Chapter 7b - Point Estimation and Sampling Distributions

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Chater 7b - Point Estimation and Samling Distributions Chater 7 (b) Point Estimation and Samling Distributions Point estimation is a form of statistical inference.

as the oint estimator of the oulation s is the oint estimator of the oulation standard deviation. is the oint estimator of the oulation roortion.

1 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 the samle to comute a value of a samle statistic that serves as an estimate of a oulation arameter. We refer to mean. as the oint estimator of the oulation s is the oint estimator of the oulation standard deviation. is the oint estimator of the oulation roortion. 1 Point Estimation Out of a total number of 2500 emloyees, a simle random samle of 30 emloyees and corresonding data on annual salary and management training rogram articiation are shown in the given table. 1, 2.. n is used to denote annual salary of the emloyees and the articiation in the management training rogram is indicated by a yes / no. Annual Salary ($) Management Training Program Annual Salary ($) Management Training Program Annual Salary ($) Management Training Program X 1 = YES 45, YES 45, YES X 2 = YES 57, NO 51, YES 49, YES 55, YES 54, NO 49, YES 51, NO 50, NO 47, NO 56, NO 52, NO 55, YES 51, YES 50, NO 49, YES 52, NO 52, NO 51, YES YES 50, YES 50, YES 51, YES 55, YES 55, YES 52, YES 57, NO 2 Point Estimation Poulation Parameter = Poulation mean annual salary = Poulation standard deviation for annual salary = Poulation roortion having comleted MTP Summary of Point Estimates Obtained from a Simle Random Samle Parameter value Point estimator Point estimate $ 51,800 $ 51,814 $ 4,000 s = Samle standard deviation for annual salary $ 3, Practical Advice 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. Samling Distribution of Process of Statistical Inference Poulation with mean =? A simle random samle of n elements is selected from the oulation. 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. The value of is used to make inferences about the value of. The samle data rovide a value for the samle mean

2 Chater 7b - Point Estimation and Samling Distributions Samling Distribution of Samling Distribution of Standard Deviation of We will use the following notation to define the standard deviation of the samling distribution of. where: µ = the oulation mean When the eected value of the oint estimator equals the oulation arameter, we say the oint estimator is unbiased. = the standard deviation of = the standard deviation of the oulation n = the samle size N = the oulation size 7 8 Samling Distribution 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. In cases where the oulation is highly skewed or outliers are resent, samles of size 50 may be needed Central Limit Theorem Note: n/n = 30/2500 =.012 Because samle is less than 5% of the oulation size, We ignore the finite oulation correction factor

Chater 7b - Point Estimation and Samling Distributions Suose the ersonnel director believes the samle mean will be an accetable estimate of oulation mean if the samle mean is within $500 of the

68 What is the robability that the samle mean comuted using a simle random samle of 30 emloyees will be within $500 of oulation mean?

3 Chater 7b - Point Estimation and Samling Distributions Suose the ersonnel director believes the samle mean will be an accetable estimate of oulation mean if the samle mean is within $500 of the oulation mean. Ste 1: Calculate the z-value at the uer endoint of the interval. z = (52,300 51,800)/ =.68 What is the robability that the samle mean comuted using a simle random samle of 30 emloyees will be within $500 of oulation mean? Ste 2: Find the area under the curve to the left of the uer endoint. P(z <.68) = Cumulative Probabilities for the Standard Normal Distribution Z Function used - NORM.DIST We do not have to make searate comutation of z value. Evaluating NORM.DIST function at each end oint of the interval rovides cumulative robability at the secified end oint of the interval. The result obtained using NORM.DIST is more accurate

Chater 7b - Point Estimation and Samling Distributions µ 19 20 21 22 Samling Distribution of The samling distribution of is the robability distribution of all ossible values of the samle roortion ṗ

4 Chater 7b - Point Estimation and Samling Distributions µ Samling Distribution of The samling distribution of is the robability distribution of all ossible values of the samle roortion ṗ Eected Value of E( ) where: = the oulation roortion Samling Distribution of 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

Chater 7b - Point Estimation and Samling Distributions Form of the Samling Distribution of The samling distribution of can be aroimated by a normal distribution whenever the samle size is large

5 Chater 7b - Point Estimation and Samling Distributions Form of the Samling Distribution of 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 For the EAI study we know that the oulation roortion of emloyees who articiated in the management training rogram is = 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). What is the robability that a simle random samle of 30 emloyees will rovide an estimate of the oulation roortion of emloyees attending management rogram that is within lus or minus.05 of the actual oulation roortion? For our eamle, with n = 30 and =.6, the normal distribution is an accetable aroimation because: n = 30(.6) = 18 > 5 And n(1 - ) = 30(.4) = 12 > 5 Ste 1: Calculate the z-value at the uer endoint of the interval. z = ( )/.0894 =.56 Ste 2: Find the area under the curve to the left of the uer endoint. P(z <.56) = Z Ste 3: Calculate the z-value at the lower endoint of the interval. z = ( )/.0894 = -.56 Ste 4: Find the area under the curve to the left of the lower endoint. P(z < -.56) = Ste 5: Calculate the area under the curve between the lower and uer endoints of the interval. P(-.56 < z <.56) = P(z <.56) - P(z < -.56) = =

Chater 7b - Point Estimation and Samling Distributions Other Tyes of Samling Stratified Random Samling Cluster Samling The oulation is first divided into grous of elements called strata.

31 32 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 then select every n/nth element that follows in the oulation list. Advantage: The samle usually will be easier to identify than it would be if simle random samling were used.

Items are included in the samle without known robabilities of being selected. The samle is identified rimarily by convenience.

6 Chater 7b - Point Estimation and Samling Distributions Other Tyes of Samling Stratified Random Samling Cluster 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) 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. This method has the roerties of a simle random samle, esecially if the list of the oulation elements is a random ordering. 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. 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 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. Eamle: A rofessor conducting research might use student volunteers to constitute a samle. Advantage: Samle selection and data collection are relatively easy. Disadvantage: It is imossible to determine how reresentative of the oulation the samle is 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. Eamle: A reorter might samle three or four senators, judging them as reflecting the general oinion of the senate. 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. Recommendation 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

Chapter 7, Part B Sampling and Sampling Distributions

Chapter 7, Part B Sampling and Sampling Distributions 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