CHAPTER 6 CONTINUOUS PROBABILITY DISTRIBUTIONS

Slides Prepared byJOHN S. LOUCKSSt. Edwards University 2002 South-Western/Thomson Learning # Slide#Chapter 7Sampling and Sampling DistributionsSimple Random SamplingPoint EstimationIntroduction to Sampling DistributionsSampling Distribution of Sampling Distribution ofProperties of Point EstimatorsOther Sampling Methods

n = 100n = 30 # Slide#Statistical InferenceThe purpose of statistical inference is to obtain information about a population from information contained in a sample.A population is the set of all the elements of interest.A sample is a subset of the population.The sample results provide only estimates of the values of the population characteristics.A parameter is a numerical characteristic of a population.With proper sampling methods, the sample results will provide good estimates of the population characteristics. # Slide#Simple Random SamplingFinite PopulationA simple random sample 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.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#Infinite PopulationA simple random sample from an infinite population is a sample selected such that the following conditions are satisfied.Each element selected comes from the same population.Each element is selected independently.The population is usually considered infinite if it involves an ongoing process that makes listing or counting every element impossible.The random number selection procedure cannot be used for infinite populations.Simple Random Sampling # Slide#Point EstimationIn 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 as the point estimator of the population mean .s is the point estimator of the population standard deviation . is the point estimator of the population proportion p.

# Slide#Sampling ErrorThe absolute difference between an unbiased point estimate and the corresponding population parameter is called the sampling error.Sampling error is the result of using a subset of the population (the sample), and not the entire population to develop estimates.The sampling errors are:for sample mean | s - s |for sample standard deviationfor sample proportion

# Slide#Example: St. AndrewsSt. Andrews University receives 900 applicationsannually from prospective students. The applicationforms contain a variety of information including theindividuals scholastic aptitude test (SAT) score andwhether or not the individual desires on-campus housing. # Slide#Example: St. AndrewsThe director of admissions would like to know thefollowing information:the average SAT score for the applicants, andthe proportion of applicants that want to live on campus.We will now look at three alternatives for obtainingthe desired information.Conducting a census of the entire 900 applicantsSelecting a sample of 30 applicants, using a random number table Selecting a sample of 30 applicants, using computer-generated random numbers # Slide#Taking a Census of the 900 ApplicantsSAT ScoresPopulation Mean

Population Standard Deviation

Applicants Wanting On-Campus HousingPopulation Proportion

Example: St. Andrews # Slide#Example: St. AndrewsTake a Sample of 30Applicants Using a Random Number TableSince the finite population has 900 elements, we will need 3-digit random numbers to randomly select applicants numbered from 1 to 900.We will use the last three digits of the 5-digit random numbers in the third column of a random number table. The numbers we draw will be the numbers of the applicants we will sample unlessthe random number is greater than 900 orthe random number has already been used.We will continue to draw random numbers until wehave selected 30 applicants for our sample. # Slide#Example: St. AndrewsUse of Random Numbers for Sampling

3-DigitApplicant Random Number Included in Sample 744 No. 744 436 No. 436 865 No. 865 790 No. 790 835 No. 835 902 Number exceeds 900 190 No. 190 436 Number already used etc. etc.

# Slide#Sample Data

RandomNo. Number Applicant SAT Score On-Campus 1 744 Connie Reyman 1025 Yes 2 436 William Fox 950 Yes 3 865 Fabian Avante 1090 No 4 790 Eric Paxton 1120 Yes 5 835 Winona Wheeler 1015 No . . . . . 30 685 Kevin Cossack 965 No Example: St. Andrews # Slide#Example: St. AndrewsTake a Sample of 30 Applicants Using Computer-Generated Random NumbersExcel provides a function for generating random numbers in its worksheet.900 random numbers are generated, one for each applicant in the population.Then we choose the 30 applicants corresponding to the 30 smallest random numbers as our sample.Each of the 900 applicants have the same probability of being included. # Slide#Using Excel to Selecta Simple Random SampleFormula Worksheet

Note: Rows 10-901 are not shown. # Slide#Using Excel to Selecta Simple Random SampleValue Worksheet

Note: Rows 10-901 are not shown. # Slide#Using Excel to Selecta Simple Random SampleValue Worksheet (Sorted)

Note: Rows 10-901 are not shown. # Slide#Point Estimates as Point Estimator of

s as Point Estimator of

as Point Estimator of p

Note: Different random numbers would have identified a different sample which would have resulted in different point estimates.

