Lecture4-2 - Sampling and Sampling Distributions II

Lecture4-2 - Sampling and Sampling Distributions II -...

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Unformatted text preview: Sampling and Sampling Distributions AS&W – Chapter 7 p Sampling Distribution of Properties of Point Estimators Other Sampling Methods 1 Sampling Distribution ofp Making Inferences about a Population Proportion Population with proportion p=? The value of p is used to make inferences about the value of p. A simple random sample of n elements is selected from the population. The sample data provide a value for p the sample proportion . 2 Sampling Distribution ofp p The sampling distribution of is the probability distribution of all possible values of the sample proportion p . Expected Value ofp E( p) p where: p = the population proportion 3 Sampling Distribution ofp p Standard Deviation of Finite Population p(1 p) N n p n N 1 Infinite Population p(1 p) p n p is referred to as the standard error of the proportion. • A finite population is treated as being infinite if n/N < .05. 4 Form of the Sampling Distribution ofp p The sampling distribution of can be approximated by a normal distribution whenever the sample size is large. The sample size is considered large whenever these conditions are satisfied: np > 5 and n(1 – p) > 5 5 Form of the Sampling Distribution ofp For values of p near .50, sample sizes as small as 10 permit a normal approximation. With very small (approaching 0) or very large (approaching 1) values of p, much larger samples are needed. 6 Sampling Distribution ofp Example: Robert’s College Recall that 72% of the prospective students applying to Robert’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? 7 Sampling Distribution ofp For our example, with n = 30 and p = .72, the normal distribution is an acceptable approximation because: np = 30(.72) = 21.6 > 5 and n(1 - p) = 30(.28) = 8.4 > 5 • The sample of 30 is treated as being infinite since n/N = 30/900 < .05. 8 Sampling Distribution ofp Sampling Distribution of p E(p).72 .72(1 .72) p .082 30 p 9 Sampling Distribution ofp Step 1: Calculate the z-value at the upper endpoint the interval. z = (.77 .72)/.082 = .61 Step 2: Find the area under the curve to the left of th upper endpoint. P(z < .61) = .7291 10 Sampling Distribution ofp Cumulative Probabilities for the Standard Normal Distribution z . .00 . .01 . .02 . .5 .6915 .6950 .6985 .6 .7257 .7291 .7324 .7 .7580 .7611 .7642 .8 .7881 .7910 .7939 .9 .8159 .8186 .8212 . . . . .03 . .04 . .05 . .06 . .7019 .7054 .7088 .7123 .7357 .7389 .7422 .7454 .7673 .7704 .7734 .7764 .7967 .7995 .8023 .8051 .07 . .08 . .09 . .7157 .7190 .7224 .7486 .7517 .7549 .7794 .7823 .7852 .8078 .8106 .8133 .8238 .8264 .8289 .8315 .8340 .8365 .8389 . . . . . . . 11 Sampling Distribution ofp Sampling Distribution of p p .082 Area = .7291 p .72 .77 12 Sampling Distribution ofp Step 3: Calculate the z-value at the lower endpoint o the interval. z = (.67 .72)/.082 = - .61 Step 4: Find the area under the curve to the left of th lower endpoint. P(z < -.61) = P(z > .61) = 1 P(z < .61) = 1 . 7291 = .2709 13 Sampling Distribution ofp Sampling Distribution of p p .082 Area = .2709 p .67 .72 14 Sampling Distribution ofp 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 The probability that the sample proportion of applican applica wanting on-campus housing will be within +/-.05 of th actual population proportion : P(.67 < p< .77) = .4582 15 Sampling Distribution ofp Sampling Distribution of p p .082 Area = .4582 p .67 .72 .77 16 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. Let θ be the population parameter of interest Then is the sample statistic or the point estimator of θ ˆ Unbiased Efficiency Consistency 17 Properties of Point Estimators Unbiased 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. ˆ Sampling Distribution of ˆ Unbiased Estimator E (ˆ) ˆ Biased Estimator 18 Properties of Point Estimators Efficiency 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. The point estimator with the smaller standard deviation is said to have greater relative efficiency than the other. 19 Properties of Point Estimators Efficiency Sampling Distribution of ˆ2 Sampling Distribution of ˆ1 ˆ First point estimator is relatively more efficient 20 Properties of Point Estimators Consistency 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. Recall the standard deviation (error)x of x n 21 Other Sampling Methods Stratified Random Sampling Cluster Sampling Systematic Sampling Convenience Sampling Judgment Sampling 22 Stratified Random Sampling The The population population is is first first divided divided into into groups groups of of elements elements called called strata strata.. Each Each element element in in the the population population belongs belongs to to one one and and only only one one stratum. stratum. Best Best results results are are obtained obtained when when the the elements elements within within each each stratum stratum are are as as much much alike alike as as possible possible (i.e. (i.e. aa homogeneous homogeneous group group). ). 