CHAPTER 8. Confidence Interval Estimation LEARNING OBJECTIVES. USING Saxon Home Improvement

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1 CHAPTER 8 Cofidece Iterval Estimatio USING Saxo Home Improvemet 8.1 CONFIDENCE INTERVAL ESTIMATION FOR THE MEAN (* KNOWN) 8.2 CONFIDENCE INTERVAL ESTIMATION FOR THE MEAN (* UNKNOWN) Studet s t Distributio Properties of the t Distributio The Cocept of Degrees of Freedom The Cofidece Iterval Statemet 8.3 CONFIDENCE INTERVAL ESTIMATION FOR THE PROPORTION 8.4 DETERMINING SAMPLE SIZE Sample Size Determiatio for the Mea Sample Size Determiatio for the Proportio 8.5 APPLICATIONS OF CONFIDENCE INTERVAL ESTIMATION IN AUDITING Estimatig the Populatio Total Amout Differece Estimatio Oe-Sided Cofidece Iterval Estimatio of the Rate of Nocompliace with Iteral Cotrols 8.7 (CD-ROM TOPIC) ESTIMATION AND SAMPLE SIZE DETERMINATION FOR FINITE POPULATIONS EXCEL COMPANION TO CHAPTER 8 E8.1 Computig the Cofidece Iterval Estimate for the Mea (* Kow) E8.2 Computig the Cofidece Iterval Estimate for the Mea (* Ukow) E8.3 Computig the Cofidece Iterval Estimate for the Proportio E8.4 Computig the Sample Size Needed for Estimatig the Mea E8.5 Computig the Sample Size Needed for Estimatig the Proportio E8.6 Computig the Cofidece Iterval Estimate for the Populatio Total E8.7 Computig the Cofidece Iterval Estimate for the Total Differece E8.8 Computig Fiite Populatio Correctio Factors 8.6 CONFIDENCE INTERVAL ESTIMATION AND ETHICAL ISSUES LEARNING OBJECTIVES I this chapter, you lear: * To costruct ad iterpret cofidece iterval estimates for the mea ad the proportio * How to determie the sample size ecessary to develop a cofidece iterval for the mea or proportio * How to use cofidece iterval estimates i auditig Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

2 284 CHAPTER EIGHT Cofidece Iterval Estimatio Usig Saxo Home Improvemet Saxo Home Improvemet distributes home improvemet supplies i the ortheaster Uited States. As a compay accoutat, you are resposible for the accuracy of the itegrated ivetory maagemet ad sales iformatio system. You could review the cotets of each ad every record to check the accuracy of this system, but such a detailed review would be time-cosumig ad costly. A better approach would be to use statistical iferece techiques to draw coclusios about the populatio of all records from a relatively small sample collected durig a audit. At the ed of each moth, you could select a sample of the sales ivoices to determie the followig: * The mea dollar amout listed o the sales ivoices for the moth. * The total dollar amout listed o the sales ivoices for the moth. * Ay differeces betwee the dollar amouts o the sales ivoices ad the amouts etered ito the sales iformatio system. * The frequecy of occurrece of errors that violate the iteral cotrol policy of the warehouse. Such errors iclude makig a shipmet whe there is o authorized warehouse removal slip, failure to iclude the correct accout umber, ad shippig the icorrect home improvemet item. How accurate are the results from the samples ad how do you use this iformatio? Are the sample sizes large eough to give you the iformatio you eed? Statistical iferece is the process of usig sample results to draw coclusios about the characteristics of a populatio. Iferetial statistics eables you to estimate ukow populatio characteristics such as a populatio mea or a populatio proportio. Two types of estimates are used to estimate populatio parameters: poit estimates ad iterval estimates. A poit estimate is the value of a sigle sample statistic. A cofidece iterval estimate is a rage of umbers, called a iterval, costructed aroud the poit estimate. The cofidece iterval is costructed such that the probability that the populatio parameter is located somewhere withi the iterval is kow. Suppose you would like to estimate the mea GPA of all the studets at your uiversity. The mea GPA for all the studets is a ukow populatio mea, deoted by. You select a sample of studets ad fid that the sample mea is The sample mea, X = 2. 80, is a poit estimate of the populatio mea,. How accurate is 2.80? To aswer this questio, you must costruct a cofidece iterval estimate. I this chapter, you will lear how to costruct ad iterpret cofidece iterval estimates. Recall that the sample mea, X, is a poit estimate of the populatio mea,. However, the sample mea varies from sample to sample because it depeds o the items selected i the sample. By takig ito accout the kow variability from sample to sample (see Sectio 7.4 o the samplig distributio of the mea), you ca develop the iterval estimate for the populatio mea. The iterval costructed should have a specified cofidece of correctly estimatig the value of the populatio parameter. I other words, there is a specified cofidece that is somewhere i the rage of umbers defied by the iterval. Suppose that after studyig this chapter, you fid that a 95% cofidece iterval for the mea GPA at your uiversity is ( ). You ca iterpret this iterval estimate by statig that you are 95% cofidet that the mea GPA at your uiversity is betwee 2.75 ad There is a 5% chace that the mea GPA is below 2.75 or above After learig about the cofidece iterval for the mea, you will lear how to develop a iterval estimate for the populatio proportio. The you will lear how large a sample to select whe costructig cofidece itervals ad how to perform several importat estimatio procedures accoutats use whe performig audits. Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

