Chapter 7 Sampling and Sampling Distributions
7.1 Statistical Inference
· The 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.
· With proper sampling methods, the sample results will provide “good” estimates of the population characteristics.
· A parameter is a numerical characteristic of a population.
7.2 Simple Random Sampling
· Finite Population
§ Finite populations are often defined by lists such as: Organization membership roster, Credit card account numbers, Inventory product numbers
§ A simple random sample of size n 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.
· Infinite Population
§ Infinite populations are often defined by an ongoing process whereby the elements of the population consist of items generated as though the process would operate indefinitely.
§ A 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.
§ In the case of infinite populations, it is impossible to obtain a list of all elements in the population.
§ The random number selection procedure cannot be used for infinite populations.
7.3 Point Estimation
· In 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.
Sampling Error
· When the expected value of a point estimator is equal to the population parameter, the point estimator is said to be unbiased.
· The absolute value of the 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.
· Statistical methods can be used to make probability statements about the size of the sampling error.
· The sampling errors are:
for sample mean
for sample standard deviation
for sample proportion
Example: St. Andrew’s
St. Andrew’s University receives 900 applications annually from prospective students. The application
forms contain a variety of information including the individual’s scholastic aptitude test (SAT) score and
whether or not the individual desires on-campus housing.
The director of admissions would like to know the following information:
• the average SAT score for the applicants, and
• the proportion of applicants that want to live on campus.
We will now look at three alternatives for obtaining the desired information.
• Conducting a census of the entire 900 applicants
• Selecting a sample of 30 applicants, using a random number table
• Selecting a sample of 30 applicants, using computer-generated random numbers
Taking a Census of the 900 Applicants
• SAT Scores
• Population Mean
• Population Standard Deviation
• Applicants Wanting On-Campus Housing
• Population Proportion
Take a Sample of 30Applicants Using a Random Number Table
Since 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 (p. 261). The numbers we draw will be the numbers of the applicants we will sample unless
• the random number is greater than 900 or
• the random number has already been used.
We will continue to draw random numbers until we have selected 30 applicants for our sample.
Use of Random Numbers for Sampling (Table 7.1 p. 261)
3-Digit Applicant
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.
Sample Data
Random
No. 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
Take a Sample of 30 Applicants Using Computer-Generated Random Numbers
• Excel 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.
Using Excel to Select a Simple Random Sample ( = RANDBETWEEN(1,900) )
Formula Worksheet
Using Excel to Select a Simple Random Sample
Value Worksheet (Sorted)
Point Estimates
· as the point estimator of
· s as the point estimator of
· as the point estimator of p
Note: Different random numbers would have identified a different sample which would have resulted in different point estimates.
Sampling Distribution of
Process of Statistical Inference
Sampling Distribution of
The sampling distribution of is the probability distribution of all possible values of the sample mean . .
Expected Value of
E( x ) =
where:
= the population mean
Sampling Distribution 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.
Form of the Sampling Distribution of
· When the population has a normal distribution, the sampling distribution of is normally distributed
for any sample size
· In most applications, the sampling distribution of can be approximated by a normal distribution
whenever the sample is size 30 or more.
· In cases where the population is highly skewed or outliers are present, samples of size 50 may be
needed.
Example: St. Andrew’s
o Sampling Distribution of for the SAT Scores
What 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?
Step 1: Calculate the z-value at the upper endpoint of the interval: z = (1000 - 990)/14.6= .68
Step 2: Find the area under the curve to the left of the upper endpoint: P(z < .68) = .7517
Step 3: Calculate the z-value at the lower endpoint of the interval: z = (980 - 990)/14.6= - .68
Step 4: Find the area under the curve to the left of the lower endpoint: P(z < -.68) = .2483
Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval:
P(-.68 < z < .68) = P(z < .68) - P(z < -.68) = .7517 - .2483=.5034
The probability that the sample mean SAT score will be between 980 and 1000 is: P(980 < < 1000) = .5034
Relationship Between the Sample Size and the Sampling Distribution of
n Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered.
n E( ) = µ regardless of the sample size. In our example, E( ) remains at 990.
n Whenever the sample size is increased, the standard error of the mean is decreased. With the increase in the sample size to n = 100, the standard error of the mean is decreased to:
Sampling Distribution of
· The sampling distribution of is the probability distribution of all possible values of the sample proportion
Expected Value of :
where:
p = the population proportion
· Standard Deviation of
Finite Population Infinite Population
• is referred to as the standard error of the proportion.
Example: St. Andrew’s
Sampling Distribution of for In-State Residents
The normal probability distribution is an acceptable approximation since np = 30(.72) = 21.6 > 5 and n(1 - p) = 30(.28) = 8.4 > 5.
What 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?
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.
Properties of Point Estimators
Before 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.
· Unbiasedness
o If 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.
· Efficiency
o Given 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.
o The point estimator with the smaller standard deviation is said to have greater relative efficiency than the other.
· Consistency
o A 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.
Other Sampling Methods
Stratified Random Sampling
o The population is first divided into groups of elements called strata.
o Each element in the population belongs to one and only one stratum.
o Best results are obtained when the elements within each stratum are as much alike as possible (i.e. homogeneous group).
o A simple random sample is taken from each stratum.
o Formulas are available for combining the stratum sample results into one population parameter estimate.
o Advantage: 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.
Cluster Sampling
o The population is first divided into separate groups of elements called clusters.
o Ideally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group).
o A simple random sample of the clusters is then taken.
o All elements within each sampled (chosen) cluster form the sample.
o Advantage: The close proximity of elements can be cost effective (I.e. many sample observations can be obtained in a short time).
o 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.
Systematic Sampling
o If 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.
o We randomly select one of the first n/N elements from the population list.
o We then select every n/Nth element that follows in the population list.
o This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering.
o Systematic Sampling
o Advantage: 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.
Convenience Sampling
o It is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected.
o The sample is identified primarily by convenience.
o Advantage: Sample selection and data collection are relatively easy.
o 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.
Judgment Sampling
o The 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.
o It is a nonprobability sampling technique.
o Advantage: It is a relatively easy way of selecting a sample.
o 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.