### CHAPTER 6 CONTINUOUS PROBABILITY DISTRIBUTIONS

```IS 310
Statistics
CSU
Long Beach
Slide 1
Sampling and Sampling Distributions
In many instances, one cannot study an entire population. Main
reasons are cost, time and effort involved in studying the entire
population. Often, it is not even necessary to study each and
every element of the population.
Consider a manufacturing assembly line that produces thousands
or millions of items of a product. To determine the quality of
this product, is it necessary to inspect each item of the product?
The answer is obviously no. In such a case, one selects a subset
of the population, called a sample, and inspects each item in the
sample. Based on the findings from the sample, one makes
For example, if one finds 3 percent of the items in the sample as
defective, the conclusion is made that 3 percent of the items in
the population is defective.
Slide 2
Sampling and Sampling Distribution
Consider another example. Goodyear tire manufacturer
wants to know the mean (or average) life of its new
brand of tires. One way is testing and wearing out
each tire manufactured. Obviously, this does not
make sense.
Goodyear takes a sample of tires, tests and wears out
each of these tires and then calculates the mean (or
average) life of the sampled tires. Suppose, the mean
life is calculated as 42,000 miles. Based on this
sample, it is concluded that the mean life all new
brand of tires (that is population) is 42,000 miles.
Slide 3
Sampling and Sampling Distribution
In the previous two examples, we dealt with
Mean (or average) and
Proportion
Slide 4
How to Select a Sample
There are several methods to select a sample from a
population. One of the most common sampling
methods is Simple Random Sampling. This sampling
is accomplished in many ways: using a random
number table or putting all names in a hat and
pulling a name from the hat until sample size is
reached.
Refer to Table 7.1 (10-Page 261; 11-Page 269). This is a
Random Number Table.
Slide 5
Simple Random Sampling:
Finite Population

Finite populations are often defined by lists such as:
• Organization membership roster
• Credit card account numbers
• Inventory product numbers

A simple random sample of size n from a finite
population of size N is a sample selected such that
each possible sample of size n has the same
probability of being selected.
Slide 6
Simple Random Sampling:
Finite Population
 Replacing each sampled element before selecting
subsequent elements is called sampling with
replacement.
 Sampling without replacement is the procedure
used most often.
 In large sampling projects, computer-generated
random numbers are often used to automate the
sample selection process.
Slide 7
Sample and Point Estimation

Now that we know how to select a sample, let’s use
the sample to estimate population characteristics
(mean, and proportion). Using sample data to
estimate a population mean or proportion is known
as
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Point Estimation
Slide 8
Point Estimation
In point estimation we use the data from the sample
to compute a value of a sample statistic that serves
as an estimate of a population parameter.
We refer to x as the point estimator of the population
mean .
s is the point estimator of the population standard
deviation .
p is
the point estimator of the population proportion p.
Slide 9
Point Estimation
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Example Problem

Refer to Table 7.2 (10-Page 265; 11-Page 274). Using
the sample data of this table, we can calculate the
point estimates for population mean, population
standard deviation and population proportion.
_
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x = 51,814
s = 3,348
p = 0.63
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Slide 10
Sampling Distributions
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If we take several samples and calculate the point
estimates, these estimates will be different. Each
sample will provide a different value for:
_
_
x
s
and p
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Refer to Table 7.4 (10-Page 268; 11-Page 277).

Since these values are different, they are random
variables. They have means or expected values,
standard deviations and probability distributions.
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Slide 11
Sampling Distributions
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If we consider the case of mean ( x ), the probability
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distribution of x is called Sampling Distribution of x.
Slide 12
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Sampling Distribution of x
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_
Now that we know that x have different values, what
_
are the Expected Value of x and its standard
deviation?
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E( x ) = µ
Formula 7.1 (10-Page 270; 11-Page 279)
σ =σ/√n
_
x
Formula 7.2 (10-Page 271; 11-Page 280)
This is called standard error of the
mean
Slide 13
Central Limit Theorem

Central Limit Theorem is a very important concept in
statistics.