3 8.1: Cofidece Iterval Estimatio for the Mea (* Kow) CONFIDENCE INTERVAL ESTIMATION FOR THE MEAN (* KNOWN) I Sectio 7.4, you used the Cetral Limit Theorem ad kowledge of the populatio distributio to determie the percetage of sample meas that fall withi certai distaces of the populatio mea. For istace, i the cereal-fill example used throughout Chapter 7 (see Example 7.6 o page 268), 95% of all sample meas are betwee ad grams. This statemet is based o deductive reasoig. However, iductive reasoig is what you eed here. You eed iductive reasoig because, i statistical iferece, you use the results of a sigle sample to draw coclusios about the populatio, ot vice versa. Suppose that i the cerealfill example, you wish to estimate the ukow populatio mea, usig the iformatio from oly a sample. Thus, rather tha take -, ( 1. 96)( */( ) to fid the upper ad lower limits aroud, as i Sectio 7.4, you substitute the sample mea, X, for the ukow ad use X, ( )(*/( ) as a iterval to estimate the ukow. Although i practice you select a sigle sample of size ad compute the mea, X, i order to uderstad the full meaig of the iterval estimate, you eed to examie a hypothetical set of all possible samples of values. Suppose that a sample of = 25 boxes has a mea of grams. The iterval developed to estimate is , ( 1. 96)( 15)/( 25), or The estimate of is Because the populatio mea, (equal to 368), is icluded withi the iterval, this sample results i a correct statemet about (see Figure 8.1). FIGURE 8.1 Cofidece iterval estimates for five differet samples of = 25 take from a populatio where = 368 ad * = X 1 = X 2 = X 3 = X 4 = X 5 = To cotiue this hypothetical example, suppose that for a differet sample of = 25 boxes, the mea is The iterval developed from this sample is , ( 1. 96)( 15)/( 25) or The estimate is (equal to 368), is also icluded withi this iterval, this state- Because the populatio mea, met about is correct. Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

4 286 CHAPTER EIGHT Cofidece Iterval Estimatio Now, before you begi to thik that correct statemets about are always made by developig a cofidece iterval estimate, suppose a third hypothetical sample of = 25 boxes is selected ad the sample mea is equal to 360 grams. The iterval developed here is 360 ( 1. 96)( 15)/( 25), or I this case, the estimate of is This estimate is ot a correct statemet because the populatio mea,, is ot icluded i the iterval developed from this sample (see Figure 8.1). Thus, for some samples, the iterval estimate of is correct, but for others it is icorrect. I practice, oly oe sample is selected, ad because the populatio mea is ukow, you caot determie whether the iterval estimate is correct. To resolve this dilemma of sometimes havig a iterval that provides a correct estimate ad sometimes havig a iterval that provides a icorrect estimate, you eed to determie the proportio of samples producig itervals that result i correct statemets about the populatio mea,. To do this, cosider two other hypothetical samples: the case i which X = grams ad the case i which X = grams. If X = , the iterval is ( 1. 96)( 15)/( 25), or This leads to the followig iterval: Because the populatio mea of 368 is at the upper limit of the iterval, the statemet is a correct oe (see Figure 8.1). Whe X = , the iterval is ( 1. 96)( 15)/( 25), or The iterval for the sample mea is I this case, because the populatio mea of 368 is icluded at the lower limit of the iterval, the statemet is correct. I Figure 8.1, you see that whe the sample mea falls aywhere betwee ad grams, the populatio mea is icluded somewhere withi the iterval. I Example 7.6 o page 268, you foud that 95% of the sample meas fall betwee ad grams. Therefore, 95% of all samples of = 25 boxes have sample meas that iclude the populatio mea withi the iterval developed. Because, i practice, you select oly oe sample ad is ukow, you ever kow for sure whether your specific iterval icludes the populatio mea. However, if you take all possible samples of ad compute their sample meas, 95% of the itervals will iclude the populatio mea, ad oly 5% of them will ot. I other words, you have 95% cofidece that the populatio mea is somewhere i your iterval. Cosider oce agai, the first sample discussed i this sectio. A sample of = 25 boxes had a sample mea of grams. The iterval costructed to estimate is: ( 1. 96)( 15)/( 25) The iterval from to is referred to as a 95% cofidece iterval. I am 95% cofidet that the mea amout of cereal i the populatio of boxes is somewhere betwee ad grams. Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

5 8.1: Cofidece Iterval Estimatio for the Mea ( Kow) 287 I some situatios, you might wat a higher degree of cofidece (such as 99%) of icludig the populatio mea withi the iterval. I other cases, you might accept less cofidece (such as 90%) of correctly estimatig the populatio mea. I geeral, the level of cofidece is symbolized by (1 ) 100%, where is the proportio i the tails of the distributio that is outside the cofidece iterval. The proportio i the upper tail of the distributio is /2, ad the proportio i the lower tail of the distributio is /2. You use Equatio (8.1) to costruct a (1 ) 100% cofidece iterval estimate of the mea with kow. CONFIDENCE INTERVAL FOR THE MEAN ( KNOWN) X Z or X Z X + Z where Z = the value correspodig to a cumulative area of 1 ormal distributio (that is, a upper-tail probability of /2). (8.1) /2 from the stadardized The value of Z eeded for costructig a cofidece iterval is called the critical value for the distributio. 95% cofidece correspods to a value of The critical Z value correspodig to a cumulative area of is 1.96 because there is i the upper tail of the distributio ad the cumulative area less tha Z = 1.96 is There is a differet critical value for each level of cofidece, 1. A level of cofidece of 95% leads to a Z value of 1.96 (see Figure 8.2). 99% cofidece correspods to a value of The Z value is approximately 2.58 because the upper-tail area is ad the cumulative area less tha Z = 2.58 is (see Figure 8.3). FIGURE 8.2 Normal curve for determiig the Z value eeded for 95% cofidece X Z FIGURE 8.3 Normal curve for determiig the Z value eeded for 99% cofidece X Z Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

