean tuition cost at state universities throughout the USA is 4,260 USD per
year (2002 year figures). Use this value as the population mean and assume that
the population standard deviation is 900 USD. Suppose that a random sample of
50 state universities will be selected.
A-) Show the sampling distribution of x̄ (where x̄ is the sample mean tuition cost
for the 50 state universities)
B-) What is the probability that the random sample will provide a sample mean
within 250 USD of the population mean?
C-) What is the probability that the simple random sample will provide a sample
mean within 100 USD of the population mean?
Example 4:
Suppose we have selected a random sample of n=36 observations from a population with
mean equal to 80 and standard deviation equal to 6.
Q: Find the probability that x̄ will be larger than 82.
Example 5: Ping-Pong Balls
The diameter of a brand of Ping-Pong balls is approximately normally distributed, with a
mean of 1.30 inches and a standard deviation of 0.04 inch. If you select a random sample
of 16 Ping-Pong balls,
A-) What is the sampling distribution of the sample mean?
B-) What is the probability that sample mean is less than 1.28 inches?
C-) What is the probability that sample mean is between 1.31 and 1.33 inches?
D-) The probability is 60% that sample mean will be between what two values,
symmetrically distributed around the population mean?
Example 6: E-Mails
Time spent using e-mail per session is normally distributed, with a mean of 8 minutes
and a standard deviation of 2 minutes. If you select a random sample of 25 sessions,
A-) What is the probability that sample mean is between 7.8 and 8.2 minutes?
B-) What is the probability that sample mean is between 7.5 and 8.0 minutes?
C-) If you select a random sample of 100 sessions, what is the probability that sample
mean is between 7.8 and 8.2 minutes?
D-) Explain the difference in the results of (A) and (C).
Types of Survey Errors
• Coverage error
Excluded from frame
• Non response error
• Sampling error
• Measurement error
Follow up on
nonresponses
Random differences from
sample to sample
Bad or leading question
Population
Distribution
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Sampling
Distribution
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Sample
?
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X
Standard Normal
Distribution
Standardize
x
Z
Sampling Distribution Properties
Larger sample
size
As n increases,
σx
decreases
Smaller sample
size
μ
x
Sampling Distribution
Properties
μx  μ
(i.e.
Normal Population
Distribution
x is unbiased )
Variation:
σ
σx 
n
μ
x
μx
x
Normal Sampling
Distribution
(has the same mean)
How Large is Large Enough?
• For most distributions, n ≥ 30 will give a
sampling distribution that is nearly normal
• For fairly symmetric distributions, n ≥ 15
• For normal population distributions, the
sampling distribution of the mean is always
normally distributed
Exercise - 1
A package-filling process at a Cement company fills bags of cement
to an average weight of µ but µ changes from time to time. The
standard deviation is σ = 3 pounds. A sample of 25 bags has been
taken and their mean was found to be 150 pounds.
Assume that the weights of the bags are normally distributed.
Find the 90% confidence limits for µ.
Exercise - 3
An economist is interested in studying the incomes of consumers in
a particular region. The population standard deviation is known to
be $1,000. A random sample of 50 individuals resulted in an
average income of $15,000.
What is the upper end point in a 99% confidence interval for the
average income?
Exercise - 4
An economist is interested in studying the incomes of consumers in
a particular region. The population standard deviation is known to
be $1,000. A random sample of 50 individuals resulted in an
average income of $15,000.
What is the width of the 90% confidence interval?
Exercise - 5
The head librarian at the Library of Congress has asked her assistant
for an interval estimate of the mean number of books checked out
each day. The assistant provides the following interval estimate:
from 740 to 920 books per day. If the head librarian knows that the
population standard deviation is 150 books checked out per day,
and she asked her assistant for a 95% confidence interval,
approximately how large a sample did her assistant use to
determine the interval estimate?