# Chapter 7 Inferences Based on a Single Sample: Tests of Hypothesis

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1 Chpter 7 Inferences Bsed on Single Smple: Tests of ypothesis 7.1 The null hypothesis is the "sttus quo" hypothesis, while the lterntive hypothesis is the reserch hypothesis. 7. The test sttistic is used to decide whether or not to reject the null hypothesis in fvor of the lterntive hypothesis. 7.3 The "level of significnce" of test is. This is the probbility tht the test sttistic will fll in the rejection region when the null hypothesis is true. 7.4 A Type I error is rejecting the null hypothesis when it is true. A Type II error is ccepting the null hypothesis when it is flse. the probbility of committing Type I error. the probbility of committing Type II error. 7.5 The four possible results re: 1. Rejecting the null hypothesis when it is true. This would be Type I error.. Accepting the null hypothesis when it is true. This would be correct decision. 3. Rejecting the null hypothesis when it is flse. This would be correct decision. 4. Accepting the null hypothesis when it is flse. This would be Type II error. 7.6 We cn compute mesure of relibility for rejecting the null hypothesis when it is true. This mesure of relibility is the probbility of rejecting the null hypothesis when it is true which is. owever, it is generlly not possible to compute mesure of relibility for ccepting the null hypothesis when it is flse. We would hve to compute the probbility of ccepting the null hypothesis when it is flse,, for every vlue of the prmeter in the lterntive hypothesis. 7.7 When you reject the null hypothesis in fvor of the lterntive hypothesis, this does not prove the lterntive hypothesis is correct. We re11 % confident tht there is sufficient evidence to conclude tht the lterntive hypothesis is correct If we were to repetedly drw smples from the popultion nd perform the test ech time, pproximtely 1 1 % of the tests performed would yield the correct decision. 33 Copyright 18 Person Eduction, Inc.

2 Inferences Bsed on Single Smple: Tests of ypothesis 333 b. c. d. e. f. g. Pz Pz P z P z P z 1.645or z P z.575or z Let p = proportion of Americn dults who pick professionl footbll s their fvorite sport. To see if the proportion of Americn dults who pick professionl footbll s their fvorite sport differs from.4, we test: Copyright 18 Person Eduction, Inc.

3 334 Chpter 7 : p.4 : p Let verge gin in green fees, lessons, or equipment expenditures for prticipting golf fcilities. The null nd lterntive hypotheses would be: : \$, 4 : \$, 4 b. The.5 is the Type I error rte. This mens tht the probbility of concluding tht the verge gin in fees, lessons, or equipment expenditures for prticiption golf fcilities exceeds \$,4 when in fct, the verge is \$,4 is.5. c. The rejection region requires.5 in the upper til of the z-distribution. From Tble II, Appendix D, z The rejection region is z Let p = student lon defult rte in this yer. To see if the student lon defult rte is less thn.1, we test: : p.1 : p The prmeter of interest is p = proportion of deceitful speech recognized by the vtr. b. To test the clim tht the vtr cn detect deceitful speech 75% of the time, we test: : p.75 : p.75 c. A Type I error is rejecting when is true. For this exercise, tht would be concluding tht the proportion of deceitful speech recognized by the vtr is different from.75 when the proportion is equl to.75. d. A Type II error is ccepting when is flse. For this exercise, tht would be concluding tht the proportion of deceitful speech recognized by the vtr equl to.75 when the proportion differs from Let men cloric content of Virgini school lunches. To test the clim tht fter the testing period ended, the verge cloric content dropped, we test: : 863 : Let verge Libor rte for 1-yer lons. Since mny Western bnks think tht the reported verge Libor rte (1.%) is too high, they wnt to show tht the verge is less thn 1.. The pproprite hypotheses would be: : 1. : 1. Copyright 18 Person Eduction, Inc.