6 288 CHAPTER EIGHT Cofidece Iterval Estimatio Now that various levels of cofidece have bee cosidered, why ot make the cofidece level as close to 100% as possible? Before doig so, you eed to realize that ay icrease i the level of cofidece is achieved oly by wideig (ad makig less precise) the cofidece iterval. There is o free luch here. You would have more cofidece that the populatio mea is withi a broader rage of values; however, this might make the iterpretatio of the cofidece iterval less useful. The trade-off betwee the width of the cofidece iterval ad the level of cofidece is discussed i greater depth i the cotext of determiig the sample size i Sectio 8.4. Example 8.1 illustrates the applicatio of the cofidece iterval estimate. EXA MPL E 8.1 ESTIMATING THE MEAN PAPER LENGTH WITH 95% CONFIDENCE A paper maufacturer has a productio process that operates cotiuously throughout a etire productio shift. The paper is expected to have a mea legth of 11 iches, ad the stadard deviatio of the legth is 0.02 ich. At periodic itervals, a sample is selected to determie whether the mea paper legth is still equal to 11 iches or whether somethig has goe wrog i the productio process to chage the legth of the paper produced. You select a radom sample of 100 sheets, ad the mea paper legth is iches. Costruct a 95% cofidece iterval estimate for the populatio mea paper legth. SOLUTION Usig Equatio (8.1) o page 287, with Z = 1.96 for 95% cofidece, X Z = (. ) = Thus, with 95% cofidece, you coclude that the populatio mea is betwee ad iches. Because the iterval icludes 11, the value idicatig that the productio process is workig properly, you have o reaso to believe that aythig is wrog with the productio process. To see the effect of usig a 99% cofidece iterval, examie Example 8.2. EXA MPL E 8.2 ESTIMATING THE MEAN PAPER LENGTH WITH 99% CONFIDENCE Costruct a 99% cofidece iterval estimate for the populatio mea paper legth. SOLUTION Usig Equatio (8.1) o page 287, with Z = 2.58 for 99% cofidece, X Z = (. ) = Oce agai, because 11 is icluded withi this wider iterval, you have o reaso to believe that aythig is wrog with the productio process. Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

7 8.1: Cofidece Iterval Estimatio for the Mea ( Kow) 289 As discussed i sectio 7.4, the samplig distributio of X is ormally distributed if the populatio of X is a ormal distributio. Ad, if the populatio of X is ot a ormal distributio, the Cetral Limit Theorem esures that X is ormally distributed whe is large. However, whe dealig with a small sample size ad a populatio of X that is ot a ormal distributio, the samplig distributio of X is ot ormally distributed ad therefore the cofidece iterval discussed i this sectio is iappropriate. I practice, however, as log as the sample size is large eough ad the populatio is ot very skewed, you ca use the cofidece iterval defied i Equatio 8.1 to estimate the populatio mea whe is kow. To assess the assumptio of ormality, you ca evaluate the shape of the sample data by usig a histogram, stem-ad-leaf display, box-ad-whisker plot, or ormal probability plot. PROBLEMS FOR SECTION 8.1 Learig the Basics PH Grade ASSIST PH Grade ASSIST 8.1 If X = 85, = 8, ad = 64, costruct a 95% cofidece iterval estimate of the populatio mea,. 8.2 If X = 125, = 24, ad = 36, costruct a 99% cofidece iterval estimate of the populatio mea,. 8.3 A market researcher states that she has 95% cofidece that the mea mothly sales of a product are betwee $170,000 ad $200,000. Explai the meaig of this statemet. 8.4 Why is it ot possible i Example 8.1 o page 288 to have 100% cofidece? Explai. 8.5 From the results of Example 8.1 o page 288 regardig paper productio, is it true that 95% of the sample meas will fall betwee ad iches? Explai. 8.6 Is it true i Example 8.1 o page 288 that you do ot kow for sure whether the populatio mea is betwee ad iches? Explai. Applyig the Cocepts PH Grade ASSIST 8.7 The maager of a pait supply store wats to estimate the actual amout of pait cotaied i 1-gallo cas purchased from a atioally kow maufacturer. The maufacturer s specificatios state that the stadard deviatio of the amout of pait is equal to 0.02 gallo. A radom sample of 50 cas is selected, ad the sample mea amout of pait per 1-gallo ca is gallo. a. Costruct a 99% cofidece iterval estimate of the populatio mea amout of pait icluded i a 1-gallo ca. b. O the basis of these results, do you thik the maager has a right to complai to the maufacturer? Why? c. Must you assume that the populatio amout of pait per ca is ormally distributed here? Explai. d. Costruct a 95% cofidece iterval estimate. How does this chage your aswer to (b)? PH Grade ASSIST SELF Test 8.8 The quality cotrol maager at a light bulb factory eeds to estimate the mea life of a large shipmet of light bulbs. The stadard deviatio is 100 hours. A radom sample of 64 light bulbs idicated a sample mea life of 350 hours. a. Costruct a 95% cofidece iterval estimate of the populatio mea life of light bulbs i this shipmet. b. Do you thik that the maufacturer has the right to state that the light bulbs last a average of 400 hours? Explai. c. Must you assume that the populatio of light bulb life is ormally distributed? Explai. d. Suppose that the stadard deviatio chages to 80 hours. What are your aswers i (a) ad (b)? PH Grade ASSIST 8.9 The ispectio divisio of the Lee Couty Weights ad Measures Departmet wats to estimate the actual amout of soft drik i 2-liter bottles at the local bottlig plat of a large atioally kow soft-drik compay. The bottlig plat has iformed the ispectio divisio that the populatio stadard deviatio for 2-liter bottles is 0.05 liter. A radom sample of liter bottles at this bottlig plat idicates a sample mea of 1.99 liters. a. Costruct a 95% cofidece iterval estimate of the populatio mea amout of soft drik i each bottle. b. Must you assume that the populatio of soft-drik fill is ormally distributed? Explai. c. Explai why a value of 2.02 liters for a sigle bottle is ot uusual, eve though it is outside the cofidece iterval you calculated. d. Suppose that the sample mea is 1.97 liters. What is your aswer to (a)? Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

