1 1 Slide Chapter 7 (b) – Point Estimation and Sampling Distributions Point estimation is a form of statistical inference. Point estimation is a form of statistical inference. In point estimation we use the data from the sample In point estimation we use the data from the sample to compute a value of a sample statistic that serves to compute a value of a sample statistic that serves as an estimate of a population parameter. as an estimate of a population parameter. In point estimation we use the data from the sample In point estimation we use the data from the sample to compute a value of a sample statistic that serves to 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 as the point estimator of the population We refer to as the point estimator of the population mean . mean . We refer to as the point estimator of the population We refer to as the point estimator of the population mean . mean . s is the point estimator of the population standard s is the point estimator of the population standard deviation . deviation . s is the point estimator of the population standard s is the point estimator of the population standard deviation . deviation . is the point estimator of the population proportion p. is the point estimator of the population proportion p.

2 2 Slide Example: St. Andrew’s College Example: St. Andrew’s College Point Estimation  St. Andrew’s College received 900 applications 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.  At a meeting in a few hours, the Director of Admissions would like to announce the average SAT score and the proportion of applicants that want to live on campus, for the population of 900 applicants.  However, the necessary data on the applicants have not yet been entered in the college’s computerized database. So, the Director decides to estimate the values of the population parameters of interest based on sample statistics. The sample of 30 applicants selected earlier with Excel’s RAND function will be used.

3 3 Slide Excel Value Worksheet (Sorted) Note: Rows are not shown. Point Estimation Using Excel AB 1 Applicant Number Random Number CD SAT Score On-Campus Housing 1207No 1143Yes 1091Yes 1108No 1227Yes 982Yes 1363Yes 1108No

4 4 Slide 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 

5 5 Slide Population Mean SAT Score Population Standard Deviation for SAT Score Population Proportion Wanting On-Campus Housing Once all the data for the 900 applicants were entered in the college’s database, the values of the population parameters of interest were calculated. Point Estimation

6 6 Slide Population Parameter Point Estimator Point Estimate Parameter Value  = Population mean SAT score  = Population std. deviation for SAT score 87.4s = Sample stan- dard deviation for SAT score 85.2 p = Population pro- portion wanting campus housing Summary of Point Estimates Obtained from a Simple Random Sample

7 7 Slide Practical Advice The target population is the population we want to The target population is the population we want to make inferences about. make inferences about. The target population is the population we want to The target population is the population we want to make inferences about. make inferences about. Whenever a sample is used to make inferences about Whenever a sample is used to make inferences about a population, we should make sure that the targeted a population, we should make sure that the targeted population and the sampled population are in close population and the sampled population are in close agreement. agreement. Whenever a sample is used to make inferences about Whenever a sample is used to make inferences about a population, we should make sure that the targeted a population, we should make sure that the targeted population and the sampled population are in close population and the sampled population are in close agreement. agreement. The sampled population is the population from The sampled population is the population from which the sample is actually taken. which the sample is actually taken. The sampled population is the population from The sampled population is the population from which the sample is actually taken. which the sample is actually taken.

8 8 Slide 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

9 9 Slide where:  = the population mean When the expected value of the point estimator equals the population parameter, we say the point estimator is unbiased. Sampling Distribution of

10 Slide Sampling Distribution of We will use the following notation to define the We will use the following notation to define the standard deviation of the sampling distribution of.  = the standard deviation of  = the standard deviation of the population n = the sample size N = the population size Standard Deviation of Standard Deviation of

11 Slide 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 population is the finite population correction factor. correction factor. Standard Deviation of Standard Deviation of

12 Slide In cases where the population is highly skewed or outliers are present, samples of size 50 may be needed.

13 Slide Central Limit Theorem

14 Slide  Example: St. Andrew’s College

15 Slide Step 1: Calculate the z -value at the upper endpoint of the interval. the interval. z = ( )/15.96 =.63 P ( z <.63) =.7357 Step 2: Find the area under the curve to the left of the upper endpoint. upper endpoint. Sampling Distribution of n Example: St. Andrew’s College 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  ? 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  ? Be between 1687 and 1707?

16 Slide Cumulative Probabilities for the Standard Normal Distribution z Example: St. Andrew’s College

17 Slide Area =.7357 n Example: St. Andrew’s College

18 Slide Area =.2643 Step 3: Calculate the z-value at the lower endpoint of the interval. the interval. z = ( )/15.96 = -.63 Step 4: Find the area under the curve to the left of the lower endpoint. lower endpoint. P(z < -.63) =.2643

19 Slide 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) = P(z <.68) - P(z < -.68) = =.4714 The probability that the sample mean SAT score will be between 1687 and 1707 is: Example: St. Andrew’s College

20 Slide Area =.4714 Example: St. Andrew’s College

21 Slide 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.  Example: St. Andrew’s College

22 Slide Example: St. Andrew’s College

23 Slide Example: St. Andrew’s College Example: St. Andrew’s College

24 Slide Area =.7776 Example: St. Andrew’s College

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

26 Slide is referred to as the standard error of is referred to as the standard error of the proportion. the proportion. Sampling Distribution of Finite Population Infinite Population Standard Deviation of Standard Deviation of is the finite population is the finite population correction factor. correction factor.