4 Inferences Bsed on Single Smple: Tests of ypothesis Since the compny must give proof the drug is sfe, the null hypothesis would be the drug is unsfe. The lterntive hypothesis would be the drug is sfe. b. A Type I error would be concluding the drug is sfe when it is not sfe. A Type II error would be concluding the drug is not sfe when it is. is the probbility of concluding the drug is sfe when it is not. is the probbility of concluding the drug is not sfe when it is. c. In this problem, it would be more importnt for to be smll. We would wnt the probbility of concluding the drug is sfe when it is not to be s smll s possible A Type I error would be concluding the proposed user is unuthorized when, in fct, the proposed user is uthorized. A Type II error would be concluding the proposed user is uthorized when, in fct, the proposed user is unuthorized. In this cse, more serious error would be Type II error. One would not wnt to conclude tht the proposed user is uthorized when he/she is not. b. The Type I error rte is 1%. This mens tht the probbility of concluding the proposed user is unuthorized when, in fct, the proposed user is uthorized is.1. The Type II error rte is.5%. This mens tht the probbility of concluding the proposed user is uthorized when, in fct, the proposed user is unuthorized is.5. c. The Type I error rte is.1%. This mens tht the probbility of concluding the proposed user is unuthorized when, in fct, the proposed user is uthorized is.1. The Type II error rte is.5%. This mens tht the probbility of concluding the proposed user is uthorized when, in fct, the proposed user is unuthorized is A Type I error is rejecting the null hypothesis when it is true. In murder tril, we would be concluding tht the ccused is guilty when, in fct, he/she is innocent. A Type II error is ccepting the null hypothesis when it is flse. In this cse, we would be concluding tht the ccused is innocent when, in fct, he/she is guilty. b. Both errors re serious. owever, if n innocent person is found guilty of murder nd is put to deth, there is no wy to correct the error. On the other hnd, if guilty person is set free, he/she could murder gin. c. In jury tril, is ssumed to be smller thn. The only wy to convict the ccused is for unnimous decision of guilt. Thus, the probbility of convicting n innocent person is set to be smll. d. In order to get unnimous vote to convict, there hs to be overwhelming evidence of guilt. The probbility of getting unnimous vote of guilt if the person is relly innocent will be very smll. e. If jury is prejudiced ginst guilty verdict, the vlue of will decrese. The probbility of convicting n innocent person will be even smller if the jury if prejudiced ginst guilty verdict. f. If jury is prejudiced ginst guilty verdict, the vlue of will increse. The probbility of declring guilty person innocent will be lrger if the jury is prejudiced ginst guilty verdict. Copyright 18 Person Eduction, Inc.

5 336 Chpter The null hypothesis is: o : There is no intrusion. b. The lterntive hypothesis is: : There is n intrusion. 1 c. P(wrning no intrusion) P(no wrning intrusion) p Pz b. p Pz c. p Pz Pz We will reject if the p-vlue...6.5, do not reject. b..1.5, do not reject. c..1.5, reject. d..1.5, reject. e..51.5, do not reject. f..4.5, reject Since the p-vlue.1 is greter thn.5, is not rejected. b. Since the p-vlue.5 is less thn.1, is rejected. c. Since the p-vlue.1is less thn.1, is rejected. d. Since the p-vlue.5 is greter thn.5, is not rejected. e. Since the p-vlue.45 is greter thn.1, is not rejected. 7. x z / 1 x p-vlue p P z (Using Tble II, Appendix D) Since the p-vlue is so lrge, would not be rejected for ny resonble vlue of. There is no evidence to indicte the men is greter thn p p Pz P z -vlue (using Tble II, Appendix D) Copyright 18 Person Eduction, Inc.

6 Inferences Bsed on Single Smple: Tests of ypothesis 337 The probbility of observing test sttistic of.17 or nything more unusul if the true men is 1 is.15. Since this probbility is so smll, there is evidence tht the true men is greter thn 1. x First, find the vlue of the test sttistic z / 5 x p-vlue p P z 1.6 or z 1.6 P z There is no evidence to reject for p p Pz Pz (using Tble II, Appendix D) -vlue (using Tble II, Appendix D) 7.6. The p-vlue reported by SPSS is for two-tiled test. Thus, Pz Pz this one-tiled test, the p-vlue p Pz / For Since the p-vlue.516.5, is not rejected. There is insufficient evidence to indicte 75 t.5. b. For this one-tiled test, the p-vlue Pz Since Pz P z /.516, Since the p-vlue p , is not rejected. There is insufficient evidence to indicte 75 t.1. c. For this one-tiled test, the p Pz -vlue /.516. Since the p-vlue p.516.1, is rejected. There is sufficient evidence to indicte 75 t.1. d. For this two-tiled test, the p-vlue.13. Since the p-vlue.13.1, is not rejected. There is insufficient evidence to indicte 75 t The smllest vlue of for which the null hypothesis would be rejected is just greter thn By the Centrl Limit Theorem, the smpling distribution of x is pproximtely norml with x nd x 1. n 4 x b. The test sttistic is z.5. n 4 c. The p-vlue is p Pz Copyright 18 Person Eduction, Inc.