8 290 CHAPTER EIGHT Cofidece Iterval Estimatio 8.2 CONFIDENCE INTERVAL ESTIMATION FOR THE MEAN ( UNKNOWN) Just as the mea of the populatio,, is usually ukow, you rarely kow the actual stadard deviatio of the populatio,. Therefore, you ofte eed to costruct a cofidece iterval estimate of, usig oly the sample statistics X ad S. Studet s t Distributio At the begiig of the twetieth cetury, William S. Gosset, a statisticia for Guiess Breweries i Irelad (see referece 3) wated to make ifereces about the mea whe was ukow. Because Guiess employees were ot permitted to publish research work uder their ow ames, Gosset adopted the pseudoym Studet. The distributio that he developed is kow as Studet s t distributio ad is commoly referred to as the t distributio. If the radom variable X is ormally distributed, the the followig statistic has a t distributio with 1 degrees of freedom: t = X S This expressio has the same form as the Z statistic i Equatio (7.4) o page 266, except that S is used to estimate the ukow. The cocept of degrees of freedom is discussed further o pages Properties of the t Distributio I appearace, the t distributio is very similar to the stadardized ormal distributio. Both distributios are bell shaped. However, the t distributio has more area i the tails ad less i the ceter tha does the stadardized ormal distributio (see Figure 8.4). Because S is used to estimate the ukow, the values of t are more variable tha those for Z. FIGURE 8.4 Stadardized ormal distributio ad t distributio for 5 degrees of freedom Stadardized ormal distributio t distributio for 5 degrees of freedom The degrees of freedom, 1, are directly related to the sample size,. As the sample size ad degrees of freedom icrease, S becomes a better estimate of, ad the t distributio gradually approaches the stadardized ormal distributio, util the two are virtually idetical. With a sample size of about 120 or more, S estimates precisely eough that there is little differece betwee the t ad Z distributios. As stated earlier, the t distributio assumes that the radom variable X is ormally distributed. I practice, however, as log as the sample size is large eough ad the populatio is ot very skewed, you ca use the t distributio to estimate the populatio mea whe is ukow. Whe dealig with a small sample size ad a skewed populatio distributio, the validity of the cofidece iterval is a cocer. To assess the assumptio of ormality, you ca Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

9 8.2: Cofidece Iterval Estimatio for the Mea ( Ukow) 291 evaluate the shape of the sample data by usig a histogram, stem-ad-leaf display, box-adwhisker plot, or ormal probability plot. You fid the critical values of t for the appropriate degrees of freedom from the table of the t distributio (see Table E.3). The colums of the table represet the area i the upper tail of the t distributio. The rows of the table represet the degrees of freedom. The cells of the table represet the particular t value for each specific degree of freedom. For example, with 99 degrees of freedom, if you wat 95% cofidece, you fid the appropriate value of t, as show i Table 8.1. The 95% cofidece level meas that 2.5% of the values (a area of 0.025) are i each tail of the distributio. Lookig i the colum for a upper-tail area of ad i the row correspodig to 99 degrees of freedom gives you a critical value for t of Because t is a symmetrical distributio with a mea of 0, if the upper-tail value is , the value for the lower-tail area (lower 0.025) is A t value of meas that the probability that t is less tha is 0.025, or 2.5% (see Figure 8.5). Note that for a 95% cofidece iterval, you will always use a upper-tail area of Similarly, for a 99% cofidece iterval, use 0.005, for 98% use 0.01, 90% use 0.05, ad 80% use TABLE 8.1 Determiig the Critical Value from the t Table for a Area of i Each Tail with 99 Degrees of Freedom Upper-Tail Areas Degrees of Freedom Source: Extracted from Table E.3. FIGURE 8.5 t distributio with 99 degrees of freedom = t 99 The Cocept of Degrees of Freedom I Chapter 3 you leared that the umerator of the sample variace, S 2 [see Equatio (3.9) o page 107], requires the computatio of i= 1 ( X X ) i 2 Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

10 292 CHAPTER EIGHT Cofidece Iterval Estimatio I order to compute S 2, you first eed to kow X. Therefore, oly 1 of the sample values are free to vary. This meas that you have 1 degrees of freedom. For example, suppose a sample of five values has a mea of 20. How may values do you eed to kow before you ca determie the remaider of the values? The fact that = 5 ad X = 20 also tells you that X i i= 1 = 100 because X i i= 1 = X Thus, whe you kow four of the values, the fifth oe is ot free to vary because the sum must add to 100. For example, if four of the values are 18, 24, 19, ad 16, the fifth value must be 23 so that the sum equals 100. The Cofidece Iterval Statemet Equatio (8.2) defies the (1 ) 100% cofidece iterval estimate for the mea with ukow. CONFIDENCE INTERVAL FOR THE MEAN ( or X t 1 S UNKNOWN) X t S X + t 1 1 S (8.2) where t 1 is the critical value of the t distributio, with area of /2 i the upper tail. 1 degrees of freedom for a To illustrate the applicatio of the cofidece iterval estimate for the mea whe the stadard deviatio,, is ukow, recall the Saxo Home Improvemet Compay Usig Statistics sceario preseted o page 284. You wated to estimate the mea dollar amout listed o the sales ivoices for the moth. You select a sample of 100 sales ivoices from the populatio of sales ivoices durig the moth, ad the sample mea of the 100 sales ivoices is $110.27, with a sample stadard deviatio of $ For 95% cofidece, the critical value from the t distributio (as show i Table 8.1) is Usig Equatio (8.2), A Microsoft Excel worksheet for these data is preseted i Figure 8.6. X t 1 S = ( ) 100 = $ $ Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

