IS 310 Business Statistics CSU Long Beach

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IS 310 – Business StatisticsIS 310 – Business Statistics

IS 310

Business Statistic

sCSU

Long Beach

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Sampling and Sampling DistributionsSampling and Sampling Distributions

In many instances, one cannot study an entire population. In many instances, one cannot study an entire population. Main reasons are cost, time and effort involved in studying Main reasons are cost, time and effort involved in studying the entire population. Often, it is not even necessary to the entire population. Often, it is not even necessary to study each and every element of the population. study each and every element of the population.

Consider a manufacturing assembly line that produces Consider a manufacturing assembly line that produces thousands or millions of items of a product. To determine thousands or millions of items of a product. To determine the quality of this product, is it necessary to inspect each the quality of this product, is it necessary to inspect each item of the product? The answer is obviously no. In such item of the product? The answer is obviously no. In such a case, one selects a subset of the population, called a a case, one selects a subset of the population, called a sample, and inspects each item in the sample. Based on sample, and inspects each item in the sample. Based on the findings from the sample, one makes conclusion about the findings from the sample, one makes conclusion about the entire population.the entire population.

For example, if one finds 3 percent of the items in the For example, if one finds 3 percent of the items in the sample as defective, the conclusion is made that 3 sample as defective, the conclusion is made that 3 percent of the items in the population is defective.percent of the items in the population is defective.

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Sampling and Sampling DistributionSampling and Sampling Distribution

Consider another example. Goodyear tire Consider another example. Goodyear tire manufacturer wants to know the manufacturer wants to know the mean mean (or (or averageaverage) life of its new brand of tires. One way ) life of its new brand of tires. One way is testing and wearing out each tire is testing and wearing out each tire manufactured. Obviously, this does not make manufactured. Obviously, this does not make sense.sense.

Goodyear takes a Goodyear takes a sample of tiressample of tires, tests and wears , tests and wears out each of these tires and then calculates the out each of these tires and then calculates the mean (or average) life of the sampled tires. mean (or average) life of the sampled tires. Suppose, the mean life is calculated as 42,000 Suppose, the mean life is calculated as 42,000 miles. Based on this sample, it is concluded miles. Based on this sample, it is concluded that the mean life that the mean life all new brand of tires all new brand of tires (that is (that is population) is 42,000 miles.population) is 42,000 miles.

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Sampling and Sampling DistributionSampling and Sampling Distribution

In the previous two examples, we dealt withIn the previous two examples, we dealt with

Mean (or average) andMean (or average) and

ProportionProportion

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How to Select a SampleHow to Select a Sample

There are several methods to select a sample There are several methods to select a sample from a population. One of the most common from a population. One of the most common sampling methods is sampling methods is Simple Random Simple Random SamplingSampling. This sampling is accomplished in . This sampling is accomplished in many ways: using a random number table or many ways: using a random number table or putting all names in a hat and pulling a name putting all names in a hat and pulling a name from the hat until sample size is reached. from the hat until sample size is reached.

Refer to Table 7.1 (10-Page 261; 11-Page 269). Refer to Table 7.1 (10-Page 261; 11-Page 269). This is a Random Number Table. This is a Random Number Table.

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Simple Random Sampling:Simple Random Sampling:Finite PopulationFinite Population

Finite populationsFinite populations are often defined by lists such as: are often defined by lists such as:

• Organization membership rosterOrganization membership roster

• Credit card account numbersCredit card account numbers

• Inventory product numbersInventory product numbers

A A simple random sample of size simple random sample of size nn from a from a finitefinite

population of size population of size NN is a sample selected is a sample selected such thatsuch that

each possible sample of size each possible sample of size nn has the same has the same

probability of being selected.probability of being selected.

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Simple Random Sampling:Simple Random Sampling:Finite PopulationFinite Population

In large sampling projects, computer-generatedIn large sampling projects, computer-generated random numbersrandom numbers are often used to automate the are often used to automate the sample selection process.sample selection process.