7 338 Chpter 7 d. The rejection region requires.1in the upper til of the z-distribution. From Tble II, Appendix D, z The rejection region is z.33. e. Since the p-vlue is less thn p.6.1 indicte the true men is greter thn 7 t.1., is rejected. There is sufficient evidence to f. Since the observed vlue of the test sttistics flls in the rejection region z.5.33, is rejected. There is sufficient evidence to indicte the true men is greter thn 7 t.1. g. Yes, the conclusions in prts e nd f gree The decision rule is to reject if x 7 Therefore, reject if x 7 x. Recll tht z. The decision rule in terms of z is to reject if z.14. b. Pz P z x 7 55 cn be written s reject if z / 81 x x 7.3. : 1 : 1 x x 11 1 The test sttistic is z 1.67 x / n 6/ 1 The rejection region requires.5 in the upper til of the z-distribution. From Tble II, Appendix D, z The rejection region is z Since the observed vlue of the test sttistic flls in the rejection region, z , is rejected. There is sufficient evidence to indicte the true popultion men is greter thn 1 t.5. b. : 1 : 1 x 11 1 The test sttistic is z 1.67 x 6 / 1 The rejection region requires /.5/.5in ech til of the z-distribution. From Tble II, Appendix D, z The rejection region is z 1.96 or z Since the observed vlue of the test sttistic does not fll in the rejection region, z , is not rejected. There is insufficient evidence to indicte does not equl 1 t.5. Copyright 18 Person Eduction, Inc.

8 Inferences Bsed on Single Smple: Tests of ypothesis 339 c. In prt, we rejected nd concluded the men ws greter thn 1. In prt b, we did not reject. There ws insufficient evidence to conclude the men ws different from 1. Becuse the lterntive hypothesis in prt is more specific thn the one in b, it is esier to reject :.36 :.36 x The test sttistic is z / 64 x The rejection region requires.1 in the lower til of the z-distribution. From Tble II, Appendix D, z The rejection region is z 1.8. Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte the men is less thn.36 t.1. b. :.36 :.36 The test sttistic is z 1.61 (see prt ). The rejection region requires /.1/.5in the ech til of the z-distribution. From Tble II, Appendix D, z The rejection region is z or z Since the observed vlue of the test sttistic does not fll in the rejection region z 1.61< 1.645, is not rejected. There is insufficient evidence to indicte the men is different from.36 t Let true men level of support. To determine if the true men level of support differs from 75, we test: : 75 : 75 b. For this problem, Type I error would be concluding the true men level of support differs from 75 when, in fct, the true men level of support is 75. For this problem, Type II error would be concluding the true men level of support equls 75 when, in fct, the true men level of support differs from 75. c. The test sttistic is z nd the p-vlue is p.1. d. Since the p-vlue is less thn p.1.5, is rejected. There is sufficient evidence to indicte the true men level of support differs from 75 t.5. e. We do not need to mke ny ssumptions bout the distribution of support levels. The smple size is very lrge n 99. Thus, the Centrl Limit Theorem holds nd no ssumptions re necessry Let true men willingness to et the brnd of sliced pples. To determine if the true men willingness to et the brnd of sliced pples exceeds 3, we test: Copyright 18 Person Eduction, Inc.

9 34 Chpter 7 : 3 : 3 x The test sttistic is z n 48 The rejection region requires.5 in the upper til of the z-distribution. From Tble II, Appendix D, z The rejection region is z Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte tht the true men willingness to et the brnd of sliced pples exceeds 3 t.5. b. Even though the willingness to et scores re not normlly distributed, the test in prt is vlid. Becuse the smple size is so lrge n 48, the Centrl Limit Theorem pplies Let true men rtio of fup/fumic. To determine if the true men rtio differs from 1, we test: : 1 : 1 b. Looking simply t the vlue of x does not tke the vrition in the smpling process into ccount. We need to see if vlue of.37 x is unusul if the true men vlue of the rtio is 1. c. From the printout, the test sttistic is z 47.9 nd the p-vlue is p.. d. Suppose we select.5. The probbility of rejecting the null hypothesis when it is true is.5. If this cse, the probbility of concluding the men rtio is different from 1 when the true men rtio is equl to 1. e. Since the p-vlue is less thn p..5, is rejected. There is sufficient evidence to indicte the men rtio is different from 1. f. Becuse the smple size is so lrge n 416, the Centrl Limit Theorem pplies. We must ssume tht the smple ws rndom smple Let true men fcil width-to-height rtio. To determine if the true men fcil width-to-height rtio differs from., we test: :. :. x The test sttistic is z n 55 Copyright 18 Person Eduction, Inc.