11 8.2: Cofidece Iterval Estimatio for the Mea ( Ukow) 293 FIGURE 8.6 Microsoft Excel worksheet to compute a cofidece iterval estimate for the mea sales ivoice amout for the Saxo Home Improvemet Compay See Sectio E8.2 to create this. Thus, with 95% cofidece, you coclude that the mea amout of all the sales ivoices is betwee $ ad $ The 95% cofidece level idicates that if you selected all possible samples of 100 (somethig that is ever doe i practice), 95% of the itervals developed would iclude the populatio mea somewhere withi the iterval. The validity of this cofidece iterval estimate depeds o the assumptio of ormality for the distributio of the amout of the sales ivoices. With a sample of 100, the ormality assumptio is ot overly restrictive, ad the use of the t distributio is likely appropriate. Example 8.3 further illustrates how you costruct the cofidece iterval for a mea whe the populatio stadard deviatio is ukow. EXA MPL E 8.3 TABLE 8.2 Force (i Pouds) Required to Break the Isulator ESTIMATING THE MEAN FORCE REQUIRED TO BREAK ELECTRIC INSULATORS A maufacturig compay produces electric isulators. If the isulators break whe i use, a short circuit is likely. To test the stregth of the isulators, you carry out destructive testig to determie how much force is required to break the isulators. You measure force by observig how may pouds are applied to the isulator before it breaks. Table 8.2 lists 30 values from this experimet, which are located i the file force.xls. Costruct a 95% cofidece iterval estimate for the populatio mea force required to break the isulator. 1,870 1,728 1,656 1,610 1,634 1,784 1,522 1,696 1,592 1,662 1,866 1,764 1,734 1,662 1,734 1,774 1,550 1,756 1,762 1,866 1,820 1,744 1,788 1,688 1,810 1,752 1,680 1,810 1,652 1,736 SOLUTION Figure 8.7 shows that the sample mea is X = 1, pouds ad the sample stadard deviatio is S = pouds. Usig Equatio (8.2) o page 292 to costruct the cofidece iterval, you eed to determie the critical value from the t table for a area of i FIGURE 8.7 Microsoft Excel cofidece iterval estimate for the mea amout of force required to break electric isulators See Sectio E8.2 to create this. Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

12 294 CHAPTER EIGHT Cofidece Iterval Estimatio FIGURE 8.8 Microsoft Excel ormal probability plot for the amout of force required to break electric isulators each tail, with 29 degrees of freedom. From Table E.3, you see that t 29 = Thus, usig X = 1, , S = 89.55, = 30, ad t 29 = , X t 1 S You coclude with 95% cofidece that the mea breakig force required for the populatio of isulators is betwee 1, ad 1, pouds. The validity of this cofidece iterval estimate depeds o the assumptio that the force required is ormally distributed. Remember, however, that you ca slightly relax this assumptio for large sample sizes. Thus, with a sample of 30, you ca use the t distributio eve if the amout of force required is oly slightly left skewed. From the ormal probability plot displayed i Figure 8.8 or the box-ad-whisker plot displayed i Figure 8.9, the amout of force required appears slightly left skewed. Thus, the t distributio is appropriate for these data = 1, ( ) 30 = 1, , , See Sectio E6.2 to create this. FIGURE 8.9 Microsoft Excel boxad-whisker plot for the amout of force required to break electric isulators See Sectio E3.4 to create this. Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