Sampling without replacementSampling without replacement is the procedure is the procedure used most often.used most often.

Replacing each sampled element before selectingReplacing each sampled element before selecting subsequent elements is called subsequent elements is called sampling withsampling with replacementreplacement..

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Sample and Point EstimationSample and Point Estimation

Now that we know how to select a sample, Now that we know how to select a sample, let’s use the sample to estimate population let’s use the sample to estimate population characteristics (mean, and proportion). Using characteristics (mean, and proportion). Using sample data to estimate a population mean or sample data to estimate a population mean or proportion is known as proportion is known as

Point EstimationPoint Estimation

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ss is the is the point estimatorpoint estimator of the population standard of the population standard deviation deviation .. ss is the is the point estimatorpoint estimator of the population standard of the population standard deviation deviation ..

In In point estimationpoint estimation we use the data from the sample we use the data from the sample to compute a value of a sample statistic that servesto compute a value of a sample statistic that serves as an estimate of a population parameter.as an estimate of a population parameter.

In In point estimationpoint estimation we use the data from the sample we use the data from the sample to compute a value of a sample statistic that servesto compute a value of a sample statistic that serves as an estimate of a population parameter.as an estimate of a population parameter.


We refer to We refer to as the as the point estimatorpoint estimator of the population of the population mean mean .. We refer to We refer to as the as the point estimatorpoint estimator of the population of the population mean mean ..

xx

is the is the point estimatorpoint estimator of the population proportion of the population proportion pp.. is the is the point estimatorpoint estimator of the population proportion of the population proportion pp..pp

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Example Problem Example Problem

Refer to Table 7.2 (10-Page 265; 11-Page 274). Refer to Table 7.2 (10-Page 265; 11-Page 274). Using the sample data of this table, we can Using the sample data of this table, we can calculate the point estimates for population calculate the point estimates for population mean, population standard deviation and mean, population standard deviation and population proportion.population proportion.

_ __ _ x = 51,814 s = 3,348 p = 0.63x = 51,814 s = 3,348 p = 0.63

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Sampling DistributionsSampling Distributions

If we take several samples and calculate the point If we take several samples and calculate the point estimates, these estimates will be different. Each estimates, these estimates will be different. Each sample will provide a different value for:sample will provide a different value for:

_ __ _ x s and px s and p

Refer to Table 7.4 (10-Page 268; 11-Page 277).Refer to Table 7.4 (10-Page 268; 11-Page 277).

Since these values are different, they are random Since these values are different, they are random variables. They have means or expected values, variables. They have means or expected values, standard deviations and probability distributions.standard deviations and probability distributions.

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Sampling DistributionsSampling Distributions

__ If we consider the case of mean ( x ), the If we consider the case of mean ( x ), the

probability probability _ _

_ _ distribution of x is called Sampling Distribution distribution of x is called Sampling Distribution

of x.of x.

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__Sampling Distribution of x Sampling Distribution of x

__ Now that we know that x have different values, what Now that we know that x have different values, what __ are the Expected Value of x and its standard are the Expected Value of x and its standard

deviation?deviation? __ E( x ) = µ Formula 7.1 (10-Page 270; 11-Page E( x ) = µ Formula 7.1 (10-Page 270; 11-Page

279)279)

σσ = = σσ / √ n Formula 7.2 (10-Page 271; 11-Page / √ n Formula 7.2 (10-Page 271; 11-Page 280)280)

_ This is called standard error of the_ This is called standard error of the x meanx mean

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Central Limit TheoremCentral Limit Theorem

Central Limit Theorem is a very important Central Limit Theorem is a very important concept in statistics.concept in statistics.

