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# BNAD277 Chapter 7 Notes BNAD277

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This 6 page Class Notes was uploaded by Kristin Koelewyn on Thursday February 4, 2016. The Class Notes belongs to BNAD277 at University of Arizona taught by Dr. S. Umashankar in Spring 2016. Since its upload, it has received 75 views. For similar materials see Business Statistics in Business at University of Arizona.

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Date Created: 02/04/16

Bnad277: Chapter 7 Notes Sampling and Sampling Distributions - Introduction o An element is the entity on which data are collected o A population is a collection of all the elements of interest o A sample is a subset of the population o The sampled population is the population from which the sample is drawn o A frame is a list of the elements that the sample will be selected from o The reason we select a sample is to collect data to answer a research question about a population o The sample results provide only estimates of the values of the population characteristics o The reason is simply that the sample contains only a portion of the population o With proper sampling methods, the sample results can provide good estimates of the population characteristics - Selecting a sample from a Finite Population: o Finite populations are often defined by lists such as: ▯ Organization membership roster ▯ Credit card account numbers ▯ Inventory product numbers o 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. o Replacing each sampled element before selecting subsequent elements is called sampling with replacement. o Sampling without replacement is the procedure used most often. o In large sampling projects, computer-generated random numbers are often used to automate the sample selection process. ▯ Example: St. Andrew’s College received 900 applications for admission in the upcoming year from prospective students. The applicants were numbered, from 1 to 900, as their applications arrived. The Director of Admissions would like to select a simple random sample of 30 applicants. • Step 1: Assign a random number to each of the 900 applicants (generated by Excel’s RAND function). • Step 2: Select the 30 applicants corresponding to the 30 smallest random numbers. - Sampling from an Infinite Population: o Sometimes we want to select a sample, but find it is not possible to obtain a list of all elements in the population. o As a result, we cannot construct a frame for the population. o Hence, we cannot use the random number selection procedure. o Most often this situation occurs in infinite population cases. o Populations are often generated by an ongoing process where there is no upper limit on the number of units that can be generated. o Some examples of on-going processes, with infinite populations, are: ▯ parts being manufactured on a production line ▯ transactions occurring at a bank ▯ telephone calls arriving at a technical help desk ▯ customers entering a store o In the case of an infinite population, we must select a random sample in order to make valid statistical inferences about the population from which the sample is taken. o A random sample from an infinite population is a sample selected such that the following conditions are satisfied. ▯ Each element selected comes from the population of interest. ▯ Each element is selected independently. - Point Estimation: o Point estimation is a form of statistical inference o 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. o We refer to x bar as the point estimator of the population mean mu, s is the point estimator of the population standard deviation, and p bar is the point estimator of the population proportion p. - Practical Advice: o The target population is the population we want to make inferences about. o The sampled population is the population from which the sample is actually taken. o Whenever a sample is used to make inferences about a population, we should make sure that the targeted population and the sampled population is in close agreement. - Sampling Distribution of x bar: o The sampling distribution of x bar us the probability distribution of all possible values of the sample mean x bar. o Expected value of x bar: ▯ When the expected value of the point estimator equals the population parameter, we say the point estimator is unbiased. o Standard deviation of x bar: o Finite Population: o Infinite Population: o When the population has a normal distribution, the sampling distribution of x bar is normally distributed for any sample size. o In most applications, the sampling distribution of x bar can be approximated by a normal distribution whenever the sample is size 30 or more. o In cases where the population is highly skewed or outliers are present, samples of size 50 may be needed. - Central Limit Theorem: o When the population from which we are selecting a random sample does not have a normal distribution, the central limit theorem is helpful in identifying the shape of the sampling distribution x bar. ▯ Example: 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 m? • Step 1: Calculate the z-value at the upper endpoint of the interval. Z=(1707-1697)/15.96=.63 • Step 2: Find the area under the curve to the left of the upper endpoint. P(z<.63)=.7357 • Step 3: Calculate the z-value at the lower endpoint of the interval. Z=(1687-1697)/15.96=-.63 • Step 4: Find the area under the curve to the left of the lower endpoint. P(z<-.63)=.2643 • 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) =.7357-.2643=.4714 - Sampling Distribution of p bar: o Expected value of p bar: o Finite Population: o Infinite Population: o Example: Recall that 72% of the prospective students applying to St. Andrew’s College desire on-campus housing. 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: ▯ Step 1: Calculate the z-value at the upper endpoint of the interval. z=(.77-.72)/.082=.61 ▯ Step 2: Find the area under the curve to the left of the upper endpoint. P(z<.61)=.7291 ▯ Step 3: Calculate the z-value at the lower endpoint of the interval. z=(.67-.72)/.082=-.61 ▯ Step 4: Find the area under the curve to the left of the lower endpoint. P(z<-.61)=.2709 ▯ Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval. P(-.61<z<.61) = P(z<.61)- P(z<.-61) = .7291-.2709 = .4582 • Probability that proportion of applicants will be within +/- .05: - Stratified Random Sampling: o The population id 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. o A 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. o 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. o A simple random sample of the clusters is then taken. o All elements within each sampled (chosen) cluster from the sample. o Example: A primary application is area sampling, where clusters are city blocks or other well-defined areas. o Advantage: The close proximity of the elements can be cost effective. o Disadvantage: This method generally requires a larger total sample size than a simple or stratified random sampling. - 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 Advantage: The sample usually will be easier to identify than it would be if a simple random sampling were used. o Example: Selecting every 100 listing in a telephone book after the first randomly selected listing. - Convenience Sampling: o It is a non-probability sampling technique. Items are included in the sample without known probabilities of being selected. o The sample is identified primarily by convenience. o Example: A professor conducting research might use student volunteers to constitute a sample. 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. - 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 non-probability sampling technique. o Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate. 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.

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