13 8.2: Cofidece Iterval Estimatio for the Mea ( Ukow) 295 PROBLEMS FOR SECTION 8.2 Learig the Basics PH Grade ASSIST 8.10 Determie the critical value of t i each of the followig circumstaces: a. 1 = 0.95, = 10 b. 1 = 0.99, = 10 c. 1 = 0.95, = 32 d. 1 = 0.95, = 65 e. 1 = 0.90, = 16 PH Grade 8.11 If X = 75, S = 24, ad = 36, ad assumig that the populatio is ormally distributed, ASSIST costruct a 95% cofidece iterval estimate of the populatio mea,. PH Grade 8.12 If X = 50, S = 15, ad = 16, ad assumig that the populatio is ormally distributed, ASSIST costruct a 99% cofidece iterval estimate of the populatio mea, Costruct a 95% cofidece iterval estimate for the populatio mea, based o each of the followig sets of data, assumig that the populatio is ormally distributed: Set 1 1, 1, 1, 1, 8, 8, 8, 8 Set 2 1, 2, 3, 4, 5, 6, 7, 8 Explai why these data sets have differet cofidece itervals eve though they have the same mea ad rage Costruct a 95% cofidece iterval for the populatio mea, based o the umbers 1, 2, 3, 4, 5, 6, ad 20. Chage the umber 20 to 7 ad recalculate the cofidece iterval. Usig these results, describe the effect of a outlier (that is, a extreme value) o the cofidece iterval. Applyig the Cocepts 8.15 A statioery store wats to estimate the mea retail value of greetig cards that it has i its ivetory. A radom sample of 100 greetig cards idicates a mea value of $2.55 ad a stadard deviatio of $0.44. a. Assumig a ormal distributio, costruct a 95% cofidece iterval estimate of the mea value of all greetig cards i the store s ivetory. b. Suppose there were 2,500 greetig cards i the store s ivetory. How are the results i (a) useful i assistig the store ower to estimate the total value of her ivetory? SELF Test 8.16 Southside Hospital i Bay Shore, New York, commoly coducts stress tests to study the heart muscle after a perso has a heart attack. Members of the diagostic imagig departmet coducted a quality improvemet project to try to reduce the tur- aroud time for stress tests. Turaroud time is defied as the time from whe the test is ordered to whe the radiologist sigs off o the test results. Iitially, the mea turaroud time for a stress test was 68 hours. After icorporatig chages ito the stress-test process, the quality improvemet team collected a sample of 50 turaroud times. I this sample, the mea turaroud time was 32 hours, with a stadard deviatio of 9 hours (Extracted from E. Godi, D. Rave, C. Sweetapple, ad F. R. Del Guidice, Faster Test Results, Quality Progress, Jauary 2004, 37(1), pp ). a. Costruct a 95% cofidece iterval for the populatio mea turaroud time. b. Iterpret the iterval costructed i (a). c. Do you thik the quality improvemet project was a success? Explai. PH Grade ASSIST 8.17 The U.S. Departmet of Trasportatio requires tire maufacturers to provide tire performace iformatio o the sidewall of the tire to better iform prospective customers whe makig purchasig decisios. Oe very importat measure of tire performace is the tread wear idex, which idicates the tire s resistace to tread wear compared with a tire graded with a base of 100. This meas that a tire with a grade of 200 should last twice as log, o average, as a tire graded with a base of 100. A cosumer orgaizatio wats to estimate the actual tread wear idex of a brad ame of tires graded 200 that are produced by a certai maufacturer. A radom sample of = 18 idicates a sample mea tread wear idex of ad a sample stadard deviatio of a. Assumig that the populatio of tread wear idices is ormally distributed, costruct a 95% cofidece iterval estimate of the populatio mea tread wear idex for tires produced by this maufacturer uder this brad ame. b. Do you thik that the cosumer orgaizatio should accuse the maufacturer of producig tires that do ot meet the performace iformatio provided o the sidewall of the tire? Explai. c. Explai why a observed tread wear idex of 210 for a particular tire is ot uusual, eve though it is outside the cofidece iterval developed i (a) The followig data (stored i the file bakcost1.xls) represet the bouced check fee, i dollars, charged by a sample of 23 baks for direct-deposit customers who maitai a $100 balace: Source: Extracted from The New Face of Bakig, Cosumer Reports, Jue Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

14 296 CHAPTER EIGHT Cofidece Iterval Estimatio a. Costruct a 95% cofidece iterval for the populatio mea bouced check fee. b. Iterpret the iterval costructed i (a) 8.19 The data i the file chicke.xls represet the total fat, i grams per servig, for a sample of 20 chicke sadwiches from fast-food chais. The data are as follows: Source: Extracted from Fast Food: Addig Health to the Meu, Cosumer Reports, September 2004, pp a. Costruct a 95% cofidece iterval for the populatio mea total fat, i grams per servig. b. Iterpret the iterval costructed i (a) Oe of the major measures of the quality of service provided by ay orgaizatio is the speed with which it respods to customer complaits. A large family-held departmet store sellig furiture ad floorig, icludig carpet, had udergoe a major expasio i the past several years. I particular, the floorig departmet had expaded from 2 istallatio crews to a istallatio supervisor, a measurer, ad 15 istallatio crews. Last year, there were 50 complaits cocerig carpet istallatio. The followig data, i the file furiture.xls, represet the umber of days betwee the receipt of a complait ad the resolutio of the complait: a. Costruct a 95% cofidece iterval estimate of the mea umber of days betwee the receipt of a complait ad the resolutio of the complait. b. What assumptio must you make about the populatio distributio i (a)? c. Do you thik that the assumptio made i (b) is seriously violated? Explai. d. What effect might your coclusio i (c) have o the validity of the results i (a)? 8.21 I New York State, savigs baks are permitted to sell a form of life isurace called savigs bak life isurace (SBLI). The approval process cosists of uderwritig, which icludes a review of the applicatio, a medical iformatio bureau check, possible requests for additioal medical iformatio ad medical exams, ad a policy compilatio stage i which the policy pages are geerated ad set to the bak for delivery. The ability to deliver approved policies to customers i a timely maer is critical to the profitability of this service to the bak. Durig a period of oe moth, a radom sample of 27 approved policies was selected, ad the total processig time, i days, was as show below ad stored i the file isurace.xls: a. Costruct a 95% cofidece iterval estimate of the mea processig time. b. What assumptio must you make about the populatio distributio i (a)? c. Do you thik that the assumptio made i (b) is seriously violated? Explai The data i the file batterylife.xls represet the battery life (i shots) for three-pixel digital cameras: Source: Extracted from Cameras: More Features i the Mix, Cosumer Reports, July 2005, pp a. Costruct a 95% cofidece iterval for the populatio mea battery life (i shots). b. What assumptio do you eed to make about the populatio of iterest to costruct the iterval i (a)? c. Give the data preseted, do you thik the assumptio eeded i (a) is valid? Explai Oe operatio of a mill is to cut pieces of steel ito parts that are used later i the frame for frot seats i a automobile. The steel is cut with a diamod saw ad requires the resultig parts to be withi ich of the legth specified by the automobile compay. The measuremet reported from a sample of 100 steel parts (ad stored i the file steel.xls) is the differece, i iches, betwee the actual legth of the steel part, as measured by a laser measuremet device, ad the specified legth of the steel part. For example, the first observatio, , represets a steel part that is ich shorter tha the specified legth. a. Costruct a 95% cofidece iterval estimate of the mea differece betwee the actual legth of the steel part ad the specified legth of the steel part. b. What assumptio must you make about the populatio distributio i (a)? c. Do you thik that the assumptio made i (b) is seriously violated? Explai. d. Compare the coclusios reached i (a) with those of Problem 2.23 o page CONFIDENCE INTERVAL ESTIMATION FOR THE PROPORTION This sectio exteds the cocept of the cofidece iterval to categorical data. Here you are cocered with estimatig the proportio of items i a populatio havig a certai characteristic of iterest. The ukow populatio proportio is represeted by the Greek letter *. The Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