If we select random samples of size n If we select random samples of size n from a population, the sampling from a population, the sampling distribution of the sampledistribution of the sample

__ mean (x) can be approximated by a mean (x) can be approximated by a

normal distribution as the sample size normal distribution as the sample size becomes largebecomes large

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Use of Central Limit TheoremUse of Central Limit Theorem

Problem #26 (10-Page 279; 11-Page 288)Problem #26 (10-Page 279; 11-Page 288)

Given: µ = $939 Given: µ = $939 σσ = 245 n = 30 (1 = 245 n = 30 (1stst case) case) = / √n = 245/√30 = 44.71= / √n = 245/√30 = 44.71 -- x _x _ a. P( 914 < x < 964) a. P( 914 < x < 964)

Convert 914 and 964 to z-valuesConvert 914 and 964 to z-values z = (914 – 939)/44.71 = - 0.56 z = (964 – 939)/44.71 = 0.56z = (914 – 939)/44.71 = - 0.56 z = (964 – 939)/44.71 = 0.56 __ P( 914 < x < 964) = P( -0.56 < z < 0.56) = 0.4246P( 914 < x < 964) = P( -0.56 < z < 0.56) = 0.4246

b. The probability value increases with a larger sample size.b. The probability value increases with a larger sample size.

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Differences Between Chapter 6 and Differences Between Chapter 6 and Chapter 7Chapter 7

Chapter 6:Chapter 6:

P(100 < x < 200)P(100 < x < 200) Use the following formula:Use the following formula: z = (x - µ)/z = (x - µ)/σσ

Chapter 7:Chapter 7: __ P( 100 < x < 200)P( 100 < x < 200) Use the following formula:Use the following formula: __ z = [(x - µ)/(z = [(x - µ)/(σσ/√n)]/√n)]

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Sample Problem:Sample Problem: The regular savings accounts of a large bank have The regular savings accounts of a large bank have

a mean balance of $750 (µ = 750) and a standard a mean balance of $750 (µ = 750) and a standard deviation of $120 (deviation of $120 (σσ = 120). A sample of 36 = 120). A sample of 36 accounts is selected. accounts is selected.

Find the following:Find the following: a. Probability of any single account balance beinga. Probability of any single account balance being between $720 and $780.between $720 and $780.

b. Probability of the mean of a sample of 36 b. Probability of the mean of a sample of 36 accountsaccounts

being between $720 and $780. being between $720 and $780.

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In first part of the problem, we deal with Chapter 6In first part of the problem, we deal with Chapter 6 P( 720 < x < 780) = ?P( 720 < x < 780) = ? z = (720-750)/120 z = (780-750)/120z = (720-750)/120 z = (780-750)/120 = - 0.25 = 0.25= - 0.25 = 0.25 P(-0.25 < z < 0.25) = 0.1974P(-0.25 < z < 0.25) = 0.1974

In the second part of the problem, we deal with Chapter In the second part of the problem, we deal with Chapter 77

__ P(720 < x < 780) = ?P(720 < x < 780) = ? z = (720-750)/(120/√36) z = (780-750)/(120/√36)z = (720-750)/(120/√36) z = (780-750)/(120/√36) = -1.5 = 1.5= -1.5 = 1.5 P(-1.5 < z < 1.5) = 0.8664P(-1.5 < z < 1.5) = 0.8664

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Relationship Between Sample Size and Relationship Between Sample Size and Sampling DistributionSampling Distribution

If we look at Formula 7.2 (Page 280), we know If we look at Formula 7.2 (Page 280), we know that the standard error of the mean will be that the standard error of the mean will be lower if we increase the size of the sample. lower if we increase the size of the sample. Lower the standard error of the mean, the Lower the standard error of the mean, the better is the estimate of the population mean.better is the estimate of the population mean.

Using the example of EAI managers, let’s use a Using the example of EAI managers, let’s use a sample size of 100 rather than 30. The sample size of 100 rather than 30. The standard error of the mean is reduced to 400 standard error of the mean is reduced to 400 from 730.3.from 730.3.

