8.7: Estimation and Sample Size Determination for …

1 : Estimation and Sample size Determination for Finite PopulationsCD8-1 FIGURE interval estimate forthe mean sales invoice amountwith the finite populationcorrection for Saxon : Estimation AND Sample size Determination FOR FINITE POPULATIONSE stimating the MeanIn section thefinite population correction (fpc) factoris used to reduce the standard errorby a value equal to. When developing confidence interval estimates for popu-lation parameters, the fpc factor is used when samples are selected without replacement. Thus,the (1 ) 100% confidence interval estimate for the mean is calculated as in Equation( ).CONFIDENCE INTERVAL FOR A MEAN ( UNKNOWN) FOR A FINITE population ( )To illustrate the finite population correction factor, refer to the confidence interval estimate forthe mean developed for Saxon Plumbing Company in section Suppose that in this month thereare 5,000 sales invoices.

2 Using = $ ,S= $ ,N= 5,000,n= 100, and with 95% confi-dence,t99= From equation ( )Figure illustrates the PHStat output for this 271 984228 951005 0001005 0001110 275 744 0 99110 275 69104 58115 96.(.).,,..(.)..$.$. = XXtSnNnNn 11()/()NnN 1In this case, because the Sample is a very small fraction of the population , the correction factor has aminimal effect on the width of the confidence interval. To examine the effect of the correction factorwhen the Sample size is more than 5% of the population size , see Example THE MEAN FORCEFOR INSULATORSCD8-2CD MATERIALIn Example , a Sample of 30 insulators was selected. Suppose a population of 300 insulators wereproduced by the company.

3 Set up a 95% confidence interval estimate of the population the finite population correction factor, with = 1, pounds,S= ,n= 30,N= 300, and t29= (for 95% confidence):Here, because 10% of the population is to be sampled, the fpc factor has a small effect on the confi-dence interval the ProportionIn sampling without replacement, the (1 ) 100% confidence interval estimate of the propor-tion is defined in Equation ( ).CONFIDENCE INTERVAL ESTIMATE FOR THE PROPORTION USING THE FINITEPOPULATION CORRECTION FACTOR( )To illustrate the use of the finite population correction factor when developing a confidence intervalestimate of the population proportion, consider again the estimate developed for Saxon HomeImprovement Company in section For these data,N= 5,000,n= 100,ps= 10/100 = , andwith 95% confidence,Z= Using Equation ( ),In this case, because the Sample is a very small fraction of the population , the fpc factor has virtuallyno effect on the confidence interval = = = ().

4 (.)( .)( .),,.(.)(.)(.)..11010196010 0901005 0001005 0001010196 003 0990 1005820 04180 1582pZppnNnNsss ()11 XtSnNnNn = = = 111 723 42 045289 55303003030011 723 433 44 0 95031 723 431 7761 691 621 755 18,. (. ).,..(. ),..,.,. : Estimation and Sample size Determination for Finite PopulationsCD8-3 Determining the Sample SizeJust as the fpc factor is used to develop confidence interval estimates, it also is used to determinesample size when sampling without replacement. For example, in estimating the mean, the samplingerror isand in estimating the proportion, the sampling error isTo determine the Sample size in estimating the mean or the proportion from Equations ( ) and ( ),where n0is the Sample size without considering the finite population correction factor.

5 Applying thefpc factor results in the actual Sample size n, computed as in Equation ( ). Sample size Determination USING THE FINITE population CORRECTION FACTOR( )In determining the Sample size for Saxon Home Improvement Company, a Sample size of 97 wasneeded (rounded up from ) for the mean and a Sample of 100 (rounded up from ) wasneeded for the proportion. Using the fpc factor in Equation ( ) for the mean, with N= 5,000,e= $5,S= $25, and Z= (for 95% confidence), leads toThus,n= the fpc factor in Equation ( ) for the proportion, with N= 5,000,e= ,p= ,and Z= (for 95% confidence),Thus,n= satisfy both requirements simultaneously with one Sample , the larger Sample size of 99 isneeded.

6 PHStat output is displayed in Figure + =(.)(, ).(, ).99 96 5 00099 965 000198 02n=+ =(.)(, ).(, ).96 04 5 00096 045 000194 24nnNnN=+ 001()nZenZppe02220221== and ()eZppnNnN= ()11eZnNnN= 1 FIGURE size for estimating themean sales invoice amount withthe finite population correctionfor the Saxon HomeImprovement CompanyCD8-4CD MATERIALL earning the If= 75,S= 24,n= 36, and N= 200, set up a 95% confi-dence interval estimate of the population mean if sam-pling is done Consider a population of 1,000 where the standard deviationis assumed equal to 20. What Sample size would be requiredif sampling is done withoutreplacement if you desire 95%confidence and a sampling error of 5?

7 Applying the The quality control manager at a lightbulb factory needs toestimate the mean life of a large shipment of lightbulbs. Theprocess standard deviation is known to be 100 that the shipment contains a total of 2,000 light-bulbs and that sampling is done Set up a 95% confidence interval estimate of the popula-tion mean life of lightbulbs in this shipment if a randomsample of 50 lightbulbs selected from the shipment indi-cates a Sample average life of 350 Determine the Sample size needed to estimate the averagelife to within 20 hours with 95% What are your answers to (a) and (b) if the shipmentcontains 1,000 lightbulbs? A survey is planned to determine the mean annual familymedical expenses of employees of a large company.

8 Themanagement of the company wishes to be 95% confidentthat the Sample average is correct to within $50 of the trueaverage annual family medical expenses. A pilot study indi-cates that the standard deviation is estimated as $400. Howlarge a Sample size is necessary if the company has 3,000employees and if sampling is done withoutreplacement? The manager of a bank that has 1,000 depositors in a smallcity wants to determine the proportion of its depositors withmore than one account at the Set up a 90% confidence interval estimate of the popu-lation proportion of the bank s depositors who havemore than one account at the bank if a random sampleof 100 depositors is selectedwithoutreplacement and30 state that they have more than one account at A bank manager wants to be 90% confident of being cor-rect to within of the true population proportion ofXdepositors who have more than one account at the Sample size is needed if sampling is done withoutreplacement?

9 C. What are your answers to (a) and (b) if the bank has2,000 depositors? An automobile dealer wants to estimate the proportion of cus-tomers who still own the cars they purchased 5 years records indicate that the population of owners is 4, Set up a 95% confidence interval estimate of the popula-tion proportion of all customers who still own their cars5 years after they were purchased if a random Sample of200 customers selected withoutreplacement from theautomobile dealer s records indicate that 82 still own carsthat were purchased 5 years What Sample size is necessary to estimate the true pro-portion to within with 95% confidence?c. What are your answers to (a) and (b) if the populationconsists of 6,000 owners?

10 The inspection division of the Lee County Weights andMeasures Department is interested in estimating the actualamount of soft drink that is placed in 2-liter bottles at thelocal bottling plant of a large nationally known soft-drinkcompany. The population consists of 2,000 bottles. The bot-tling plant has informed the inspection division that thestandard deviation for 2-liter bottles is Set up a 95% confidence interval estimate of the popula-tion mean amount of soft drink per bottle if a randomsample of one hundred 2-liter bottles obtained withoutreplacement from this bottling plant indicates a sampleaverage of Determine the Sample size necessary to estimate the pop-ulation mean amount to within liter with 95% What are your answers to (a) and (b) if the populationconsists of 1,000 bottles?