15 8.3: Cofidece Iterval Estimatio for the Proportio 297 poit estimate for is the sample proportio, p = X/, where is the sample size ad X is the umber of items i the sample havig the characteristic of iterest. Equatio (8.3) defies the cofidece iterval estimate for the populatio proportio. CONFIDENCE INTERVAL ESTIMATE FOR THE PROPORTION p Z p ( 1 p ) or p Z p ( 1 p ) p + Z p ( 1 p ) (8.3) where p = Sample proportio = = populatio proportio Z = critical value from the stadardized ormal distributio = sample size assumig that both X ad X are greater tha 5 X = Number of items havig the characteristic Sample size You ca use the cofidece iterval estimate of the proportio defied i Equatio (8.3) to estimate the proportio of sales ivoices that cotai errors (see the Usig Statistics sceario o page 284). Suppose that i a sample of 100 sales ivoices, 10 cotai errors. Thus, for these data, p = X/ = 10/100 = Usig Equatio (8.3) ad Z = 1.96 for 95% cofidece, p Z p ( 1 p ) = ( 1. 96) = ( 1. 96)( 0. 03) = ( 0. 10)( 0. 90) Therefore, you have 95% cofidece that betwee 4.12% ad 15.88% of all the sales ivoices cotai errors. Figure 8.10 shows a Microsoft Excel worksheet for these data. FIGURE 8.10 Microsoft Excel worksheet to costruct a cofidece iterval estimate for the proportio of sales ivoices that cotai errors See Sectio E8.3 to create this. Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

16 298 CHAPTER EIGHT Cofidece Iterval Estimatio Example 8.4 illustrates aother applicatio of a cofidece iterval estimate for the proportio. EXA MPL E 8.4 ESTIMATING THE PROPORTION OF NONCONFORMING NEWSPAPERS PRINTED A large ewspaper wats to estimate the proportio of ewspapers prited that have a ocoformig attribute, such as excessive ruboff, improper page setup, missig pages, or duplicate pages. A radom sample of 200 ewspapers is selected from all the ewspapers prited durig a sigle day. For this sample of 200, 35 cotai some type of ocoformace. Costruct ad iterpret a 90% cofidece iterval for the proportio of ewspapers prited durig the day that have a ocoformig attribute. SOLUTION Usig Equatio (8.3), p 35 = = , ad with a 90% level of cofidece Z = p Z p ( 1 p ) = ( ) = ( )( ) = ( )( ) 200 You coclude with 90% cofidece that betwee 13.08% ad 21.92% of the ewspapers prited o that day have some type of ocoformace. Equatio (8.3) cotais a Z statistic because you ca use the ormal distributio to approximate the biomial distributio whe the sample size is sufficietly large. I Example 8.4, the cofidece iterval usig Z provides a excellet approximatio for the populatio proportio because both X ad X are greater tha 5. However, if you do ot have a sufficietly large sample size, you should use the biomial distributio rather tha Equatio (8.3) (see refereces 1, 2, ad 6). The exact cofidece itervals for various sample sizes ad proportios of successes have bee tabulated by Fisher ad Yates (referece 2). PROBLEMS FOR SECTION 8.3 Learig the Basics PH Grade ASSIST PH Grade ASSIST 8.24 If = 200 ad X = 50, costruct a 95% cofidece iterval estimate of the populatio proportio If = 400 ad X = 25, costruct a 99% cofidece iterval estimate of the populatio proportio. Applyig the Cocepts PH Grade ASSIST SELF Test 8.26 The telephoe compay wats to estimate the proportio of households that would purchase a additioal telephoe lie if it were made available at a substatially reduced istallatio cost. A radom sample of 500 households is selected. The results idicate that 135 of the households would purchase the additioal telephoe lie at a reduced istallatio cost. Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