Example: St. Andrews # Slide#Sampling Distribution ofProcess of Statistical Inference Population with meanm = ?A simple random sampleof n elements is selectedfrom the population.

The sample data provide a value forthe sample mean .

The value of is used tomake inferences aboutthe value of m.

# Slide#The sampling distribution of is the probability distribution of all possible values of the sample mean .Expected Value of

E( ) = where: = the population mean Sampling Distribution of

# Slide#Standard Deviation of

Finite Population Infinite Population

A finite population is treated as being infinite if n/N < .05. is the finite correction factor. is referred to as the standard error of the mean.

Sampling Distribution of

# Slide#If we use a large (n > 30) simple random sample, the central limit theorem enables us to conclude that the sampling distribution of can be approximated by a normal probability distribution.

When the simple random sample is small (n < 30), the sampling distribution of can be considered normal only if we assume the population has a normal probability distribution.

Sampling Distribution of

# Slide#Sampling Distribution of for the SAT Scores

Example: St. Andrews

# Slide#Sampling Distribution of for the SAT ScoresWhat is the probability that a simple random sample of 30 applicants will provide an estimate of the population mean SAT score that is within plus or minus 10 of the actual population mean ? In other words, what is the probability that will be between 980 and 1000?

Example: St. Andrews

# Slide#Sampling Distribution of for the SAT Scores

Using the standard normal probability table with z = 10/14.6= .68, we have area = (.2518)(2) = .5036

Sampling distribution of

1000980990Area = .2518Area = .2518Example: St. Andrews

# Slide#The sampling distribution of is the probability distribution of all possible values of the sample proportion

Expected Value of

where:p = the population proportion

Sampling Distribution of

# Slide#Sampling Distribution of

Standard Deviation of

Finite Population Infinite Population

is referred to as the standard error of the proportion. # Slide#Sampling Distribution of for In-State Residents

The normal probability distribution is an acceptable approximation since np = 30(.72) = 21.6 > 5 andn(1 - p) = 30(.28) = 8.4 > 5.

Example: St. Andrews

# Slide#Sampling Distribution of for In-State ResidentsWhat is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicants desiring on-campus housing that is within plus or minus .05 of the actual population proportion?In other words, what is the probability that will be between .67 and .77?

Example: St. Andrews

# Slide#Sampling Distribution of for In-State Residents

For z = .05/.082 = .61, the area = (.2291)(2) = .4582.The probability is .4582 that the sample proportion will be within +/-.05 of the actual population proportion.Sampling distribution of 0.770.670.72Area = .2291Area = .2291

Example: St. Andrews

# Slide#Properties of Point EstimatorsBefore using a sample statistic as a point estimator, statisticians check to see whether the sample statistic has the following properties associated with good point estimators.UnbiasednessEfficiencyConsistency # Slide#Properties of Point EstimatorsUnbiasednessIf the expected value of the sample statistic is equal to the population parameter being estimated, the sample statistic is said to be an unbiased estimator of the population parameter. # Slide#Properties of Point EstimatorsEfficiencyGiven the choice of two unbiased estimators of the same population parameter, we would prefer to use the point estimator with the smaller standard deviation, since it tends to provide estimates closer to the population parameter.The point estimator with the smaller standard deviation is said to have greater relative efficiency than the other. # Slide#Properties of Point EstimatorsConsistencyA point estimator is consistent if the values of the point estimator tend to become closer to the population parameter as the sample size becomes larger. # Slide#Other Sampling MethodsStratified Random SamplingCluster SamplingSystematic SamplingConvenience SamplingJudgment Sampling # Slide#Stratified Random SamplingThe population is first divided into groups of elements called strata.Each element in the population belongs to one and only one stratum.Best results are obtained when the elements within each stratum are as much alike as possible (i.e. homogeneous group).A simple random sample is taken from each stratum.Formulas are available for combining the stratum sample results into one population parameter estimate.

