Chapter 7 Sampling and Sampling Distributions Sampling Distribution of Sampling Distribution of Introduction to Sampling Distributions Introduction to Sampling Distributions Point Estimation Point Estimation Simple Random Sampling Simple Random Sampling

Example: St. Andrew’s St. Andrew’s College receives St. Andrew’s College receives 900 applications annually from prospective students. The application form contains a variety of information including the individual’s scholastic aptitude test (SAT) score and whether or not the individual desires on-campus housing.

Example: St. Andrew’s The director of admissions The director of admissions would like to know the following information: the average SAT score for the average SAT score for the 900 applicants, and the 900 applicants, and the proportion of the proportion of applicants that want to live on campus.

Example: St. Andrew’s We will now look at three alternatives for obtaining the desired information. n Conducting a census of the entire 900 applicants entire 900 applicants n Selecting a sample of 30 applicants, using a random number table n Selecting a sample of 30 applicants, using Excel

Conducting a Census n If the relevant data for the entire 900 applicants were in the college’s database, the population parameters of interest could be calculated using the formulas presented in Chapter 3. n We will assume for the moment that conducting a census is practical in this example.

Conducting a Census n Population Mean SAT Score n Population Standard Deviation for SAT Score n Population Proportion Wanting On-Campus Housing

as Point Estimator of  as Point Estimator of  n as Point Estimator of p Point Estimation Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. s as Point Estimator of  s as Point Estimator of 

PopulationParameterPointEstimatorPointEstimateParameterValue  = Population mean SAT score SAT score  = Population std. deviation for deviation for SAT score SAT score 80 s = Sample std. s = Sample std. deviation for deviation for SAT score SAT score75.2 p = Population pro- portion wanting portion wanting campus housing campus housing Summary of Point Estimates Obtained from a Simple Random Sample = Sample mean = Sample mean SAT score SAT score = Sample pro- = Sample pro- portion wanting portion wanting campus housing campus housing

n Process of Statistical Inference The value of is used to make inferences about the value of . The sample data provide a value for the sample mean. A simple random sample of n elements is selected from the population. Population with mean  = ? Sampling Distribution of

The sampling distribution of is the probability distribution of all possible values of the sample mean. Sampling Distribution of where:  = the population mean  = the population mean E ( ) =  Expected Value of

Sampling Distribution of Finite Population Infinite Population is referred to as the standard error of the is referred to as the standard error of the mean. mean. A finite population is treated as being A finite population is treated as being infinite if n / N <.05. infinite if n / N <.05. is the finite correction factor. is the finite correction factor. Standard Deviation of

Form of the Sampling Distribution of 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 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 distribution.

Sampling Distribution of for SAT Scores SamplingDistributionof

With a mean SAT score of 990 and a standard deviation of 80, 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 +/  10 of the actual population mean  ? In other words, what is the probability that will be In other words, what is the probability that will be between 980 and 1000? Sampling Distribution of for SAT Scores

Step 1: Calculate the z -value at the upper endpoint of the interval. the interval. z = (1000  990)/14.6= Step 2: Find the area under the curve between the mean and the upper endpoint. and the upper endpoint. Sampling Distribution of for SAT Scores

Probabilities for the Standard Normal Distribution the Standard Normal Distribution

Sampling Distribution of for SAT Scores 990SamplingDistributionof1000 Area =.2517

Step 3: Calculate the z -value at the lower endpoint of the interval. the interval. Step 4: Find the area under the curve between the mean and the lower endpoint. and the lower endpoint. z = (980  990)/14.6= -.68 =.2517 Sampling Distribution of for SAT Scores

990SamplingDistributionof980 Area =.2517

Sampling Distribution of for SAT Scores Area =.2517 SamplingDistributionof1000

Sampling Distribution of for SAT Scores Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval. the lower and upper endpoints of the interval. P (-.68 < z <.68) = =.2517 .2517 =.5034 The probability that the sample mean SAT score will be between 980 and 1000 is: P (980 < < 1000) =.5034

Sampling Distribution of for SAT Scores Area =.5034 SamplingDistributionof

Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of Suppose we select a simple random sample of 100 Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered. applicants instead of the 30 originally considered. E ( ) =  regardless of the sample size. In our E ( ) =  regardless of the sample size. In our example, E ( ) remains at 990. example, E ( ) remains at 990. Whenever the sample size is increased, the standard Whenever the sample size is increased, the standard error of the mean is decreased. With the increase error of the mean is decreased. With the increase in the sample size to n = 100, the standard error of the in the sample size to n = 100, the standard error of the mean is decreased to: mean is decreased to:

Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of With n = 30, With n = 100,

Recall that when n = 30, P (980 < < 1000) = Recall that when n = 30, P (980 < < 1000) = Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of We follow the same steps to solve for P (980 < < 1000) We follow the same steps to solve for P (980 < < 1000) when n = 100 as we showed earlier when n = 30. when n = 100 as we showed earlier when n = 30. Now, with n = 100, P (980 < < 1000) = Now, with n = 100, P (980 < < 1000) = Because the sampling distribution with n = 100 has a Because the sampling distribution with n = 100 has a smaller standard error, the values of have less smaller standard error, the values of have less variability and tend to be closer to the population variability and tend to be closer to the population mean than the values of with n = 30. mean than the values of with n = 30.

Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of Area =.7888 SamplingDistributionof

Chapter 7 Sampling and Sampling Distributions Other Sampling Methods Other Sampling Methods Sampling Distribution of Sampling Distribution of

A simple random sample of n elements is selected from the population. Population with proportion p = ? n Making Inferences about a Population Proportion The sample data provide a value for the sample proportion. The value of is used to make inferences about the value of p. Sampling Distribution of

where: p = the population proportion The sampling distribution of is the probability distribution of all possible values of the sample proportion. Expected Value of

is referred to as the standard error of the is referred to as the standard error of the proportion. Sampling Distribution of Finite Population Infinite Population Standard Deviation of A finite population is treated as being A finite population is treated as being infinite if n / N <.05. infinite if n / N <.05.

Recall that 72% of the Recall that 72% of the prospective students applying to St. Andrew’s College desire on-campus housing. n Example: St. Andrew’s College Sampling Distribution of What is the probability that What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicant desiring on-campus housing that is within plus or minus.05 of the actual population proportion?

SamplingDistributionof Sampling Distribution of

Step 1: Calculate the z -value at the upper endpoint of the interval. the interval. z = (.77 .72)/.082 = Step 2: Find the area under the curve between the mean and upper endpoint. and upper endpoint. Sampling Distribution of

Probabilities for the Standard Normal Distribution the Standard Normal Distribution Sampling Distribution of

Area =.2291 SamplingDistributionof Sampling Distribution of

Step 3: Calculate the z -value at the lower endpoint of the interval. the interval. Step 4: Find the area under the curve between the mean and the lower endpoint. and the lower endpoint. z = (.67 .72)/.082 = Sampling Distribution of

Area =.2291 SamplingDistributionof Sampling Distribution of

P (.67 < <.77) =.4582 Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval. the lower and upper endpoints of the interval. P (-.61 < z <.61) = =.2291 .2291 =.4582 The probability that the sample proportion of applicants wanting on-campus housing will be within +/-.05 of the actual population proportion : Sampling Distribution of

Area =.4582 SamplingDistributionof Sampling Distribution of

Other Sampling Methods n Stratified Random Sampling n Cluster Sampling n Systematic Sampling n Convenience Sampling n Judgment Sampling

The population is first divided into groups of The population is first divided into groups of elements called strata. elements called strata. The population is first divided into groups of The population is first divided into groups of elements called strata. elements called strata. Stratified Random Sampling Each element in the population belongs to one and Each element in the population belongs to one and only one stratum. only one stratum. Each element in the population belongs to one and Each element in the population belongs to one and only one stratum. only one stratum. Best results are obtained when the elements within Best results are obtained when the elements within each stratum are as much alike as possible each stratum are as much alike as possible (i.e. a homogeneous group). (i.e. a homogeneous group). Best results are obtained when the elements within Best results are obtained when the elements within each stratum are as much alike as possible each stratum are as much alike as possible (i.e. a homogeneous group). (i.e. a homogeneous group).

Stratified Random Sampling A simple random sample is taken from each stratum. A simple random sample is taken from each stratum. Formulas are available for combining the stratum Formulas are available for combining the stratum sample results into one population parameter sample results into one population parameter estimate. estimate. Formulas are available for combining the stratum Formulas are available for combining the stratum sample results into one population parameter sample results into one population parameter estimate. estimate. Advantage: If strata are homogeneous, this method Advantage: If strata are homogeneous, this method is as “precise” as simple random sampling but with is as “precise” as simple random sampling but with a smaller total sample size. a smaller total sample size. Advantage: If strata are homogeneous, this method Advantage: If strata are homogeneous, this method is as “precise” as simple random sampling but with is as “precise” as simple random sampling but with a smaller total sample size. a smaller total sample size. Example: The basis for forming the strata might be Example: The basis for forming the strata might be department, location, age, industry type, and so on. department, location, age, industry type, and so on. Example: The basis for forming the strata might be Example: The basis for forming the strata might be department, location, age, industry type, and so on. department, location, age, industry type, and so on.