17 8.4: Determiig Sample Size 299 a. Costruct a 99% cofidece iterval estimate of the populatio proportio of households that would purchase the additioal telephoe lie. b. How would the maager i charge of promotioal programs cocerig residetial customers use the results i (a)? 8.27 Accordig to the Ceter for Work-Life Policy, a survey of 500 highly educated wome who left careers for family reasos foud that 66% wated to retur to work (Extracted from A. M. Chaker ad H. Stout, After Years Off, Wome Struggle to Revive Careers, The Wall Street Joural, May 6, 2004, p. A1). a. Costruct a 95% cofidece iterval for the populatio proportio of highly educated wome who left careers for family reasos who wat to retur to work. b. Iterpret the iterval i (a) I a survey coducted for America Express, 27% of small busiess owers idicated that they ever check i with the office whe o vacatio ( Sapshots, usatoday.com, April 18, 2006.). The article did ot disclose the sample size used i the study. a. Suppose that the survey was based o 500 small busiess owers. Costruct a 95% cofidece iterval estimate for the populatio proportio of small busiess owers who ever check i with the office whe o vacatio. b. Suppose that the survey was based o 1,000 small busiess owers. Costruct a 95% cofidece iterval estimate for the populatio proportio of small busiess owers who ever check i with the office whe o vacatio. c. Discuss the effect of sample size o the cofidece iterval estimate The umber of older cosumers i the Uited States is growig, ad they are becomig a eve bigger ecoomic force. May feel overwhelmed whe cofroted with the task of selectig ivestmets, bakig services, health care providers, or phoe service providers. A telephoe survey of 1,900 older cosumers foud that 27% said they did t have eough time to be good moey maagers (Extracted from Seiors Cofused by Fiacial Choices Study, msbc.com, May 6, 2004). a. Costruct a 95% cofidece iterval for the populatio proportio of older cosumers who do t thik they have eough time to be good moey maagers. b. Iterpret the iterval i (a) A survey of 705 workers (USA Today Sapshots, March 21, 2006, p. 1B) were asked how much they used the Iteret at work. 423 said they used it withi limits, ad 183 said that they did ot use the Iteret at work. a. Costruct a 95% cofidece iterval for the proportio of all workers who used the Iteret withi limits. b. Costruct a 95% cofidece iterval for the proportio of all workers who did ot use the Iteret at work Whe do Americas decide what to make for dier? A olie survey (N. Hellmich, Americas Go for the Quick Fix for Dier, USA Today, February 14, 2005, p. 1B) idicated that 74% of Americas decided either at the last miute or that day. Suppose that the survey was based o 500 respodets. a. Costruct a 95% cofidece iterval for the proportio of Americas who decided what to make for dier either at the last miute or that day. b. Costruct a 99% cofidece iterval for the proportio of Americas who decided what to make for dier either at the last miute or that day. c. Which iterval is wider? Explai why this is true I a survey of 894 respodets with salaries below $100,000 per year, 367 idicated that the primary reaso for stayig o their job was iterestig job resposibilities ( What Is the Primary Reaso for Stayig o Your Job? USA Today Sapshots, October 5, 2005, p. 1B). a. Costruct a 95% cofidece iterval for the proportio of all workers whose primary reaso for stayig o their job was iterestig job resposibilities. b. Iterpret the iterval costructed i (a) A large umber of compaies are tryig to reduce the cost of prescriptio drug beefits by requirig employees to purchase drugs through a madatory mailorder program. I a survey of 600 employers, 126 idicated that they either had a madatory mail-order program i place or were adoptig oe by the ed of 2004 (B. Martiez, Forcig Employees to Buy Drugs via Mail, The Wall Street Joural, February 18, 2004, p. 1B). a. Costruct a 95% cofidece iterval for the populatio proportio of employers who had a madatory mail-order program i place or were adoptig oe by the ed of b. Costruct a 99% cofidece iterval for the populatio proportio of employers who had a madatory mail-order program i place or were adoptig oe by the ed of c. Iterpret the itervals i (a) ad (b). d. Discuss the effect o the cofidece iterval estimate whe you chage the level of cofidece. 8.4 DETERMINING SAMPLE SIZE I each example of cofidece iterval estimatio so far i this chapter, the sample size was reported alog with the results with little discussio with regard to the width of the resultig cofidece iterval. I the busiess world, sample sizes are determied prior to data Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.

18 300 CHAPTER EIGHT Cofidece Iterval Estimatio collectio to esure that the cofidece iterval is arrow eough to be useful i makig decisios. Determiig the proper sample size is a complicated procedure, subject to the costraits of budget, time, ad the amout of acceptable samplig error. I the Saxo Home Improvemet example, you wat to estimate the mea dollar amout of the sales ivoices, you must determie i advace how large a samplig error to allow i estimatig the populatio mea. You must also determie i advace the level of cofidece (that is, 90%, 95%, or 99%) to use i estimatig the populatio parameter. Sample Size Determiatio for the Mea To develop a equatio for determiig the appropriate sample size eeded whe costructig a cofidece iterval estimate of the mea, recall Equatio (8.1) o page 287: X Z 1 I this cotext, some statisticias refer to e as the margi of error. The amout added to or subtracted from X is equal to half the width of the iterval. This quatity represets the amout of imprecisio i the estimate that results from samplig error. The samplig error, 1 e, is defied as e = Z Solvig for gives the sample size eeded to costruct the appropriate cofidece iterval estimate for the mea. Appropriate meas that the resultig iterval will have a acceptable amout of samplig error. SAMPLE SIZE DETERMINATION FOR THE MEAN The sample size,, is equal to the product of the Z value squared ad the variace,, squared, divided by the square of the samplig error, e, = Z 2 2 e 2 (8.4) 2 You use Z istead of t because, to determie the critical value of t, you eed to kow the sample size, but you do ot kow it yet. For most studies, the sample size eeded is large eough that the stadardized ormal distributio is a good approximatio of the t distributio. To determie the sample size, you must kow three factors: 1. The desired cofidece level, which determies the value of Z, the critical value from the stadardized ormal distributio 2 2. The acceptable samplig error, e 3. The stadard deviatio, I some busiess-to-busiess relatioships that require estimatio of importat parameters, legal cotracts specify acceptable levels of samplig error ad the cofidece level required. For compaies i the food or drug sectors, govermet regulatios ofte specify samplig errors ad cofidece levels. I geeral, however, it is usually ot easy to specify Statistics for Maagers Usig Microsoft Excel, Fifth Editio, by David M. Levie, Mark L. Bereso, ad Timothy C. Krehbiel. Published by Pretice Hall. Copyright 2008 by Pearso Educatio, Ic.