9.8: THE POWER OF A TEST


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1 9.8: The Power of a Test CD : THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based oly o sample evidece were defied. Recall from sectio 9.1 that represets the probability that the ull hypothesis is rejected whe i fact it is true ad should ot be rejected, ad represets the probability that the ull hypothesis is ot rejected whe i fact it is false ad should be rejected. The power of the test, which is 1 (that is, the complemet of ), idicates the sesitivity of the statistical test i detectig chages that have occurred by measurig the probability of rejectig the ull hypothesis whe i fact it is false ad should be rejected. The power of the statistical test depeds o how differet the actual populatio mea really is from the value beig hypothesized (uder H 0 ), the value of used, ad the sample size. If there is a large differece betwee the actual populatio mea ad the hypothesized mea, the power of the test will be much greater tha if the differece betwee the actual populatio mea ad the hypothesized mea is small. Selectig a larger value of makes it easier to reject H 0 ad therefore icreases the power of a test. Icreasig the sample size icreases the precisio i the estimates ad therefore icreases the ability to detect differeces i the parameters ad icreases the power of a test. I this sectio, the cerealboxfillig process is examied i order to further develop the cocept of the power of a statistical test. Suppose that the fillig process is subject to periodic ispectio from a represetative of the local office of cosumer affairs. It is this represetative s job to detect the possible short weightig of boxes, a situatio i which cereal boxes are sold at less tha the specified 368 grams. Thus, the represetative is iterested i determiig whether there is evidece that the cereal boxes have a average amout that is less tha 368 grams. The ull ad alterative hypotheses are set up as follows: H 0 :µ 368 (fillig process is workig properly) H 1 : µ < 368 (fillig process is ot workig properly) The represetative of the office of cosumer affairs is willig to accept the compay s claim that the stadard deviatio over the etire packagig process is equal to grams. Therefore, the test is appropriate. If the level of sigificace of 0.05 is selected ad a radom sample of boxes is obtaied, the value of that eables you to reject the ull hypothesis is foud from Equatio (9.1) as follows with used i place of : L L µ L µ L µ + Because this is a oetail test with a level of sigificace of 0.05, the value of equal to stadard deviatios below the hypothesized mea is obtaied from Table E.2 (see Figure 9.27). Therefore, L ( ) ( )
2 CD92 CD MATERIAL L µ FIGURE 9.27 Determiig the lower critical value for a oetail test for a populatio mea at the 0.05 level of sigificace Regio of Rejectio L Regio of Norejectio The decisio rule for this oetail test is Reject H0 if < ; otherwise do ot reject H. The decisio rule states that if a radom sample of boxes reveals a sample mea of less tha grams, the ull hypothesis is rejected, ad the represetative cocludes that the process is ot workig properly. The power of the test measures the probability of cocludig that the process is ot workig properly for differig values of the true populatio mea. Suppose that you wat to determie the chace of rejectig the ull hypothesis whe the populatio mea is actually 360 grams. O the basis of the decisio rule, the probability or area uder the ormal curve below grams eeds to be determied. From the cetral limit theorem ad the assumptio of ormality i the populatio, you ca assume that the samplig distributio of the mea follows a ormal distributio. Therefore, the area uder the ormal curve below grams ca be expressed i stadard deviatio uits, because you are fidig the probability of rejectig the ull hypothesis whe the true populatio mea has shifted to 360 grams. Usig Equatio (9.1), 0 where µ 1 is the actual populatio mea. Thus, µ From Table E.2, there is a 84.61% chace of observig a value less tha This is the power of the test or area below (see Figure 9.28). The probability ( ) that the ull hypothesis (µ 368) will ot be rejected is Thus, the probability of committig a Type II error is.39%.
