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Lecture 5: Sampling Methods and the Central Limit Theorem

North South University School of BusinessSlide 1 of 29

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


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


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


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:


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.


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.


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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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)


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.


EXAMPLESuppose you want to determine the views


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.


sample.

Sampling Error

The sampling error is the differencebetween a sample statistic and itscorresponding population parameter.


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


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?


Example 1 (contd)


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.


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


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


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


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.


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.


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


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


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


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


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.


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.


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)


Problems 15, 17, 18 (Page 274) (Page 281)

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