27 Slide Recall that 72% of the prospective students applying Recall that 72% of the prospective students applying to St. Andrew’s College desire on-campus housing. Example: St. Andrew’s College Sampling Distribution of 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? 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? For our example, with n = 30 and p =.72, the normal distribution is an acceptable approximation because: np = 30(.72) = 21.6 > 5 n (1 - p ) = 30(.28) = 8.4 > 5

28 Slide SamplingDistributionof Prospective Students Applying to St. Andrew’s College Desiring On-Campus Housing.

29 Slide The sampling distribution of can be approximated by a normal distribution whenever the sample size by a normal distribution whenever the sample size is large enough to satisfy the two conditions: is large enough to satisfy the two conditions: np > 5 and n (1 – p ) > 5 np > 5 and n (1 – p ) > 5... because when these conditions are satisfied, the... because when these conditions are satisfied, the probability distribution of x in the sample proportion, probability distribution of x in the sample proportion, = x / n, can be approximated by normal distribution = x / n, can be approximated by normal distribution (and because n is a constant). (and because n is a constant). Form of the Sampling Distribution of

30 Slide Step 1: Calculate the z -value at the upper endpoint of the interval. of the interval. z = (.77 .72)/.082 =.61 P ( z <.61) =.7291 Step 2: Find the area under the curve to the left of the upper endpoint. the upper endpoint. Step 3: Calculate the z -value at the lower endpoint of the interval. the interval. z = (.67 .72)/.082 = -.61 Step 4: Find the area under the curve to the left of the lower endpoint. lower endpoint. P ( z < -.61) =.2709

31 Slide 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) = P ( z <.61)  P ( z < -.61) =.7291 .2709 =.4582 The probability that the sample proportion of applicants wanting on-campus housing will be within +/-.05 of the actual population proportion :

32 Slide Area =.4582 SamplingDistributionof Sampling Distribution of Sampling Distribution of Example: St. Andrew’s College

33 Slide 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). Other Types of Sampling

34 Slide 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.

35 Slide 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.

36 Slide 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.

37 Slide Systematic Sampling This method has the properties of a simple random sample, This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering. This method has the properties of a simple random sample, This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering. Advantage: The sample usually will be easier to identify than it Advantage: The sample usually will be easier to identify than it would be if simple random sampling were used. Advantage: The sample usually will be easier to identify than it Advantage: The sample usually will be easier to identify than it would be if simple random sampling were used. Example: Selecting every 100 th listing in a telephone book after the first randomly selected listing Example: Selecting every 100 th listing in a telephone book after the first randomly selected listing 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. 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. We randomly select one of the first n / N elements from the population list. list. We then select every n / N th element that follows in the population list.

38 Slide 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.

39 Slide 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 senators, Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate. judging them as reflecting the general opinion of the senate. Example: A reporter might sample three or four senators, Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate. judging them as reflecting the general opinion of the senate. Advantage: It is a relatively easy way of selecting a sample. Advantage: It is a relatively easy way of selecting a sample. Disadvantage: The quality of the sample results depends Disadvantage: The quality of the sample results depends on the judgment of the person selecting the sample. on the judgment of the person selecting the sample. Disadvantage: The quality of the sample results depends Disadvantage: The quality of the sample results depends on the judgment of the person selecting the sample. on the judgment of the person selecting the sample.

40 Slide Recommendation It is recommended that probability sampling methods It is recommended that probability sampling methods (simple random, stratified, cluster, or systematic) be (simple random, stratified, cluster, or systematic) be used. used. It is recommended that probability sampling methods It is recommended that probability sampling methods (simple random, stratified, cluster, or systematic) be (simple random, stratified, cluster, or systematic) be used. used. For these methods, formulas are available for For these methods, formulas are available for evaluating the “goodness” of the sample results in evaluating the “goodness” of the sample results in terms of the closeness of the results to the population terms of the closeness of the results to the population parameters being estimated. parameters being estimated. For these methods, formulas are available for For these methods, formulas are available for evaluating the “goodness” of the sample results in evaluating the “goodness” of the sample results in terms of the closeness of the results to the population terms of the closeness of the results to the population parameters being estimated. parameters being estimated. An evaluation of the goodness cannot be made with An evaluation of the goodness cannot be made with non-probability (convenience or judgment) sampling non-probability (convenience or judgment) sampling methods. methods. An evaluation of the goodness cannot be made with An evaluation of the goodness cannot be made with non-probability (convenience or judgment) sampling non-probability (convenience or judgment) sampling methods. methods.