Chapter 7 Sampling and Sampling Distributions. Introduction. Selecting a Sample. Introduction. Sampling from a Finite Population

Transcription

1 Chater 7 and s Selecting a Samle Point Estimation Introduction to s of Proerties of Point Estimators Other Methods Introduction An element is the entity on which data are collected. A oulation is a collection of all the elements of interest. A samle is a subset of the oulation. The samled oulation is the oulation from which the samle is drawn. A frame is a list of the elements that the samle will be selected from. Introduction The reason we select a samle is to collect data to answer a research question about a oulation. Selecting a Samle from a Finite Poulation from an Infinite Poulation The samle results rovide only estimates of the values of the oulation characteristics. The reason is simly that the samle contains only a ortion of the oulation. With roer samling methods, the samle results can rovide good estimates of the oulation characteristics. from a Finite Poulation Finite oulations are often defined by lists such as: Organization membershi roster Credit card account numbers Inventory roduct numbers A simle random samle of size n from a finite oulation of size N is a samle selected such that each ossible samle of size n has the same robability of being selected. from a Finite Poulation Relacing each samled element before selecting subsequent elements is called samling with relacement. without relacement is the rocedure used most often. In large samling rojects, comuter-generated random numbers are often used to automate the samle selection rocess. 1

2 from a Finite Poulation Eamle: St. Andrew s College St. Andrew s College received 900 alications for admission in the ucoming year from rosective students. The alicants were numbered, from 1 to 900, as their alications arrived. The Director of Admissions would like to select a simle random samle of 30 alicants. from a Finite Poulation Eamle: St. Andrew s College Ste 1: Assign a random number to each of the 900 alicants. The random numbers generated by Ecel s RAND function follow a uniform robability distribution between 0 and 1. Ste 2: Select the 30 alicants corresonding to the 30 smallest random numbers. from an Infinite Poulation from an Infinite Poulation Sometimes we want to select a samle, but find it is not ossible to obtain a list of all elements in the oulation. As a result, we cannot construct a frame for the oulation. Hence, we cannot use the random number selection rocedure. Most often this situation occurs in infinite oulation cases. Poulations are often generated by an ongoing rocess where there is no uer limit on the number of units that can be generated. Some eamles of on-going rocesses, with infinite oulations, are: arts being manufactured on a roduction line transactions occurring at a bank telehone calls arriving at a technical hel desk customers entering a store from an Infinite Poulation In the case of an infinite oulation, we must select a random samle in order to make valid statistical inferences about the oulation from which the samle is taken. A random samle from an infinite oulation is a samle selected such that the following conditions are satisfied. Each element selected comes from the oulation of interest. Each element is selected indeendently. Point Estimation Point estimation is a form of statistical inference. In oint estimation we use the data from the samle to comute a value of a samle statistic that serves as an estimate of a oulation arameter. We refer to as the oint estimator of the oulation mean. s is the oint estimator of the oulation standard deviation. is the oint estimator of the oulation roortion. 2

3 Point Estimation Eamle: St. Andrew s College Recall that St. Andrew s College received 900 alications from rosective students. The alication form contains a variety of information including the individual s Scholastic Atitude Test (SAT) score and whether or not the individual desires on-camus housing. At a meeting in a few hours, the Director of Admissions would like to announce the average SAT score and the roortion of alicants that want to live on camus, for the oulation of 900 alicants. Point Estimation Eamle: St. Andrew s College However, the necessary data on the alicants have not yet been entered in the college s comuterized database. So, the Director decides to estimate the values of the oulation arameters of interest based on samle statistics. The samle of 30 alicants is selected using comuter-generated random numbers. Point Estimation Point Estimation as Point Estimator of i 32, s as Point Estimator of 2 ( i ) 163,996 s as Point Estimator of Note: Different random numbers would have identified a different samle which would have resulted in different oint estimates. Once all the data for the 900 alicants were entered in the college s database, the values of the oulation arameters of interest were calculated. Poulation Mean SAT Score i Poulation Standard Deviation for SAT Score 2 ( i ) Poulation Proortion Wanting On-Camus Housing Summary of Point Estimates Obtained from a Simle Random Samle Practical Advice Poulation Parameter = Poulation mean SAT score = Poulation std. deviation for SAT score = Poulation roortion wanting camus housing Parameter Value Point Estimator Point Estimate 1090 = Samle mean 1097 SAT score 80 s = Samle std. deviation for SAT score = Samle roortion.68 wanting camus housing The target oulation is the oulation we want to make inferences about. The samled oulation is the oulation from which the samle is actually taken. Whenever a samle is used to make inferences about a oulation, we should make sure that the targeted oulation and the samled oulation are in close agreement. 3

4 of Process of Statistical Inference Poulation with mean =? A simle random samle of n elements is selected from the oulation. of The samling distribution of is the robability distribution of all ossible values of the samle mean. Eected Value of E( ) = The value of is used to make inferences about the value of. The samle data rovide a value for the samle mean. where: = the oulation mean When the eected value of the oint estimator equals the oulation arameter, we say the oint estimator is unbiased. of Standard Deviation of We will use the following notation to define the standard deviation of the samling distribution of. = the standard deviation of = the standard deviation of the oulation n = the samle size N = the oulation size of Standard Deviation of Finite Poulation N n ( ) N 1 n Infinite Poulation n A finite oulation is treated as being infinite if n/n <.05. ( N n) / ( N 1) is the finite oulation correction factor. is referred to as the standard error of the mean. of When the oulation has a normal distribution, the samling distribution of is normally distributed for any samle size. of The samling distribution of can be used to rovide robability information about how close the samle mean is to the oulation mean. In most alications, the samling distribution of can be aroimated by a normal distribution whenever the samle is size 30 or more. In cases where the oulation is highly skewed or outliers are resent, samles of size 50 may be needed. 4

