Chapter 9Hypothesis Testing Chapter 9 Hypothesis Testing Case Problem 1: Quality Associates, Inc. 1. The hypothesis testing results are shown below: Sample 1 Sample Size Mean Stan ar !e"iation #e"el o$ Si%ni$icance &alp'a( Critical )alue &lo*er tail( Critical )alue &upper tail( +ypot'esize "alue Stan ar ,rror -est Statistic p."alue 30 11.959 0.220 0.010 -2.576 2.576 12 0.0 0 -1.027 0.30 Sample 2 30 12.029 0.220 0.010 -2.576 2.576 12 0.0 0 0.713 0. 76 Sample 3 30 11.889 0.207 0.010 -2.576 2.576 12 0.038 -2.935 0.003 Sample 4 30 12.081 0.206 0.010 -2.576 2.576 12 0.038 2.161 0.031 Only sample 3 leads to the rejection of the hypothesis H0: µ = 12. Thus, corrective action is warranted for sample 3. The other samples indicate H0 cannot be rejected and thus from all we can tell, the process is operating satisfactorily. Sample 3 with x = 11.89 shows the process is operating below the desired mean. Sample 4 with x = 12.08 is on the high side, but the p-value of .03 is not sufficient to reject H0. 2. The sample standard deviations for all four samples are in the .20 to .22 range. It appears that the process population standard deviation assumption of .21 is good. With α = .01, z.005 = 2.576. Using the standard error of the mean σ x = σ ! n =0.0383, the upper and lower control limits are computed as follows: Upper Control Limit = 12 + 2.576 (0.0383) = 12.0987 Lower Control Limit = 12 - 2.576 (0.0383) = 11.9013 As long as a sample mean x is between these two limits, the process is in control and no corrective action is required. Note that sample 3 with a mean of 11.89 shows corrective action is necessary because the sample mean is outside the control limits. 4. Increasing the level of significance will cause the null hypothesis to be rejected more often. While this may mean quicker corrective action when the process is out of control, it also means that there will be a higher error probability of stopping the process and attempting corrective action when the process is operating satisfactorily. This would be an increase in the probability of a making a Type I error. 3. C" - 30 # 2010 Cengage $earning. %ll &ights &eser'e(. )ay not be s*anne(+ *opie( or (upli*ate(+ or poste( to a publi*ly a**essible website+ in whole or in part. e/ale stu(ents that presente( wor1 *opie( o. 7'erall+ 37 o..or both /ale an( . the internet as their own< 18 o.78 32. the internet as your own2 .22 17. >e will assu/e that this preli/inary wor1 has been (one an( that the new *olu/n is labele( 8Cheate(. the 90 business stu(ents A17. the abo'e -uestions an( 8.Chapter 9 Hypothesis Testing Case Problem 2: . All /usiness Stu ents Copie( .es 'alues into the *orrespon(ing *ells in the new *olu/n. all stu(ents+ /ale stu(ents+ an( . another stu(ent3s e4a/+ or *ollaborate( with other stu(ents on pro5e*ts that were suppose( to be *o/plete( in(i'i(ually. 7ne o. the 90 A32.uring your ti/e at 0ay'iew+ (i( you e'er *opy answers o.ollowing tables show the nu/ber an( per*entage o. .: The . the (ata set< then it is a si/ple /atter to enter an( paste 9o an( .. stu(ents that answere( 9o an( . the stu(ent answere( yes to one or /ore o. C" .es: i.or ea*h o.uring your ti/e at 0ay'iew+ (i( you e'er present wor1 *opie( o.78BC presente( wor1 *opie( o. the -uestions. the 90 stu(ents A20BC *opie( answers o.uring your ti/e at 0ay'iew+ (i( you e'er *ollaborate with other stu(ents on pro5e*ts that were suppose( to be *o/plete( in(i'i(ually2 6sing the (ata pro'i(e( we *an *o/pute the proportion o. the 90 stu(ents A 1.ollowing -uestions were use( to obtain (ata regar(ing three types o.. %ll &ights &eser'e(. another stu(ent3s e4a/< an( 29 o.es 9 Count 53 37 90 "er*ent 58. the easiest ways is to (o this is to a(( another *olu/n to the (ata set whose *ell 'alues will be 89o: i.11 These results show that 16 o.. *heating+ we nee( to *ount how /any stu(ents answere( yes to one or /ore o. %n(+ sin*e we nee( to obtain *ounts . 