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3/5/2015
1
Lecture 5: Sampling Methods and the Central Limit Theorem
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Outline
Explain why a sample is the only feasible way to learnabout a population
Describe methods to select a sample Define and construct a sampling distribution of the
sample mean
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sample mean
Explain the central limit theorem
Use the Central Limit Theorem to find probabilities ofselecting possible sample means from a specifiedpopulation
Why Sample the Population?
The physical impossibility of checking all items in the
population.
The cost of studying all the
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The destructive nature of certain
tests.
The cost of studying all the items in a population.
The adequacy of sample results in most
cases.
The time-consuming aspect of contacting the whole
population.
Major Sampling Types
Probability Sampling Non-probability Sampling
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Probability Sampling
A probability sample is a sample selected such that each
item or person in the population being studied has a
known likelihood of being included in the sample
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included in the sample.
Four Most Commonly Used Probability Sampling Methods1.Simple Random Sampling 2.Systematic Random Sampling3.Stratified Random Sampling4.Cluster Sampling
Probability Sampling Methods
Simple Random Sample A sample formulated so that each item or person in the population has the same chance of being included.
EXAMPLE:
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A population consists of 845 employees of Nitra Industries. A sample of 52 employees is to be selected from that population. The name of each
employee is written on a small slip of paper and deposited all of the slips in a box. After they have been thoroughly mixed, the first selection is made by drawing a slip out of the box without looking at it. This process is repeated
until the sample of 52 employees is chosen.
Simple Random Sample: Using Table of Random Numbers
A population consists of 845 employees of Nitra Industries. A sample of 52 employees is to be selected from that population.
A more convenient method of selecting a random sample is to use the identification number of each employee and a table of random numbers such as the one in Appendix E.
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Probability Sampling Methods (contd)
Systematic Random Sampling The items or individuals of the population are arranged in some order. A random starting point
is selected and then every kth member of the population is selected for the sample.
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EXAMPLEA population consists of 845 employees of Nitra Industries.
A sample of 52 employees is to be selected from that population. First, k is calculated as the population size divided by the sample size. For Nitra Industries,
we would select every 16th (845/52) employee list. If k is not a whole number, then round down. Random sampling is used in the selection of the first name. Then,
select every 16th name on the list thereafter.
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Probability Sampling Methods (contd)
Stratified Random Sampling: A population is first divided into subgroups, called strata, and a
sample is selected from each stratum.
EXAMPLESuppose we want to study the advertising expenditures for the 352 largest companies in the
United States to determine whether firms with high returns on equity (a measure of profitability)
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spent more of each sales dollar on advertising than firms with a low return or deficit.
To make sure that the sample is a fair representation of the 352 companies, the companies are grouped on percent return on equity and a sample proportional to the
relative size of the group is randomly selected.
Cluster Sampling: A population is divided into clusters using naturally occurring geographic or other boundaries. Then,
clusters are randomly selected and a sample is collected by randomly selecting from each cluster.
Probability Sampling Methods (contd)
EXAMPLESuppose you want to determine the views
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of residents in Oregon about state and federal environmental protection policies.
Cluster sampling can be used by subdividing the state into small units
either counties or regions, select at random say 4 regions, then take samples of the residents in each of these regions and interview them. (Note that this is a combination of cluster sampling and
simple random sampling.)
Non-Probability Sampling
In non-probability sampling, inclusion in the sample is based on the judgment of the person selecting the
sample.
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sample.
Sampling Error
The sampling error is the differencebetween a sample statistic and itscorresponding population parameter.
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For example, X
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Sampling Distribution of the Sample Mean
The sampling distribution of the sample mean is a probability distribution consisting of all possible sample means of a given sample size selected
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from a population.
Example 1
Tartus Industries has seven production employees (considered the population). Thehourly earnings of each employee are given in the table below.1. What is the population mean?2. What is the sampling distribution of the sample mean for samples of size 2?3. What is the mean of the sampling distribution?4. What observations can be made about the population and the samplingdistribution?
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Example 1 (contd)
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Example 1 (contd)
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Example 2
Partner Hours
Dunn 22
Hardy 26
Kiers 30
The law firm of Hoya and Associates has five
partners. At their weekly partners meeting each reported the number of hours they billed clients
for their services last week.