5 Central Limit Theorem When the oulation from which we are selecting a random samle does not have a normal distribution, the central limit theorem is helful in identifying the shae of the samling distribution of. CENTRAL LIMIT THEOREM In selecting random samles of size n from a oulation, the samling distribution of the samle mean can be aroimated by a normal distribution as the samle size becomes large. of Eamle: St. Andrew s College of for SAT Scores E ( ) n 30 of Eamle: St. Andrew s College What is the robability that a simle random samle of 30 alicants will rovide an estimate of the oulation mean SAT score that is within +/10 of the actual oulation mean? In other words, what is the robability that will be between 1080 and 1100? of Eamle: St. Andrew s College Ste 1: Calculate the z-value at the uer endoint of the interval. z = ( )/14.6=.68 Ste 2: Find the area under the curve to the left of the uer endoint. P(z <.68) =.7517 of of Eamle: St. Andrew s College Eamle: St. Andrew s College Cumulative Probabilities for the Standard Normal z of for SAT Scores Area =

6 of of for SAT Scores Eamle: St. Andrew s College Ste 3: Calculate the z-value at the lower endoint of the interval. z = ( )/14.6= -.68 Ste 4: Find the area under the curve to the left of the lower endoint. P(z < -.68) =.2483 Eamle: St. Andrew s College of for SAT Scores Area = of for SAT Scores Eamle: St. Andrew s College Ste 5: Calculate the area under the curve between the lower and uer endoints of the interval. P(-.68 < z <.68) = P(z <.68) - P(z < -.68) = =.5034 The robability that the samle mean SAT score will be between 1080 and 1100 is: P(1080 < < 1100) =.5034 of for SAT Scores Eamle: St. Andrew s College of for SAT Scores Area =.5034 Relationshi Between the Samle Size and the of Eamle: St. Andrew s College Suose we select a simle random samle of 100 alicants instead of the 30 originally considered. E( ) = regardless of the samle size. In our eamle, E( ) remains at Whenever the samle size is increased, the standard error of the mean is decreased. With the increase in the samle size to n = 100, the standard error of the mean is decreased from 14.6 to: n 100 Relationshi Between the Samle Size and the of Eamle: St. Andrew s College With n = 100, 8 E ( ) 1090 With n = 30,

7 Relationshi Between the Samle Size and the of Eamle: St. Andrew s College Recall that when n = 30, P(1080 < < 1100) = We follow the same stes to solve for P(1080 < < 1100) when n = 100 as we showed earlier when n = 30. Now, with n = 100, P(1080 < < 1100) = Because the samling distribution with n = 100 has a smaller standard error, the values of have less variability and tend to be closer to the oulation mean than the values of with n = 30. Relationshi Between the Samle Size and the of Eamle: St. Andrew s College of for SAT Scores Area =.7888 Making Inferences about a Poulation Proortion Poulation with roortion =? A simle random samle of n elements is selected from the oulation. The samling distribution of is the robability distribution of all ossible values of the samle roortion. Eected Value of E( ) The value of is used to make inferences about the value of. The samle data rovide a value for the samle roortion. where: = the oulation roortion Standard Deviation of Finite Poulation Infinite Poulation N n (1 ) ( 1 ) N 1 n n is referred to as the standard error of the roortion. ( N n) / ( N 1) is the finite oulation correction factor. Form of the The samling distribution of can be aroimated by a normal distribution whenever the samle size is large enough to satisfy the two conditions: n > 5 and n(1 ) > 5... because when these conditions are satisfied, the robability distribution of in the samle roortion, = /n, can be aroimated by normal distribution (and because n is a constant). 7

8 Eamle: 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? Eamle: St. Andrew s College For our eamle, with n = 30 and =.72, the normal distribution is an accetable aroimation because: n = 30(.72) = 21.6 > 5 and n(1 - ) = 30(.28) = 8.4 > 5 Eamle: St. Andrew s College Eamle: St. Andrew s College of.72(1.72) 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 E ( ).72 Eamle: St. Andrew s College Eamle: St. Andrew s College Cumulative Probabilities for the Standard Normal z of Area =

9 Eamle: St. Andrew s College 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) =.2709 of Eamle: St. Andrew s College Area = Eamle: St. Andrew s College 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 of Eamle: St. Andrew s College Area =.4582 Proerties of Point Estimators 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 Unbiased If the eected 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. 9

10 Efficiency Proerties of Point Estimators 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. Consistency Proerties of Point Estimators 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. In other words, a large samle size tends to rovide a better oint estimate than a small samle size. Other Methods Stratified Random Stratified Random Cluster Systematic Convenience Judgment 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). Stratified Random Cluster 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. Eamle: The basis for forming the strata might be deartment, location, age, industry tye, and so on. 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. 10

11 Cluster Systematic Eamle: A rimary alication is area samling, where clusters are city blocks or other well-defined areas. Advantage: The close roimity 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. 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. Systematic Convenience 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. Eamle: Selecting every 100 th listing in a telehone book after the first randomly selected listing 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. Eamle: A rofessor conducting research might use student volunteers to constitute a samle. Convenience Judgment Advantage: Samle selection and data collection are relatively easy. Disadvantage: It is imossible to determine how reresentative of the oulation the samle is. 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. Eamle: A reorter might samle three or four senators, judging them as reflecting the general oinion of the senate. 11

12 Judgment Recommendation 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. It is recommended that robability samling methods (simle random, stratified, cluster, or systematic) be used. For these methods, formulas are available for evaluating the goodness of the samle results in terms of the closeness of the results to the oulation arameters being estimated. An evaluation of the goodness cannot be made with non-robability (convenience or judgment) samling methods. End of Chater 7 12