Your favorite professional football team (I shall refer to them as the

1
Sample Sample Of size 2 Sample Of size 3
1 A,B3,1 2 A,B,C3,1,5 3
2 A,C3,5 4 A,B,D3,1,6 3.33
3 A,D3,6 4.5 A,B,E3,1,2 2
4 A,E3,2 2.5 A,C,D3,5,6 4.67
5 B,C1,5 3 A,C,E3,5,2 3.33
6 B,D1,6 3.5 A,D,E3,6,2 3.67
7 B,E1,2 1.5 B,C,D1,5,6 4
8 C,D5,6 5.5 B,C,E1,5,2 2.67
9 C,E5,2 3.5 B,D,E1,6,2 3
10 D,E6,2 4 C,D,E5,6,2 4.33
2
Sample Sample Of size 2 p Sample Proportion Sample Of size 3 p Sample Proportion
1 A,B3,1 0 A,B,C3,1,5 0
2 A,C3,5 0 A,B,D3,1,6 1/3
3 A,D3,6 0.5 A,B,E3,1,2 1/3
4 A,E3,2 0.5 A,C,D3,5,6 1/3
5 B,C1,5 0 A,C,E3,5,2 1/3
6 B,D1,6 0.5 A,D,E3,6,2 2/3
7 B,E1,2 0.5 B,C,D1,5,6 1/3
8 C,D5,6 0.5 B,C,E1,5,2 1/3
9 C,E5,2 0.5 B,D,E1,6,2 2/3
10 D,E6,2 1 C,D,E5,6,2 2/3
3
An importer of Herbs and Spices claims that
average weight of packets of Saffron is 20 grams.
However packets are actually filled to an average
weight, µ19.5 grams and standard deviation,
s1.8 gram. A random sample of 36 packets is
selected, calculate A- The probability that the
average weight is 20 grams or more B- The two
limits within which 95 of all packets weight C-
The two limits within which 95 of all weights
fall (n36) D- If the size of the random sample
was 16 instead of 36 how would this affect the
results in (a), (b) and (c)? (State any
assumptions made)
4
THE DISTRIBUTION OF THE SAMPLE MEAN CENTRAL LIMIT
THEOREM A-) FROM ANY POPULATION If
x1,x2,,xn is a random variable of size n
taken from any distribution with mean µ and
variance s2 then, for large n, the distribution
of the sample mean x is approximately normal and
x N(µ, s2/n), where x (x1x2xn )/n
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THE DISTRIBUTION OF THE SAMPLE MEAN CENTRAL LIMIT
THEOREM B-) FROM A NORMAL POPULATION If
x1,x2,,xn is a random variable of size n
taken from a normal distribution with mean µ and
variance s2 such that X N(µ, s2), then the
distribution of x is also normal and x N(µ,
s2/n), where x (x1x2xn )/n The
distribution of the sample mean (x) is known as
the sampling distribution of means and the
standard deviation of this distribution s/ vn
is known as the standard error of the mean
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Example for Correction Factor What is the value
of the finite population correction factor
when a-) n 20 and N200 ? b-) n 20 and N
2000 ?
7
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 x (where x 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?
8
Example 1 A random variable of size 15 is taken
from normal distribution with mean 60 and
standard deviation 4. Find the probability that
the mean of the sample is less than 58.
9
Example 3 If a random sample of size 30 is
taken from binomial distribution with n9 and p
0.5 Q Find the probability that the sample mean
exceeds 5.
10
Example 4 Suppose we have selected a random
sample of n36 observations from a population
with mean equal to 80 and standard deviation
equal to 6. Q Find the probability that x
will be larger than 82.
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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?
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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).
13
Example 6 ELECTION A political pollster is
conducting an analysis of sample results in order
to make predictions on election night. Assuming a
two-candidate election, if a specific candidate
receives at least 55 of the vote in the sample,
that candidate will be forecast as the winner of
the election. If you select a random sample of
100 voters, what is the probability that a
candidate will be forecast as the winner when A-)
the population percentage of her vote is 50.1
? B-) the population percentage of her vote is
60 ? C-) the population percentage of her vote
is 49 (and she will actually lose the
election)? D-) If the sample size is increased to
400, what are your answers to (A) through (C) ?
Discuss.
14
Types of Survey Errors

• Coverage error
• Non response error
• Sampling error
• Measurement error

Excluded from frame
Follow up on nonresponses
Random differences from sample to sample
Bad or leading question

15
Sampling Distribution
Standard Normal Distribution
Population Distribution
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Sample
Standardize
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Z
X
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Sampling Distribution Properties

• As n increases,
• decreases

Larger sample size
Smaller sample size
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Sampling Distribution Properties
Normal Population Distribution

• (i.e. is unbiased )

Normal Sampling Distribution (has the same mean)
Variation
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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