23 Stratified Random Sampling A A simple simple random random sample sample is is taken taken from from each each stratum. stratum. Formulas Formulas are are available available for for combining combining the the stratum stratum sample sample results results into into one one population population parameter parameter estimate. estimate. Advantage Advantage:: IfIf strata strata are are homogeneous, homogeneous, this this method method is is as as “precise” “precise” as as simple simple random random sampling sampling but but with with aa smaller smaller total total sample sample size. size. Example Example:: The The basis basis for for forming forming the the strata strata might might be be department, department, location, location, age, age, industry industry type, type, and and so so on. on. 24 Cluster Sampling The The population population is is first first divided divided into into separate separate groups groups of of elements elements called called clusters clusters.. Ideally, Ideally, each each cluster cluster is is aa representative representative small-scale small-scale version version of of the the population population (i.e. (i.e. heterogeneous heterogeneous group). group). A A simple simple random random sample sample of of the the clusters clusters is is then then taken. taken All All elements elements within within each each sampled sampled (chosen) (chosen) cluster cluster form form the the sample. sample. 25 Cluster Sampling Example Example:: A A primary primary application application is is area area sampling, sampling, where where clusters clusters are are city city blocks blocks or or other other well-defined well-defined areas. areas. Advantage Advantage:: The The close close proximity proximity of of elements elements can can be be cost cost effective effective (i.e. (i.e. many many sample sample observations observations can can be be obtained obtained in in aa short short time). time). Disadvantage Disadvantage:: This This method method generally generally requires requires aa larger larger total total sample sample size size than than simple simple or or stratified stratified random random sampling. sampling. 26 Systematic Sampling IfIf aa sample sample size size of of nn is is desired desired from from aa population population containing containing N N elements, elements, we we might might sample sample one one element element for for every every N N//nn elements elements in in the the population. population. We We randomly randomly select select one one of of the the first first N N//nn elements elements from from the the population population list. list. We We then then select select every every N N//nn th th element element that that follows follows in in the the population population list. list. 27 Systematic Sampling This This method method has has the the properties properties of of aa simple simple random random sample, sample, especially especially ifif the the list list of of the the population population elements elements is is aa random random ordering. ordering. Advantage Advantage:: The The sample sample usually usually will will be be easier easier to to identify identify than than it it would would be be ifif simple simple random random sampling sampling were were used. used. Example Example:: Selecting Selecting every every 100 100thth listing listing in in aa telephone telephone book book after after the the first first randomly randomly selected selected listing listing 28 Convenience Sampling It It is is aa nonprobability nonprobability sampling sampling technique technique.. Items Items are are included included in in the the sample sample without without known known probabilities probabilities of of being being selected. selected. The The sample sample is is identified identified primarily primarily by by convenience convenience.. Example Example:: A A professor professor conducting conducting research research might might use use student student volunteers volunteers to to constitute constitute aa sample. sample. 29 Convenience Sampling Advantage Advantage:: Sample Sample selection selection and and data data collection collection are are relatively relatively easy. easy. Disadvantage Disadvantage:: It It is is impossible impossible to to determine determine how how representative representative of of the the population population the the sample sample is. is. 30 Judgment Sampling The The person person most most knowledgeable knowledgeable on on the the subject subject of of the the study study selects selects elements elements of of the the population population that that he he or or she she feels feels are are most most representative representative of of the the population. population. It It is is aa nonprobability nonprobability sampling sampling technique technique.. Example Example:: A A reporter reporter might might sample sample three three or or four four senators, senators, judging judging them them as as reflecting reflecting the the general general opinion opinion of of the the senate. senate. 31 Judgment Sampling Advantage Advantage:: It It is is aa relatively relatively easy easy way way of of selecting selecting aa sample. sample. Disadvantage Disadvantage:: The The quality quality of of the the sample sample results results depends depends on on the the judgment judgment of of the the person person selecting selecting the the sample. sample. 32 ...
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