10 The p-vlue is p Pz Pz Inferences Bsed on Single Smple: Tests of ypothesis 341 Since the p-vlue is so smll p, will be rejected for ny resonble vlue of. There is sufficient evidence to indicte the true men fcil width-to-height rtio differs from. for Let true men rte of return of round-trip trdes. To determine if the true men rte of return of round-trip trdes is positive, we test: : : b. The rejection region requires.5 in the upper til of the z-distribution. From Tble II, Appendix D, z The rejection region is z c. probbility of mking Type I error or the probbility of rejecting when is true. Thus, probbility of concluding the true men rte of return of round-trip trdes is positive when, in fct, it is not. d. The test sttistic is t 4.73 nd the p-vlue is p.. e. Since the p-vlue is less thn p..5, is rejected. There is sufficient evidence to indicte the true men rte of return of round-trip trdes is positive t Let true men weight of golf tees. To determine if the process is not operting stisfctorily, we test: :.5 :.5 b. Using MINITAB, the descriptive sttistics re: Descriptive Sttistics: Tees Vrible N Men Medin StDev Minimum Mximum Q1 Q3 Tees Thus, x.548 nd s.3. x c. The test sttistic is z / 4 d. The p-vlue is p Pz Pz x e. The rejection region requires /.1/.5 in ech til of the z-distribution. From Tble II, Appendix D, z The rejection region is z.575 or z.575. f. Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte the process is performing in n unstisfctory mnner t.1. Copyright 18 Person Eduction, Inc.

11 34 Chpter 7 g. is the probbility of Type I error. A Type I error, in this cse, is to sy the process is unstisfctory when, in fct, it is stisfctory. The risk, then, is to the producer since he will be spending time nd money to repir process tht is not in error. is the probbility of Type II error. A Type II error, in this cse, is to sy the process is stisfctory when it, in fct, is not. This is the consumer's risk since he could unknowingly purchse defective product Let true men intension score. To determine if the true men intension score is greter thn, we test: : : x.4 The test sttistic is z n 86 The rejection region requires.1in the upper til of the z-distribution. From Tble II, Appendix D, z The rejection region is z.33. Since the observed vlue of the test sttistic does not fll in the rejection region z , is not rejected. There is insufficient evidence to indicte tht the true men intension score is greter thn t Let men estimted time to red the report. To determine if the students, on verge, overestimte the time it tkes to red the report, we test: : 48 : 48 x 6 48 The test sttistic is z / 4 The p-vlue is p Pz x (using Tble II, Appendix D) Since the p-vlue is less thn p.3.1, is rejected. There is sufficient evidence to indicte the students, on verge, overestimte the time it tkes to red the report t.1. b. Let men estimted number of pges of the report red. To determine if the students, on verge, underestimte the number of report pges red, we test: : 3 : 3 x 8 3 The test sttistic is z / 4 x Copyright 18 Person Eduction, Inc.

12 The p-vlue is p Pz Inferences Bsed on Single Smple: Tests of ypothesis (using Tble II, Appendix D) Since the p-vlue is less thn p.3.1, is rejected. There is sufficient evidence to indicte the students, on verge, underestimte the number of report pges red t.1. c. No. In both tests, the smple sizes re greter thn 3. Thus, the Centrl Limit Theorem will pply. The distribution of x is pproximtely norml regrdless of the popultion distribution Let B true men number of vouchers sold 3 minutes before the tipping point. To determine if the true men number of vouchers sold 3 minutes before the tipping point is less thn 5, we test: : 5 B : 5 B x The test sttistic is z n, 11 The rejection region requires.1 in the lower til of the z-distribution. From Tble II, Appendix D, z The rejection region is z 1.8. Since the observed vlue of the test sttistic does not fll in the rejection region z , is not rejected. There is insufficient evidence to indicte tht the true men number of vouchers sold 3 minutes before the tipping point is less thn 5 t.1. b. Let A true men number of vouchers sold 3 minutes fter the tipping point. To determine if the true men number of vouchers sold 3 minutes fter the tipping point is greter thn 1, we test: : 1 A : 1 A x The test sttistic is z n,617 The rejection region requires.1 in the upper til of the z-distribution. From Tble II, Appendix D, z The rejection region is z 1.8. Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte tht the true men number of vouchers sold 3 minutes fter the tipping point is greter thn 1 t.1. Copyright 18 Person Eduction, Inc.