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

Cluster Sampling Advantage: The close proximity of elements can be Advantage: The close proximity of elements can be cost effective (i.e. many sample observations can be cost effective (i.e. many sample observations can be obtained in a short time). obtained in a short time). Advantage: The close proximity of elements can be Advantage: The close proximity of elements can be cost effective (i.e. many sample observations can be cost effective (i.e. many sample observations can be obtained in a short time). obtained in a short time). Disadvantage: This method generally requires a Disadvantage: This method generally requires a larger total sample size than simple or stratified larger total sample size than simple or stratified random sampling. random sampling. Disadvantage: This method generally requires a Disadvantage: This method generally requires a larger total sample size than simple or stratified larger total sample size than simple or stratified random sampling. random sampling. Example: A primary application is area sampling, Example: A primary application is area sampling, where clusters are city blocks or other well-defined where clusters are city blocks or other well-defined areas. areas. Example: A primary application is area sampling, Example: A primary application is area sampling, where clusters are city blocks or other well-defined where clusters are city blocks or other well-defined areas. areas.

Systematic Sampling If a sample size of n is desired from a population If a sample size of n is desired from a population containing N elements, we might sample one containing N elements, we might sample one element for every n / N elements in the population. element for every n / N elements in the population. If a sample size of n is desired from a population If a sample size of n is desired from a population containing N elements, we might sample one containing N elements, we might sample one element for every n / N elements in the population. element for every n / N elements in the population. We randomly select one of the first n / N elements We randomly select one of the first n / N elements from the population list. from the population list. We randomly select one of the first n / N elements We randomly select one of the first n / N elements from the population list. from the population list. We then select every n / N th element that follows in We then select every n / N th element that follows in the population list. the population list. We then select every n / N th element that follows in We then select every n / N th element that follows in the population list. the population list.

Systematic Sampling This method has the properties of a simple random This method has the properties of a simple random sample, especially if the list of the population sample, especially if the list of the population elements is a random ordering. elements is a random ordering. This method has the properties of a simple random This method has the properties of a simple random sample, especially if the list of the population sample, especially if the list of the population elements is a random ordering. elements is a random ordering. Advantage: The sample usually will be easier to Advantage: The sample usually will be easier to identify than it would be if simple random sampling identify than it would be if simple random sampling were used. were used. Advantage: The sample usually will be easier to Advantage: The sample usually will be easier to identify than it would be if simple random sampling identify than it would be if simple random sampling were used. were used. Example: Selecting every 100 th listing in a telephone Example: Selecting every 100 th listing in a telephone book after the first randomly selected listing book after the first randomly selected listing Example: Selecting every 100 th listing in a telephone Example: Selecting every 100 th listing in a telephone book after the first randomly selected listing book after the first randomly selected listing

Convenience Sampling It is a nonprobability sampling technique. Items are It is a nonprobability sampling technique. Items are included in the sample without known probabilities included in the sample without known probabilities of being selected. of being selected. It is a nonprobability sampling technique. Items are It is a nonprobability sampling technique. Items are included in the sample without known probabilities included in the sample without known probabilities of being selected. of being selected. Example: A professor conducting research might use Example: A professor conducting research might use student volunteers to constitute a sample. student volunteers to constitute a sample. Example: A professor conducting research might use Example: A professor conducting research might use student volunteers to constitute a sample. student volunteers to constitute a sample. The sample is identified primarily by convenience. The sample is identified primarily by convenience.

Advantage: Sample selection and data collection are Advantage: Sample selection and data collection are relatively easy. relatively easy. Advantage: Sample selection and data collection are Advantage: Sample selection and data collection are relatively easy. relatively easy. Disadvantage: It is impossible to determine how Disadvantage: It is impossible to determine how representative of the population the sample is. representative of the population the sample is. Disadvantage: It is impossible to determine how Disadvantage: It is impossible to determine how representative of the population the sample is. representative of the population the sample is. Convenience Sampling

Judgment Sampling The person most knowledgeable on the subject of the The person most knowledgeable on the subject of the study selects elements of the population that he or study selects elements of the population that he or she feels are most representative of the population. she feels are most representative of the population. The person most knowledgeable on the subject of the The person most knowledgeable on the subject of the study selects elements of the population that he or study selects elements of the population that he or she feels are most representative of the population. she feels are most representative of the population. It is a nonprobability sampling technique. It is a nonprobability sampling technique. Example: A reporter might sample three or four Example: A reporter might sample three or four senators, judging them as reflecting the general senators, judging them as reflecting the general opinion of the senate. opinion of the senate. Example: A reporter might sample three or four Example: A reporter might sample three or four senators, judging them as reflecting the general senators, judging them as reflecting the general opinion of the senate. opinion of the senate.

Judgment Sampling Advantage: It is a relatively easy way of selecting a Advantage: It is a relatively easy way of selecting a sample. sample. Advantage: It is a relatively easy way of selecting a Advantage: It is a relatively easy way of selecting a sample. sample. Disadvantage: The quality of the sample results Disadvantage: The quality of the sample results depends on the judgment of the person selecting the depends on the judgment of the person selecting the sample. sample. Disadvantage: The quality of the sample results Disadvantage: The quality of the sample results depends on the judgment of the person selecting the depends on the judgment of the person selecting the sample. sample.