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The purpose of The purpose of statistical inferencestatistical inference is to obtain is to obtain information about a population from informationinformation about a population from information contained in a sample.contained in a sample.
The purpose of The purpose of statistical inferencestatistical inference is to obtain is to obtain information about a population from informationinformation about a population from information contained in a sample.contained in a sample.
Statistical InferenceStatistical Inference
A A populationpopulation is the set of all the elements of interest. is the set of all the elements of interest. A A populationpopulation is the set of all the elements of interest. is the set of all the elements of interest.
A A samplesample is a subset of the population. is a subset of the population. A A samplesample is a subset of the population. is a subset of the population.
The sample results provide only The sample results provide only estimatesestimates of the of the values of the population characteristics.values of the population characteristics. The sample results provide only The sample results provide only estimatesestimates of the of the values of the population characteristics.values of the population characteristics.
A A parameterparameter is a numerical characteristic of a is a numerical characteristic of a population.population. A A parameterparameter is a numerical characteristic of a is a numerical characteristic of a population.population.
With With proper sampling methodsproper sampling methods, the sample results, the sample results can provide “good” estimates of the populationcan provide “good” estimates of the population characteristics.characteristics.
With With proper sampling methodsproper sampling methods, the sample results, the sample results can provide “good” estimates of the populationcan provide “good” estimates of the population characteristics.characteristics.
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..
Infinite populations are often defined by an Infinite populations are often defined by an ongoing processongoing process whereby the elements of the whereby the elements of the population consist of items generated as though population consist of items generated as though the process would operate indefinitely.the process would operate indefinitely.
Simple Random Sampling:Simple Random Sampling:Infinite PopulationInfinite Population
A A simple random sample from an infinite populationsimple random sample from an infinite population is a sample selected such that the following conditionsis a sample selected such that the following conditions are satisfied.are satisfied.
• Each element selected comes from the sameEach element selected comes from the same population.population.
• Each element is selected independently.Each element is selected independently.
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.
Point EstimationPoint Estimation
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
Statistical methods can be used to make probabilityStatistical methods can be used to make probability statements about the size of the sampling error.statements about the size of the sampling error.
Sampling error is the result of using a subset of theSampling error is the result of using a subset of the population (the sample), and not the entirepopulation (the sample), and not the entire population.population.
The absolute value of the difference between anThe absolute value of the difference between an unbiased point estimate and the correspondingunbiased point estimate and the corresponding population parameter is called the population parameter is called the sampling errorsampling error..
When the expected value of a point estimator is equalWhen the expected value of a point estimator is equal to the population parameter, the point estimator is saidto the population parameter, the point estimator is said to be to be unbiasedunbiased..
alternatives for obtaining thealternatives for obtaining the
desired information.desired information. Conducting a census of theConducting a census of the entire 900 applicantsentire 900 applicants Selecting a sample of 30Selecting a sample of 30
If the relevant data for the entire 900 applicants If the relevant data for the entire 900 applicants were in the college’s database, the population were in the college’s database, the population parameters of interest could be calculated using parameters of interest could be calculated using the formulas presented in Chapter 3.the formulas presented in Chapter 3.
We will assume for the moment that conducting We will assume for the moment that conducting a census is practical in this example.a census is practical in this example.
The applicants were numbered, from 1 to 900, asThe applicants were numbered, from 1 to 900, as their applications arrived.their applications arrived.
She decides a sample of 30 applicants will be used.She decides a sample of 30 applicants will be used.
Furthermore, the Director of Admissions must obtainFurthermore, the Director of Admissions must obtain estimates of the population parameters of interest forestimates of the population parameters of interest for a meeting taking place in a few hours.a meeting taking place in a few hours.
Now suppose that the necessary data on theNow suppose that the necessary data on the current year’s applicants were not yet entered in thecurrent year’s applicants were not yet entered in the college’s database.college’s database.