13 344 Chpter To determine if the men point-spred error is different from, we test: : : The test sttistic is x 1.6 z 1.86 x 13.3 / 4 The rejection region requires /.1/.5 in ech til of the z-distribution. From Tble II, Appendix D, z The rejection region is z.575 or z.575. Since the observed vlue of the test sttistic does not fll in the rejection region z , is not rejected. There is insufficient evidence to indicte tht the true men point-spred error is different from t Let verge full-service fee (in thousnds of dollrs) of U.S. funerl homes in the current yer. To determine if the verge full-service fee exceeds \$6,5, we test: : 7.18 : 7.18 b. Using MINTAB, the output is: Descriptive Sttistics: FUNERAL Vrible N Men StDev Minimum Q1 Medin Q3 Mximum FUNERAL : 7.18 : 7.18 x The test sttistic is z x 36 The rejection region requires.5 in the lower til of the z-distribution. From Tble II, Appendix D, z The rejection region is z Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte the true men full-service fee of U.S. funerl homes in the current yer is less thn \$7,18 t.5. c. No. Since the smple size n 36is greter thn 3, the Centrl Limit Theorem pplies. The distribution of x is pproximtely norml regrdless of the popultion distribution To determine if the true men forecst error for buy-side nlysts is positive, we test: : : Copyright 18 Person Eduction, Inc.

14 x o.85 The test sttistic is z / 3,56 The observed p-vlue of the test is p Pz Since the p-vlue is less thn p.1 x Inferences Bsed on Single Smple: Tests of ypothesis (Using Tble II, Appendix D), is rejected. There is sufficient evidence to indicte tht the true men forecst error for buy-side nlysts is positive t.1. This mens tht the buyside nlysts re overestimting ernings. b. To determine if the true men forecst error for sell-side nlysts is negtive; we test: : : x o.5 The test sttistic is z / 58,56 x The observed p-vlue of the test is p Pz Since the p-vlue is less thn p (using Tble II, Appendix D), is rejected. There is sufficient evidence to indicte tht the true men forecst error for sell-side nlysts is negtive t.1. This mens tht the sellside nlysts re underestimting ernings To determine if the smple dt refute the mnufcturer's clim, we test: : 1 : 1 b. A Type I error is concluding the men number of solder joints inspected per second is less thn 1 when, in fct, it is 1 or more. A Type II error is concluding the men number of solder joints inspected per second is t lest 1 when, in fct, it is less thn 1. c. Using MINITAB, the descriptive sttistics re: Descriptive Sttistics: PCB Vrible N Men Medin StDev Minimum Mximum Q1 Q3 PCB : 1 : 1 x The test sttistic is z.33 x.13 / 48 The rejection region requires.5 in the lower til of the z-distribution. From Tble II, Appendix D, z The rejection region is z Copyright 18 Person Eduction, Inc.

15 346 Chpter 7 Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte the men number of inspections per second is less thn 1 t Let men externl tension level of ll mngers who engge in coopetition. To determine if the men externl tension level differs from 1.5, we test: : 1.5 : 1.5 x The test sttistic is z x 1,53 The rejection region requires /.5 /.5 in ech til of the z-distribution. From Tble II, Appendix D, z The rejection region is z 1.96 or z Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte the true men externl tension level differs from 1.5 t.5. b. The 95% confidence intervl is: 3.4 x z , 1.97 n 1, 53 c. If the hypothesized vlue of the test sttistic flls in the confidence intervl, then is not rejected. If the hypothesized vlue of the test sttistic does not fll in the confidence intervl, then is rejected. d. The observed vlue of x is very close to 1.5. Even though it is sttisticlly significnt, the men externl tension level my not be prcticlly different from No. Since the hypothesized vlue of 6, M flls in the 95% confidence intervl, it is likely cndidte for the true men. Thus, we would not reject. There is no evidence tht the men slry for mles differs from \$6,. b. To determine if the true men slry of mles with post-grdute degrees differs from \$6,, we test: : 6, : 6, x 61,34 6, The test sttistic is z.61,185 x The rejection region requires /.5/.5in ech til of the z-distribution. From Tble II, Appendix D, z The rejection region is z 1.96 or z Copyright 18 Person Eduction, Inc.

