Chapter 7, Part B Sampling and Sampling Distributions

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1 Slides Preared by JOHN S. LOUCKS St. Edward s University Slide 1 Chater 7, Part B Samling and Samling Distributions Samling Distribution of Proerties of Point Estimators Other Samling Methods Slide 2 Samling Distribution of Making Inferences about a Poulation Proortion Poulation with roortion =? A simle random samle of n elements is selected from the oulation. The value of is used to make inferences about the value of. The samle data rovide a value for the samle roortion. Slide 3 1

2 Samling Distribution of The samling distribution of is the robability distribution of all ossible values of the samle roortion. Exected Value of E ( ) = where: = the oulation roortion Slide 4 Samling Distribution of Standard Deviation of Finite Poulation Infinite Poulation ( 1 ) σ = n N n N 1 ( 1 ) σ = n σ σ is referred to as the standard error of the roortion. Slide 5 Form of the Samling Distribution of The samling distribution of can be aroximated by a normal distribution whenever the samle size is large. The samle size is considered large whenever these conditions are satisfied: n > 5 and n(1 ) > 5 Slide 6 2

3 Form of the Samling Distribution of For values of near.50, samle sizes as small as 10 ermit a normal aroximation. With very small (aroaching 0) or very large (aroaching 1) values of, much larger samles are needed. Slide 7 Samling Distribution of Examle: St. Andrew s College Recall that 72% of the rosective students alying to St. Andrew s College desire on-camus housing. What is the robability that a simle random samle of 30 alicants will rovide an estimate of the oulation roortion of alicant desiring on-camus housing that is within lus or minus.05 of the actual oulation roortion? Slide 8 Samling Distribution of For our examle, with n = 30 and =.72, the normal distribution is an accetable aroximation because: n = 30(.72) = 21.6 > 5 and n(1 - ) = 30(.28) = 8.4 > 5 Slide 9 3

4 Samling Distribution of Samling Distribution of.72(1.72) σ = = E ( ) =.72 Slide 10 Samling Distribution of Ste 1: Calculate the z-value at the uer endoint of the interval. z = ( )/.082 =.61 Ste 2: Find the area under the curve to the left of the uer endoint. P(z <.61) =.7291 Slide 11 Samling Distribution of Cumulative Probabilities for the Standard Normal Distribution z Slide 12 4

5 Samling Distribution of Samling Distribution of σ =.082 Area = Slide 13 Samling Distribution of Ste 3: Calculate the z-value at the lower endoint of the interval. z = ( )/.082 = -.61 Ste 4: Find the area under the curve to the left of the lower endoint. P(z < -.61) = P(z >.61) = 1 - P(z <.61) = =.2709 Slide 14 Samling Distribution of Samling Distribution of σ =.082 Area = Slide 15 5

6 Samling Distribution of Ste 5: Calculate the area under the curve between the lower and uer endoints of the interval. P(-.61 < z <.61) = P(z <.61) - P(z < -.61) = =.4582 The robability that the samle roortion of alicants wanting on-camus housing will be within +/-.05 of the actual oulation roortion : P(.67 < <.77) =.4582 Slide 16 Samling Distribution of Samling Distribution of σ =.082 Area = Slide 17 Proerties of Point Estimators Before using a samle statistic as a oint estimator, statisticians check to see whether the samle statistic has the following roerties associated with good oint estimators. Unbiased Efficiency Consistency Slide 18 6

7 Proerties of Point Estimators Unbiased If the exected value of the samle statistic is equal to the oulation arameter being estimated, the samle statistic is said to be an unbiased estimator of the oulation arameter. Slide 19 Proerties of Point Estimators Efficiency Given the choice of two unbiased estimators of the same oulation arameter, we would refer to use the oint estimator with the smaller standard deviation, since it tends to rovide estimates closer to the oulation arameter. The oint estimator with the smaller standard deviation is said to have greater relative efficiency than the other. Slide 20 Proerties of Point Estimators Consistency A oint estimator is consistent if the values of the oint estimator tend to become closer to the oulation arameter as the samle size becomes larger. Slide 21 7

8 Other Samling Methods Stratified Random Samling Cluster Samling Systematic Samling Convenience Samling Judgment Samling Slide 22 Stratified Random Samling The oulation is first divided into grous of elements called strata. Each element in the oulation belongs to one and only one stratum. Best results are obtained when the elements within each stratum are as much alike as ossible (i.e. a homogeneous grou). Slide 23 Stratified Random Samling A simle random samle is taken from each stratum. Formulas are available for combining the stratum samle results into one oulation arameter estimate. Advantage: If strata are homogeneous, this method is as recise as simle random samling but with a smaller total samle size. Examle: The basis for forming the strata might be deartment, location, age, industry tye, and so on. Slide 24 8

9 Cluster Samling The oulation is first divided into searate grous of elements called clusters. Ideally, each cluster is a reresentative small-scale version of the oulation (i.e. heterogeneous grou). A simle random samle of the clusters is then taken. All elements within each samled (chosen) cluster form the samle. Slide 25 Cluster Samling Examle: A rimary alication is area samling, where clusters are city blocks or other well-defined areas. Advantage: The close roximity of elements can be cost effective (i.e. many samle observations can be obtained in a short time). Disadvantage: This method generally requires a larger total samle size than simle or stratified random samling. Slide 26 Systematic Samling If a samle size of n is desired from a oulation containing N elements, we might samle one element for every n/n elements in the oulation. We randomly select one of the first n/n elements from the oulation list. We then select every n/nth element that follows in the oulation list. Slide 27 9

10 Systematic Samling This method has the roerties of a simle random samle, esecially if the list of the oulation elements is a random ordering. Advantage: The samle usually will be easier to identify than it would be if simle random samling were used. Examle: Selecting every 100 th listing in a telehone book after the first randomly selected listing Slide 28 Convenience Samling It is a nonrobability samling technique. Items are included in the samle without known robabilities of being selected. The samle is identified rimarily by convenience. Examle: A rofessor conducting research might use student volunteers to constitute a samle. Slide 29 Convenience Samling Advantage: Samle selection and data collection are relatively easy. Disadvantage: It is imossible to determine how reresentative of the oulation the samle is. Slide 30 10

11 Judgment Samling The erson most knowledgeable on the subject of the study selects elements of the oulation that he or she feels are most reresentative of the oulation. It is a nonrobability samling technique. Examle: A reorter might samle three or four senators, judging them as reflecting the general oinion of the senate. Slide 31 Judgment Samling Advantage: It is a relatively easy way of selecting a samle. Disadvantage: The quality of the samle results deends on the judgment of the erson selecting the samle. Slide 32 End of Chater 7, Part B Slide 33 11

Chapter 7b - Point Estimation and Sampling Distributions

Chapter 7b - Point Estimation and Sampling Distributions Chater 7b - Point Estimation and Samling Distributions Chater 7 (b) Point Estimation and Samling Distributions Point estimation is a form of statistical inference. In oint estimation we use the data from