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Malory 26
Tillman 22
If two partners are selected randomly, how many different
samples are possible?
Example 2 (contd)
10)!25(!2
!525 C
5 objects taken 2 at a
time.
A total of 10 different samples
Partners Total Mean 1,2 48 24 1,3 52 26 1,4 48 24
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1,4 48 24 1,5 44 22 2,3 56 28 2,4 52 26 2,5 48 24 3,4 56 28 3,5 52 26 4,5 48 24
Example 1 continued
Sample Mean Frequency Relative Frequency probability
22 1 1/10
As a sampling distribution
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24 4 4/10
26 3 3/10
28 2 2/10
Example 2 (contd)
)2(28)3(26)2(24)1(22
Compute the mean of the sample means. Compare it with the population mean.
The mean of the sample means Notice that the mean of the sample means is exactly equal to the
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2.2510
)2(28)3(26)2(24)1(22 X
The population mean
2.255
2226302622
is exactly equal to the population mean.
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Central Limit Theorem
If the population follows a normal probability distribution, then for any sample size the sampling distribution of the sample mean will also be normal.
CENTRAL LIMIT THEOREM If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal
distribution. This approximation improves with larger samples.
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If the population distribution is symmetrical (but not normal), shape of the distribution of the sample mean will emerge as normal with samples as small as 10.
If a distribution that is skewed or has thick tails, it may require samples of 30 or more to observe the normality feature.
Central Limit Theorem (contd)
The mean of the sampling distribution equal to and the variance equal to
2/n.
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n x =
The standard error of the mean is the standard deviation of the sample means given as:
Using the SamplingDistribution of the Sample Mean
Sampling distribution of the sample mean follows normaldistribution:1. If population is normal ( Sample size is not a factor)2. If population shape is unknown or non-normal, but sample
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IF SIGMA IS KNOWN
nXz
IF SIGMA IS UNKNOWN
nsXz
contains at least 30 observations
The Quality Assurance Department for Cola, Inc., maintainsrecords regarding the amount of cola in its Jumbo bottle. Theactual amount of cola in each bottle is critical, but varies a smallamount from one bottle to the next. Cola, Inc., does not wish tounder fill the bottles. On the other hand, it cannot overfill eachbottle. Its records indicate that the amount of cola follows thenormal probability distribution. The mean amount per bottle is31.2 ounces and the population standard deviation is 0.4ounces
Example 3
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ounces.
At 8 A.M. today the quality technician randomly selected 16 bottlesfrom the filling line. The mean amount of cola contained in thebottles is 31.38 ounces.
Is this an unlikely result? Is it likely the process is putting too muchsoda in the bottles? To put it another way, is the sampling errorof 0.18 ounces unusual?
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Example 3 (contd)
Step 1: Find the z-values corresponding to the sample mean of 31.38
Step 2: Find the probability of observing a Z equal to or greater than 1.80
80.1164.0$
20.3138.31 n
Xz
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Conclusion: It is unlikely, less than a 4 percent chance, we could select a sample of 16 observations from a normal population with a mean of 31.2 ounces and a population standard deviation of 0.4 ounces and find the sample mean equal to or greater than
31.38 ounces. The process is putting too much cola in the bottles.
Example 4Suppose the mean selling
price of a gallon of gasoline in the United States is $1.30.
Further, assume the distribution is positively skewed, with a standard
deviation of $0 28 What is
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deviation of $0.28. What is the probability of selecting a
sample of 35 gasoline stations and finding the sample mean
within $.08?
Example 4 (contd)
69.130.1$38.1$ Xz
Step One : Find the z-values corresponding to $1.22 and $1.38. These are the two points within
$0.08 of the population mean.
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69.13528.0$ns
z
69.13528.0$
30.1$22.1$ ns
Xz
Example 4 (contd)
9090.)4545(.2)69.169.1( zP
Step Two: determine the probability of a z-value between -1.69 and 1.69.
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We would expect about 91 percent of the sample means
to be within $0.08 of the population mean.
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Assignment-5
Problem 3 (Page 257) (Page 265) Problems 5, 7, 9 (Page 262) (Page 270)
Problems 11, 13 (Pages 269-270) (Pages 277-278)P bl 15 17 18 (P 274) (P 281)
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Problems 15, 17, 18 (Page 274) (Page 281)