16 Inferences Bsed on Single Smple: Tests of ypothesis 347 Since the observed vlue of the test sttistic does not fll in the rejection region z , is not rejected. There is insufficient evidence to indicte the true men slry of mles with postgrdute degrees differs from \$6, t.5. c. Prts nd b must gree. In both cses, two-sided test / confidence intervl is used. The z-score used in both prts is the sme, s re x nd s x. d. No. Since the hypothesized vlue of 33, F flls in the 95% confidence intervl, it is likely cndidte for the true men. Thus, we would not reject. There is no evidence tht the men slry for femles differs from \$33,. e. To determine if the true men slry of femles with post-grdute degrees differs from \$33,, we test: : 33, : 33, x 3, 7 33, The test sttistic is z x The rejection region requires /.5/.5in ech til of the z-distribution. From Tble II, Appendix D, z The rejection region is z 1.96 or z Since the observed vlue of the test sttistic does not fll in the rejection region z , is not rejected. There is insufficient evidence to indicte the true men slry of femles with post-grdute degrees differs from \$33, t.5. f. Prts d nd e must gree. In both cses, two-sided test / confidence intervl is used. The z-score used in both prts is the sme, s re x nd s x We should use the t-distribution in testing hypothesis bout popultion men if the smple size is smll, the popultion being smpled from is norml, nd the vrince of the popultion is unknown. b. Both distributions re mound-shped nd symmetric. The t-distribution is fltter thn the z- distribution Pt (Using Tble III, Appendix D, with df 6 ) b. Pt (Using Tble III, Appendix D, with df 1 ) Copyright 18 Person Eduction, Inc.

17 348 Chpter 7 c. Pt Pt (Using Tble III, Appendix D, with df 5 ) d. The probbility of Type I error is computed bove for ech of the prts The rejection region requires /.5/.5in ech til of the t-distribution with df n From Tble III, Appendix D, t The rejection region is t.16 or t.16. b. The rejection region requires.1in the upper til of the t-distribution with df n From Tble III, Appendix D, t.1.5. The rejection region is t.5. c. The rejection region requires.1 in the upper til of the t-distribution with df n From Tble III, Appendix D, t The rejection region is t d. The rejection region requires.1in the lower til of the t-distribution with df n From Tble III, Appendix D, t The rejection region is t.718. e. The rejection region requires /.1/.5in ech til of the t-distribution with df n From Tble III, Appendix D, t The rejection region is t 1.79 or t f. The rejection region requires.5 in the lower til of the t-distribution with df n From Tble III, Appendix D, t The rejection region is t : 6 : 6 x The test sttistic is t.64 s/ n 1.3/ 5 The necessry ssumption is tht the popultion is norml. The rejection region requires.5 in the lower til of the t-distribution with df n From Tble III, Appendix D, t The rejection region is t.13. Since the observed vlue of the test sttistic does not fll in the rejection region t.64.13, is not rejected. There is insufficient evidence to indicte the men is less thn 6 t.5. b. : 6 : 6 Copyright 18 Person Eduction, Inc.

18 Inferences Bsed on Single Smple: Tests of ypothesis 349 The test sttistic is t.64 (from ). The ssumption is the sme s in. The rejection region requires /.5/.5in ech til of the t-distribution with df n From Tble III, Appendix D, t The rejection region is t.776 or t.776. Since the observed vlue of the test sttistic does not fll in the rejection region t , is not rejected. There is insufficient evidence to indicte the men is different from 6 t.5. c. For prt, the p-vlue Pt.64. Using MINITAB, Cumultive Distribution Function Student's t distribution with 4 DF x P( X <= x ) The p-vlue is p For prt b, the p-vlue Pt.64 Pt.64. The p-vlue is p We must ssume tht rndom smple ws drwn from norml popultion. b. The hypotheses re: : 1, : 1, The test sttistic is t 1.89 nd the p-vlue is p.38. Since the p-vlue is so smll, there is evidence to reject. There is evidence to indicte the men is greter thn 1 for.38. c. The hypotheses re: : 1, : 1, The test sttistic is t 1.89 nd the p-vlue is There is no evidence to reject for.5. There is insufficient evidence to indicte the men is different thn 1 for.5. There is evidence to reject for.76. There is evidence to indicte the men is different thn 1 for.76. Copyright 18 Person Eduction, Inc.