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Sample ProblemSample Problem

Problem # 15 (10-Page 266-267; 11-Page 276)Problem # 15 (10-Page 266-267; 11-Page 276)

a.a. Point estimate of the mean cost per treatment with HerceptinPoint estimate of the mean cost per treatment with Herceptin

__

x = (4376+4798+5578+6446+2717+4119+4920+4237+4495+3814) / 10 = 4550x = (4376+4798+5578+6446+2717+4119+4920+4237+4495+3814) / 10 = 4550

b.b. Point estimate of the standard deviation of the cost per treatment with HerceptinPoint estimate of the standard deviation of the cost per treatment with Herceptin

Cost Per Sample Mean Deviation Squared DeviationCost Per Sample Mean Deviation Squared Deviation

Treatment from Mean from MeanTreatment from Mean from Mean

4376 4550 - 174 30,2764376 4550 - 174 30,276

4798 4550 248 61,5044798 4550 248 61,504

5578 4550 1028 1,056,7845578 4550 1028 1,056,784

6446 4550 1896 3,594,816 26446 4550 1896 3,594,816 2

2717 4550 - 1833 3,359,889 S = 9,068,620 / 2717 4550 - 1833 3,359,889 S = 9,068,620 / (10-1)(10-1)

4119 4550 - 431 185,761 = 1,007,624.444119 4550 - 431 185,761 = 1,007,624.44

4920 4550 370 136,900 S = 1003.8054920 4550 370 136,900 S = 1003.805

4237 4550 - 313 97,9694237 4550 - 313 97,969

4495 4550 - 55 3,0254495 4550 - 55 3,025

3814 4550 - 736 541,6963814 4550 - 736 541,696

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More Sample ProblemMore Sample Problem

Problem # 16 (10-Page 267; 11-Page 276)Problem # 16 (10-Page 267; 11-Page 276)

Given: n = 50Given: n = 50

a.a. Estimate of the proportion of Fortune 500 Estimate of the proportion of Fortune 500 companies based in NY = 5/50 = 0.1 or 10 companies based in NY = 5/50 = 0.1 or 10 percentpercent

c. Estimate of the proportion of Fortune 500 c. Estimate of the proportion of Fortune 500 companies not based in NY, CA, MN or WI = companies not based in NY, CA, MN or WI = 36/50 = 0.72 or 72 percent36/50 = 0.72 or 72 percent

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Other Sampling MethodsOther Sampling Methods

Stratified Random SamplingStratified Random Sampling Population is divided into groups, called strata. Population is divided into groups, called strata. Samples are selected from each strata. Useful in Samples are selected from each strata. Useful in applications where populations are diverse.applications where populations are diverse. Examples are household incomes.Examples are household incomes.

Cluster SamplingCluster Sampling If population is spread over a large geographicalIf population is spread over a large geographical area, cluster sampling is ideal. Think about area, cluster sampling is ideal. Think about universities in the US. If we want to select a universities in the US. If we want to select a sample from all universities, Cluster Sampling can besample from all universities, Cluster Sampling can be employed. employed.

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Other Sampling MethodsOther Sampling Methods

Systematic SamplingSystematic Sampling If we select every nth element from a population, we are If we select every nth element from a population, we are

using Systematic Sampling. Useful in assembly line where using Systematic Sampling. Useful in assembly line where every 10every 10thth or 15 or 15thth element can be chosen to make a sample. element can be chosen to make a sample.

Convenience SamplingConvenience Sampling When we select a sample mainly for convenience reasons, When we select a sample mainly for convenience reasons,

we are using Convenience Sampling. Think about a we are using Convenience Sampling. Think about a professor who chooses students in a study to form a professor who chooses students in a study to form a sample.sample.

Judgment SamplingJudgment Sampling When an expert selects a sample using his judgment, this is When an expert selects a sample using his judgment, this is

known as Judgment Sampling.known as Judgment Sampling.

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End of Chapter 7, Part AEnd of Chapter 7, Part A

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IS 310 Business Statistics CSU Long Beach