9o an( .es 'alues in the *olu/n na/e( Cheate(. )ay not be s*anne(+ *opie( or (upli*ate(+ or poste( to a publi*ly a**essible website+ in whole or in part.22BC *ollaborate( with other stu(ents on pro5e*ts that were suppose( to be *o/plete( in(i'i(ually. 0ut+ to esti/ate the per*entage o.11BC were in'ol'e( in so/e type o. the stu(ent answere( no to ea*h o.89 1.es 9 Collaborate( on ?n(i'i(ual "ro5e*t 9o . the internet as their own+ *opie( answers o. The . another stu(ent3s e4a/2 .or/ o..22 Cheate( 9o .es 9 Count 7 16 90 "er*ent 82. the three -uestions will appear at the beginning o. *heating.es to ea*h sur'ey -uestion+ as well as the nu/ber o. the si/plest /etho(s is to 5ust sort the original (ata so the 9o responses .31 # 2010 Cengage $earning.. the abo'e -uestions.t'ical /e'a"ior o$ /usiness Stu ents at /ay"ie* 0ni"ersity 1.ro/ ?nternet 9o . *heating: . 7ne o. stu(ents that ha'e engage( in so/e .00 Count 61 29 90 "er*ent 67.00 20. There are a nu/ber o.es 9 Count 72 18 90 "er*ent 80.e/ale stu(ents+ the (ata *an also be sorte( a**or(ing to =en(er. ways to obtain the 'alues in this new *olu/n.78 Copie( on @4a/ 9o . Chapter 9 Hypothesis Testing Male Stu ents Copie( . The /ost *o//on . all business stu(ents were in'ol'e( in this type o.75B+ were in'ol'e( in so/e type o.es 9 Collaborate( on ?n(i'i(ual "ro5e*t 9o .or /ale stu(ents A. another stu(ent3s e4a/< an( 26..1 58C. *heating is higher . 375C than .25 3. *heating. 3B *opie( answers o.es 9 Count 1 7 8 "er*ent 85.3810C. another stu(ent3s e4a/< an( 37.es 9 Count 27 21 8 "er*ent 56. the internet as their own is higher .10B+ were in'ol'e( in so/e .50 Cheate( 9o .50 37.or/ o.e/ale stu(ents A. *heating.75 Dor the 8 /ale stu(ents+ 1 .19 Cheate( 9o .e/ale stu(ents+ 21. the 8 /ale stu(ents+ or 3.es 9 Count 33 9 2 "er*ent 78.10 Dor the 2 .. the 2 .es 9 Count 39 9 8 "er*ent 81. C" . *heating.es 9 Count 33 9 2 "er*ent 78.50B *ollaborate( with other stu(ents on pro5e*ts that were suppose( to be *o/plete( in(i'i(ually.or . %ll &ights &eser'e(.32 # 2010 Cengage $earning.90 38.es 9 Collaborate( on ?n(i'i(ual "ro5e*t 9o .ro/ ?nternet 9o .75B *opie( answers o. ..ro/ ?nternet 9o . 3 Copie( on @4a/ 9o .81 26.e/ale stu(ents+ or 38.es 9 Count 26 16 2 "er*ent 61. the internet as their own< 21. The proportion stu(ents that presente( wor1 *opie( o.e/ale stu(ents A.58 Copie( on @4a/ 9o ..58B presente( wor1 *opie( o. 2 1 .or . 3 Count 31 11 2 "er*ent 73. 7'erall+ 21 o. )ay not be s*anne(+ *opie( or (upli*ate(+ or poste( to a publi*ly a**essible website+ in whole or in part..75 Count 30 18 8 "er*ent 62.or/ o. A itional Comments • • • The proportion o. *heating in'ol'e( *ollaborating on pro5e*ts that were suppose( to be *o/plete( in(i'i(ually< appro4i/ately 32B o.57 21.57 21.19B *ollaborate( with other stu(ents on pro5e*ts that were suppose( to be *o/plete( in(i'i(ually. 3B presente( wor1 *opie( o.or /ale stu(ents A.21 3C than . stu(ents in'ol'e( in so/e type o. the internet as their own< 18.25 18. 7'erall+ 16 o. 1emale Stu ents Copie( . i(en*e inter'al: . The Chronicle of Higher Education reporte( that the proportion o. *heating .or/ o.0523 n 90 z= p − p0 .3810C = 1.1017 n 90 95B Con.3810 )argin o. 375 )argin o. business stu(ents who a(/itte( to so/e . error E z.85.1 69 or .or/ o.56C = = . *heating is less than business stu(ents at other uni'ersities+ we are loo1ing . *heating was . 