19 35 Chpter Let men yers of experience for commercil suppliers of the DoD. To determine if the men yers of experience is greter thn 5, we test: : 5 : 5 b. Using MINITAB, the descriptive sttistics re: Descriptive Sttistics: Yers Vrible N Men StDev Minimum Q1 Medin Q3 Mximum Yers x The test sttistic is t.3. s 8.87 n 6 c. The rejection region requires.5 in the upper til of the t-distribution with df n From Tble III, Appendix D, t The rejection region is t.15. d. Since the observed vlue of the test sttistic flls in the rejection region t.3.15, is rejected. There is sufficient evidence to indicte the men yers of experience for commercil suppliers of the DoD is greter thn 5 t.5. e. From the printout, the p-vlue is p.49. The probbility of observing test sttistic of t.3 or nything higher if the true men is 5 is Let men number of occuptionl ccidents t ll Turkish construction sites. To determine if the men number of occuptionl ccidents is less thn 7, we test: : 7 : 7 b. The rejection region requires.1in the lower til of the t-distribution with df n From Tble III, Appendix D, t The rejection region is t x 64 7 c. The test sttistic is t.9. s 35.3 n 3 d. Since the observed vlue of the test sttistic does not fll in the rejection region t is not rejected. There is insufficient evidence to indicte the men number of occuptionl ccidents t ll Turkish construction sites is less thn 7 t.1. e. We must ssume tht rndom smple ws drwn from norml popultion To determine if the men of the trp spcing mesurements differs from 95 meters, we test:, Copyright 18 Person Eduction, Inc.

20 Inferences Bsed on Single Smple: Tests of ypothesis 351 : 95 : 95 b. The vlue of x vries from smple to smple. The next smple my yield vlue of x tht is greter thn 95. We must determine how unusul vlue of x 89.9 is if the true men is 95. x c. The test sttistic is t s 11.6 n 7 d. Using MINITAB, the results re: One-Smple T Test of mu = 95 vs not = 95 N Men StDev SE Men 95% CI T P ( , 1.68) The p-vlue is p.89. e. Suppose we pick.5. For this problem, probbility of concluding the men trp spcing is different from 95 when, in fct, the men trp spcing is equl to 95. f. Since the p-vlue is greter thn p.89.5, is not rejected. There is insufficient evidence to indicte the men trp spcing is different from 95 t.5. g. In order for the test to be vlid, the popultion of trp spcing mesurements must be norml nd the smple must be rndom. h. From Exercise 6.9, the 95% confidence intervl is (79.14, 1.616). Since 95 is contined in this intervl, there is no evidence to indicte the men trp spcing is different from 95. This grees with the test in prt f To determine if the men level of rdon exposure in the tombs is less thn 6, Bq/m 3, we test: : 6, : 6, From the printout, the test sttistic is t 1.8. Since this is one-tiled test, the p-vlue is p.96 /.48. Since the p-vlue is less thn p.48.1, is rejected. There is sufficient evidence to indicte the men level of rdon exposure is less thn 6, Bq/m 3 t Let men nnulized percentge return on investment. To determine if the men nnulized percentge return on investment is positive, we test: : : Copyright 18 Person Eduction, Inc.

21 35 Chpter 7 x b. From the printout, x nd s The test sttistic is t 5.6. s n 13 c. The p-vlue is p.1. d. Since the p-vlue is less thn p.1.5, is rejected. There is sufficient evidence to indicte the men nnulized percentge return on investment is positive t.5. e. We must ssume tht we hve selected rndom smple from the popultion nd tht the popultion of nnulized percentge return on investments for ll AAII stock screeners is normlly distributed Using MINITAB, the clcultions re: One-Smple T: Velocity Test of μ =.338 vs.338 Vrible N Men StDev SE Men 9% CI T P Velocity (.461,.786) Let men bubble rising velocity. To determine if the men bubble velocity differs from.338, we test: :.338 :.338 The test sttistic is t 8.13nd the p-vlue is p.. Since the p-vlue is less thn p..1, is rejected. There is sufficient evidence to indicte the men bubble velocity differs from.338 t.1. b. No. Becuse the p-vlue is so smll, it would be extremely unlikely tht the dt were generted t sprging rte of 3.33 x Using MINITAB, the descriptive sttistics re: Descriptive Sttistics: Choice Vrible N Men StDev Minimum Q1 Medin Q3 Mximum Choice Let men choice score for consumers shopping with flexed rms. To determine if the men choice score for consumers shopping with flexed rms is higher thn 43, we test: : 43 : 43 x The test sttistic is t s 7.78 n 11 Copyright 18 Person Eduction, Inc.