# Slide#Stratified Random SamplingAdvantage: If strata are homogeneous, this method is as precise as simple random sampling but with a smaller total sample size.Example: The basis for forming the strata might be department, location, age, industry type, etc.

# Slide#Cluster SamplingThe population is first divided into separate groups of elements called clusters.Ideally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group).A simple random sample of the clusters is then taken.All elements within each sampled (chosen) cluster form the sample. continued

# Slide#Cluster SamplingAdvantage: The close proximity of elements can be cost effective (I.e. many sample observations can be obtained in a short time).Disadvantage: This method generally requires a larger total sample size than simple or stratified random sampling. Example: A primary application is area sampling, where clusters are city blocks or other well-defined areas. # Slide#Systematic SamplingIf a sample size of n is desired from a population containing N elements, we might sample one element for every n/N elements in the population.We randomly select one of the first n/N elements from the population list.We then select every n/Nth element that follows in the population list.This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering. continued # Slide#Systematic SamplingAdvantage: The sample usually will be easier to identify than it would be if simple random sampling were used.Example: Selecting every 100th listing in a telephone book after the first randomly selected listing. # Slide#Convenience SamplingIt is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected.The sample is identified primarily by convenience.Advantage: Sample selection and data collection are relatively easy.Disadvantage: It is impossible to determine how representative of the population the sample is. Example: A professor conducting research might use student volunteers to constitute a sample. # Slide#Judgment SamplingThe person most knowledgeable on the subject of the study selects elements of the population that he or she feels are most representative of the population.It is a nonprobability sampling technique.Advantage: It is a relatively easy way of selecting a sample.Disadvantage: The quality of the sample results depends on the judgment of the person selecting the sample.Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate. # Slide#End of Chapter 7 # Slide#Sheet1ABCDD1Applicant NumberSAT ScoreOn-Campus HousingRandom Number(x-m)2f(x)211008Yes=RAND()=(A2-C$7)^2*B2321025No=RAND()=(A3-C$7)^2*B343952Yes=RAND()=(A4-C$7)^2*B4541090Yes=RAND()=(A5-C$7)^2*B5651127Yes=RAND()=(A6-C$7)^2*B6761015No=RAND()=SUM(D2:D6)87965Yes=RAND()Variance981161No=RAND()101072Yes=RAND()111066Yes=RAND()121314

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Sheet1ABCDD1Applicant NumberSAT ScoreOn-Campus HousingRandom Number(x-m)2f(x)211008Yes0.41327=(A2-C$7)^2*B2321025No0.79514=(A3-C$7)^2*B343952Yes0.66237=(A4-C$7)^2*B4541090Yes0.00234=(A5-C$7)^2*B5651127Yes0.71205=(A6-C$7)^2*B6761015No0.18037=SUM(D2:D6)87965Yes0.71607Variance981161No0.90512101072Yes0.22474111007Yes0.24680121161No0.35689131127Yes0.30231141072Yes0.7938215965Yes0.21205161025No0.36537171115No0.55101181066Yes0.35542

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Sheet1ABCDD1Applicant NumberSAT ScoreOn-Campus HousingRandom Number(x-m)2f(x)2121107No0.00027=(A2-C$7)^2*B237731043Yes0.00192=(A3-C$7)^2*B34408991Yes0.00303=(A4-C$7)^2*B45581008No0.00481=(A5-C$7)^2*B561161127Yes0.00538=(A6-C$7)^2*B67185982Yes0.00583=SUM(D2:D6)85101163Yes0.00649Variance93941008No0.0066710929No0.00785111007Yes0.43513121161No0.56998131127Yes0.57123141072Yes0.3820815965Yes0.98629161025No0.80337171115No0.15245181066Yes0.65464

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