375C = 1.1 03 or .5778 1emale Stu ents p = .3810 ± .33 ≤ -2. 111C = 1.56.1 03 n 8 95B Con.3810A1 − . %ll &ights &eser'e(. business stu(ents at 0ay'iew who a(/itte( to so/e .i(en*e inter'al: . 375A1 − .309 to . 111A1 − .or the proportion o.56A1 − .85 . 375 ± .025 pA1 − p C .ollow. Thus+ the appropriate hypothesis test is H 0 : p ≥ .025 p A1 − p C .96 = . All /usiness Stu ents p = . p = . 95B *on. error E z. stu(ents that were in'ol'e( in so/e type o. 111 )argin o. )ay not be s*anne(+ *opie( or (upli*ate(+ or poste( to a publi*ly a**essible website+ in whole or in part. error E z. . the population proportion o.i(en*e inter'als .1017 or .or e'i(en*e that p+ the population proportion+ is less than .56 H a : p < .or z C" .56. Fin*e we are intereste( in testing to see i.i(en*e inter'al: . # 2010 Cengage $earning.96 = .5128 Male Stu ents p = .0523 p0 A1 − p0 C n Fin*e the test statisti* is in the lower tail+ the p-'alue is the area un(er the *ur'e .025 p A1 − p C .23 1 to . 111 an( σp = p0 A1 − p0 C .Chapter 9 Hypothesis Testing 2.56 >e will use G E .05.2972 to . 111 − . 111 ± .1 69 n 2 95B Con.56 = = −2.5279 3.96 = . the instru*tor regar(ing *heating in*i(ents an( how they will be (ealt with.3 # 2010 Cengage $earning. 111 − . "ossible steps the (ean /ight *onsi(er in*lu(e the . 5. e. )ay not be s*anne(+ *opie( or (upli*ate(+ or poste( to a publi*ly a**essible website+ in whole or in part.. &e-uire *ourse outlines to in*lu(e a state/ent regar(ing the e4pe*tations o.or business stu(ents at other uni'ersities. There is not su.a*ulty to in'estigate how stu(ents an( .. non-business stu(ents who a(/itte( to so/e . all 0ay'iew business stu(ents are in'ol'e( in so/e type o. 7 = = −1.0022 Hypothesis Testing 0e*ause the p-'alue E . 7.05. %ll &ights &eser'e(. ≤ -1. The Chronicle of Higher Education reporte( that the proportion o.or e'i(en*e that p+ the population proportion+ is less than . *heating still in(i*ates a serious proble/ . p = .or business s*hools at other uni'ersities. 7 >e will use G E . *heating is /u*h less than the proportion .or/ o. The (ean nee(s to ta1e a((itional steps to (eal with the proble/ o.0022 ≤ . 7A1 − . ethi*al beha'ior. *heating in the *ollege.131 0e*ause the p-'alue E . The goo( news is that the proportion o. all business stu(ents at 0ay'iew that are in'ol'e( in so/e type o.0526 p0 A1 − p0 C n Fin*e the test statisti* is in the lower tail+ the p-'alue is the area un(er the *ur'e . stu(ents an( .orts to re(u*e *heating are wor1ing.or z 6sing the stan(ar( nor/al probability table+ the p-'alue E .or non-business stu(ents at other uni'ersities. C" . .or the *ollege. 111 an( σp = p0 A1 − p0 C . Continue to a(/inister the e4it sur'ey so (ata *an be obtaine( to (eter/ine i.i*ient e'i(en*e to *on*lu(e that the ethi*al beha'ior o.131 H .ollowing: • • • • Dor/ a *o//ittee o.12. *heating was .12 . 7C = = .or/ o. 7. Fponsor se/inars where business e4e*uti'es *an tal1 about the i/portan*e o. 9onetheless+ the .05 we re5e*t H 0 an( *on*lu(e that the ethi*al beha'ior o. business stu(ents at 0ay'iew who a(/itte( to so/e . stu(ents at 0ay'iew 6ni'ersity Awith respe*t to *heatingC is better than . . 7 Ha : p < . *heating is less than nonbusiness stu(ents at other uni'ersities+ we are loo1ing .0526 n 90 z= p − p0 .a*t that appro4i/ately 1B o.Chapter 9 6sing the stan(ar( nor/al probability table+ the p-'alue E . Thus+ the appropriate hypothesis test is H0 : p ≥ .05 we *annot re5e*t H 0 . Fin*e we are intereste( in testing to see i. stu(ents at 0ay'iew 6ni'ersity Awith respe*t to *heatingC is better than . *heating. the population proportion o.a*ulty *an wor1 together to re(u*e the e4tent o.