3 9.8: The Power of a Test CD93 Power.8461 FIGURE 9.28 Determiig the power of the test ad the probability of a Type II error whe µ grams.39 µ L Now that the power of the test if the populatio mea were really equal to 360 has bee determied, the power for ay other value that µ could attai ca be calculated. For example, what is the power of the test if the populatio mea is equal to 352 grams? Assumig the same stadard deviatio, sample size, ad level of sigificace, the decisio rule is Reject H 0 if < ; otherwise do ot reject H 0. Oce agai, because you are testig a hypothesis for a mea, from Equatio (9.1) If the populatio mea shifts dow to 352 grams (see Figure 9.29), the µ From Table E.2, there is a % chace of observig a value less tha This is the power of the test or area below The probability ( ) that the ull hypothesis (µ 368) will ot be rejected is Thus, the probability of committig a Type II error is oly 0.011%. Power FIGURE 9.29 Determiig the power of the test ad the probability of a Type II error whe µ grams µ L I the precedig two cases the power of the test was quite high, whereas, coversely, the chace of committig a Type II error was quite low. I the ext example, the power of the test is computed for the case i which the populatio mea is really equal to 367 grams a value that is very close to the hypothesized mea of 368 grams.
4 CD94 CD MATERIAL Oce agai, from Equatio (9.1), µ 1 If the populatio mea is really equal to 367 grams (see Figure 9.30), the Power FIGURE 9.30 Determiig the power of the test ad the probability of a Type II error whe µ grams L µ From Table E.2, observe that the probability (area uder the curve) less tha 1.31 is (or 9.51%). Because i this istace the rejectio regio is i the lower tail of the distributio, the power of the test is 9.51% ad the chace of makig a Type II error is 90.49%. Figure 9.31 illustrates the power of the test for various possible values of µ 1 (icludig the three cases examied). This is called a power curve. The computatios for the three cases are summarized i Figure FIGURE 9.31 Power curve of the cerealboxfillig process for the alterative hypothesis H 1 : µ < 368 grams Power Possible True Values for µ 1 (grams)
5 9.8: The Power of a Test CD95 Pael A Give: α.05,, Oetailed test µ (ull hypothesis is true) L 368 (1.645) Decisio rule: Reject H 0 if < ; otherwise do ot reject Regio of Rejectio α.050 Regio of Norejectio α.95 Pael B Give: α.05,, Oetailed test H 0 : µ µ (true mea shifts to 367 grams) µ Power.0951 Power Pael C Give: α.05,, Oetailed test H 0 : µ µ (true mea shifts to 360 grams) µ Power.8461 Power Pael D Give: α.05,, Oetailed test H 0 : µ µ (true mea shifts to 352 grams) µ Power Power Regio of Rejectio Regio of Norejectio FIGURE 9.32 Determiig statistical power for varyig values of the actual populatio mea
6 CD96 CD MATERIAL 3 For situatios ivolvig oetail tests i which the actual mea µ 1 really exceeds the hypothesized mea, the coverse would be true. The larger the actual mea µ 1 compared with the hypothesized mea, the greater is the power. O the other had, for twotail tests, the greater the distace betwee the actual mea µ 1 ad the hypothesized mea, the greater the power of the test. From Figure 9.31, observe that the power of this oetail test icreases sharply (ad approaches 100%) as the actual populatio mea takes o values farther below the hypothesized mea of 368 grams. Clearly, for this oetail test, the smaller the actual mea µ 1 is whe compared with the hypothesized mea, the greater will be the power to detect this disparity. 3 O the other had, for values of µ 1 close to 368 grams, the power is rather small because the test caot effectively detect small differeces betwee the actual populatio mea ad the hypothesized value of 368 grams. Note that whe the populatio mea approaches 368 grams, the power of the test approaches, the level of sigificace (which is 0.05 i this example). You ca observe the drastic chages i the power of the test for differig values of the actual populatio meas by reviewig the differet paels of Figure From paels A ad B you ca see that whe the populatio mea does ot greatly differ from 368 grams, the chace of rejectig the ull hypothesis, based o the decisio rule ivolved, is ot large. However, oce the actual populatio mea shifts substatially below the hypothesized 368 grams, the power of the test greatly icreases, approachig its maximum value of 1 (or 100%). I the discussio of the power of a statistical test, a oetail test, a level of sigificace of 0.05, ad a sample size of boxes have bee used. With this i mid, you ca determie the effect o the power of the test by varyig, oe at a time, the type of statistical test oetail versus twotail. the level of sigificace. the sample size. While these exercises are left to the reader (see problems ), three basic coclusios regardig the power of the test are summarized i Exhibit 9.6. EHIBIT 9.6 THE POWER OF A TEST There are three basic coclusios ivolved i uderstadig the power of the test. 1. A oetail test is more powerful tha a twotail test ad should be used wheever it is appropriate to specify the directio of the alterative hypothesis. 2. Because the probability of committig a Type I error ( ) ad the probability of committig a Type II error ( ) have a iverse relatioship ad the latter is the complemet of the power of the test (1 ), the ad the power of the test vary directly. A icrease i the value of the level of sigificace ( ) results i a icrease i power, ad a decrease i results i a decrease i power. 3. A icrease i the size of the sample chose results i a icrease i power. A decrease i the size of the sample selected results i a decrease i power.