Taking a Sample of 30 ApplicantsTaking a Sample of 30 Applicants
Excel’s RAND function generatesExcel’s RAND function generates random numbers between 0 and 1random numbers between 0 and 1
Excel’s RAND function generatesExcel’s RAND function generates random numbers between 0 and 1random numbers between 0 and 1
Simple Random Sampling:Simple Random Sampling:Using ExcelUsing Excel
Step 1:Step 1: Assign a random number to each of the 900 Assign a random number to each of the 900 applicants.applicants.
Step 2:Step 2: Select the 30 applicants corresponding to the Select the 30 applicants corresponding to the 30 smallest random numbers.30 smallest random numbers.
Note:Note: Different random numbers would haveDifferent random numbers would haveidentified a different sample which would haveidentified a different sample which would haveresulted in different point estimates.resulted in different point estimates.
Form of the Sampling Distribution of Form of the Sampling Distribution of xx
Central Limit Theorem: Central Limit Theorem: If we use a large (If we use a large (nn >> 30) simple random sample, the 30) simple random sample, thesampling distribution of can be approximated bysampling distribution of can be approximated bya normal distribution.a normal distribution.
x
If the simple random sample is small (If the simple random sample is small (nn < 30), < 30),the sampling distribution of can be consideredthe sampling distribution of can be considerednormal normal only if we assume the population has aonly if we assume the population has anormal distribution.normal distribution.
Relationship Between the Sample SizeRelationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of xx
Suppose we select a simple random sample of 100Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered.applicants instead of the 30 originally considered.
EE( ) = ( ) = regardless of the sample size. In regardless of the sample size. In ourour example,example, E E( ) remains at 990.( ) remains at 990.
xxxx
Whenever the sample size is increased, the standardWhenever the sample size is increased, the standard error of the mean is decreased. With the increaseerror of the mean is decreased. With the increase in the sample size to in the sample size to nn = 100, the standard error of the = 100, the standard error of the mean is decreased to:mean is decreased to:
Recall that when Recall that when nn = 30, = 30, PP(980 (980 << << 1000) = .5034. 1000) = .5034.xx
Relationship Between the Sample SizeRelationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of xx
We follow the same steps to solve for We follow the same steps to solve for PP(980 (980 << << 1000) 1000) when when nn = 100 as we showed earlier when = 100 as we showed earlier when nn = 30. = 30.
xx
Now, with Now, with nn = 100, = 100, PP(980 (980 << << 1000) = .7888. 1000) = .7888.xx
Sampling Distribution ofSampling Distribution ofpp
where:where:pp = the population proportion = the population proportion
The The sampling distribution of sampling distribution of is the probability is the probabilitydistribution of all possible values of the sampledistribution of all possible values of the sampleproportion .proportion .pp
The sampling distribution of can be approximatedThe sampling distribution of can be approximated by a normal distribution by a normal distribution whenever the sample size whenever the sample size is large.is large.
The sampling distribution of can be approximatedThe sampling distribution of can be approximated by a normal distribution by a normal distribution whenever the sample size whenever the sample size is large.is large.
pp
The sample size is considered large whenever The sample size is considered large whenever thesethese conditions are satisfied:conditions are satisfied:
The sample size is considered large whenever The sample size is considered large whenever thesethese conditions are satisfied:conditions are satisfied:
npnp >> 5 5 nn(1 – (1 – pp) ) >> 5 5andand
Form of the Sampling Distribution ofForm of the Sampling Distribution ofpp
For values of For values of pp near .50, sample sizes as near .50, sample sizes as small as 10small as 10permit a normal approximation.permit a normal approximation.
For values of For values of pp near .50, sample sizes as near .50, sample sizes as small as 10small as 10permit a normal approximation.permit a normal approximation.
With very small (approaching 0) or very large With very small (approaching 0) or very large (approaching 1) values of (approaching 1) values of pp, much larger , much larger samples are needed.samples are needed.
With very small (approaching 0) or very large With very small (approaching 0) or very large (approaching 1) values of (approaching 1) values of pp, much larger , much larger samples are needed.samples are needed.
Form of the Sampling Distribution ofForm of the Sampling Distribution ofpp