If we select random samples of size n from a
population, the sampling distribution of the sample
_
mean (x) can be approximated by a normal
distribution as the sample size becomes large
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Slide 14
Use of Central Limit Theorem

Problem #26 (10-Page 279; 11-Page 288)

Given: µ = \$939
σ = 245
n = 30 (1st case)
= / √n = 245/√30 = 44.71
x
_
a. P( 914 < x < 964)
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Convert 914 and 964 to z-values
z = (914 – 939)/44.71 = - 0.56 z = (964 – 939)/44.71 = 0.56
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P( 914 < x < 964) = P( -0.56 < z < 0.56) = 0.4246
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b. The probability value increases with a larger sample size.
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Slide 15
Differences Between Chapter 6 and Chapter 7
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Chapter 6:

P(100 < x < 200)
Use the following formula:
z = (x - µ)/σ
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Chapter 7:
_
P( 100 < x < 200)
Use the following formula:
_
z = [(x - µ)/(σ/√n)]
Slide 16
Differences Between Chapter 6 and Chapter 7
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Sample Problem:
The regular savings accounts of a large bank have a
mean balance of \$750 (µ = 750) and a standard
deviation of \$120 (σ = 120). A sample of 36 accounts
is selected.
Find the following:
a. Probability of any single account balance being
between \$720 and \$780.
b. Probability of the mean of a sample of 36 accounts
being between \$720 and \$780.
Slide 17
Differences Between Chapter 6 and Chapter 7
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In first part of the problem, we deal with Chapter 6
P( 720 < x < 780) = ?
z = (720-750)/120
z = (780-750)/120
= - 0.25
= 0.25
P(-0.25 < z < 0.25) = 0.1974
In the second part of the problem, we deal with Chapter 7
_
P(720 < x < 780) = ?
z = (720-750)/(120/√36) z = (780-750)/(120/√36)
= -1.5
= 1.5
P(-1.5 < z < 1.5) = 0.8664
Slide 18
Relationship Between Sample Size and
Sampling Distribution

If we look at Formula 7.2 (Page 280), we know that
the standard error of the mean will be lower if we
increase the size of the sample. Lower the standard
error of the mean, the better is the estimate of the
population mean.

Using the example of EAI managers, let’s use a
sample size of 100 rather than 30. The standard error
of the mean is reduced to 400 from 730.3.
Slide 19
Sample Problem
Problem # 15 (10-Page 266-267; 11-Page 276)
a.
Point estimate of the mean cost per treatment with Herceptin
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x = (4376+4798+5578+6446+2717+4119+4920+4237+4495+3814) / 10 = 4550
b.
Point estimate of the standard deviation of the cost per treatment with Herceptin
Cost Per
Sample Mean Deviation
Squared Deviation
Treatment
from Mean from Mean
4376
4550
- 174
30,276
4798
4550
248
61,504
5578
4550
1028
1,056,784
6446
4550
1896
3,594,816
2
2717
4550
- 1833
3,359,889
S = 9,068,620 / (10-1)
4119
4550
- 431
185,761
= 1,007,624.44
4920
4550
370
136,900
S = 1003.805
4237
4550
- 313
97,969
4495
4550
- 55
3,025
3814
4550
- 736
541,696
Slide 20
More Sample Problem
Problem # 16 (10-Page 267; 11-Page 276)
Given: n = 50
a. Estimate of the proportion of Fortune 500 companies
based in NY = 5/50 = 0.1 or 10 percent
c.
Estimate of the proportion of Fortune 500
companies not based in NY, CA, MN or WI = 36/50
= 0.72 or 72 percent
Slide 21
Other Sampling Methods
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Stratified Random Sampling
Population is divided into groups, called strata.
Samples are selected from each strata. Useful in
applications where populations are diverse.
Examples are household incomes.
Cluster Sampling
If population is spread over a large geographical
area, cluster sampling is ideal. Think about
universities in the US. If we want to select a
sample from all universities, Cluster Sampling can be
employed.
Slide 22
Other Sampling Methods
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Systematic Sampling
If we select every nth element from a population, we are using
Systematic Sampling. Useful in assembly line where every 10th
or 15th element can be chosen to make a sample.
Convenience Sampling
When we select a sample mainly for convenience reasons, we
are using Convenience Sampling. Think about a professor who
chooses students in a study to form a sample.
Judgment Sampling
When an expert selects a sample using his judgment, this is
known as Judgment Sampling.