7 9.8: The Power of a Test CD97 PROBLEMS FOR SECTION 9.8 Applyig the Cocepts A coioperated softdrik machie is desiged to discharge, whe it is operatig properly, at least 7 ouces of beverage per cup with a stadard deviatio of 0.2 ouce. If a radom sample of 16 cupfuls is selected by a statisticia for a cosumer testig service ad the statisticia is willig to take a risk of 0.05 of committig a Type I error, compute the power of the test ad the probability of a Type II error ( ) if the populatio average amout dispesed is actually a. 6.9 ouces per cup. b. 6.8 ouces per cup Refer to problem If the statisticia is willig to take a risk of oly 0.01 of committig a Type I error, compute the power of the test ad the probability of a Type II error ( ) if the populatio average amout dispesed is actually a. 6.9 ouces per cup. b. 6.8 ouces per cup. c. Compare the results i (a) ad (b) of this problem ad i problem What coclusio ca you reach? Refer to problem If the statisticia selects a radom sample of cupfuls ad is willig to take a risk of 0.05 of committig a Type I error, compute the power of the test ad the probability of a Type II error ( ) if the populatio average amout dispesed is actually a. 6.9 ouces per cup. b. 6.8 ouces per cup. c. Compare the results i (a) ad (b) of this problem ad i problem What coclusio ca you draw? A tire maufacturer produces tires that last, o average, at least,000 miles whe the productio process is workig properly. Based o past experiece, the stadard deviatio of the tires is assumed to be 3,500 miles. The operatios maager will stop the productio process if there is evidece that the average tire life is below,000 miles. If a radom sample of 100 tires is selected (to be subjected to destructive testig) ad the operatios maager is willig to take a risk of 0.05 of committig a Type I error, compute the power of the test ad the probability of a Type II error ( ) if the populatio average life is actually Refer to problem If the operatios maager is willig to take a risk of oly 0.01 of committig a Type I error, compute the power of the test ad the probability of a Type II error ( ) if the populatio average life is actually c. Compare the results i (a) ad (b) of this problem ad (a) ad (b) i problem What coclusio ca you draw? Refer to problem If the operatios maager selects a radom sample of tires ad is willig to take a risk of 0.05 of committig a Type I error, compute the power of the test ad the probability of a Type II error ( ) if the populatio average life is actually c. Compare the results i (a) ad (b) of this problem ad (a) ad (b) i problem What coclusio ca you draw? 9.1 Refer to problem If the operatios maager will stop the process whe there is evidece that the average life is differet from,000 miles (either less tha or greater tha) ad a radom sample of 100 tires is selected alog with a level of sigificace of 0.05, compute the power of the test ad the probability of a Type II error ( ) if the populatio average life is actually c. Compare the results i (a) ad (b) of this problem ad (a) ad (b) i problem What coclusio ca you draw?
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