22 Inferences Bsed on Single Smple: Tests of ypothesis 353 The rejection region requires.5 in the upper til of the t-distribution with df n From Tble III, Appendix D, t The rejection region is t Since the observed vlue of the test sttistic flls in the rejection region t , is rejected. There is sufficient evidence to indicte the men choice score for consumers shopping with flexed rms is higher thn 43 t Using MINITAB, the descriptive sttistics re: One-Smple T: Skid Test of mu = 45 vs < 45 95% Upper Vrible N Men StDev SE Men Bound T P Skid To determine if the men skidding distnce is less thn 45 meters, we test: : 45 : 45 x o The test sttistics is t.53. s n The rejection region requires.1 in the lower til of the t-distribution with. From Tble III, Appendix D, t The rejection region is t Since the observed vlue of the test sttistic flls in the rejection region t , is rejected. There is sufficient evidence to indicte the true men skidding distnce is less thn 45 meters t.1. There is sufficient evidence to refute the clim. 7.6 Using MINITAB, the descriptive sttistics re: Descriptive Sttistics: Dioxide Vrible Oil N Men StDev Minimum Q1 Medin Q3 Mximum Dioxide No Yes To determine if the men mount of dioxide present in wter specimens tht contin oil is less thn 3 mg/l, we test: : 3 : 3 x The test sttistic is t s.47 n 6 The rejection region requires.1 in the lower til of the t-distribution with df n From Tble III, Appendix D, t The rejection region is t Copyright 18 Person Eduction, Inc.

23 354 Chpter 7 Since the observed vlue of the test sttistic flls in the rejection region t , is rejected. There is sufficient evidence to indicte the men mount of dioxide present in wter specimens tht contin oil is less thn 3 mg/l t.1. b. To determine if the men mount of dioxide present in wter specimens tht do not contin oil is less thn 3 mg/l, we test: : 3 : 3 x.59 3 The test sttistic is t.84. s 1.54 n 1 The rejection region requires.1 in the lower til of the t-distribution with df n From Tble III, Appendix D, t The rejection region is t Since the observed vlue of the test sttistic does not fll in the rejection region t , is not rejected. There is insufficient evidence to indicte the men mount of dioxide present in wter specimens tht do not contin oil is less thn 3 mg/l t Using MINITAB, the preliminry clcultions re: One-Smple T: rdness Test of μ = 76 vs > 76 Vrible N Men StDev SE Men 95% Lower Bound T P rdness Let men hrdness of polyester composite mixture with 4% CKD weight rtio. To determine if using 4% CKD weight rtio increses the men hrdness of polyester composite mixture, we test: : 76 : 76 The test sttistic is t 3.93 nd the p-vlue is p.3. Since the p-vlue is smll p.3, is rejected. There is sufficient evidence to indicte tht using 4% CKD weight rtio increses the men hrdness of polyester composite mixture for ny vlue of x This number represents the popultion becuse it is the men of the entire n 5 popultion of sttes. b. Answers will vry. Using MINITAB, rndom smple of 5 sttes yielded Cliforni (1), Indin (), Louisin (), Michign (3), nd Montn (). c. Using MINITAB, the preliminry clcultions re: Copyright 18 Person Eduction, Inc.

24 Inferences Bsed on Single Smple: Tests of ypothesis 355 One-Smple T: Smple Test of μ = 1.4 vs 1.4 Vrible N Men StDev SE Men 95% CI T P Smple (-.419,.819) To determine if the men number of ctive nucler power plnts operting in ll sttes differs from 1.4, we test: : 1.4 : 1.4 The test sttistic is t.7 nd the p-vlue is p.949. Since the p-vlue is greter thn p.949.1, is not rejected. There is insufficient evidence to indicte the men number of ctive nucler power plnts operting in ll sttes differs from 1.4 t.1. d. Two possible resons for incorrectly rejecting the null hypothesis is (1) The popultion of the number of ctive nucler power plnts is not normlly distributed, but skewed to the left. () The smple size is very smll. It is possible to get n unusul smple with very smll smple size Using MINITAB, the descriptive sttistics for the plnts re: Descriptive Sttistics: AL1, AL Vrible N Men StDev Minimum Q1 Medin Q3 Mximum AL *.75 *.1 AL *.7 *.9 To determine if plnt 1 is violting the OSA stndrd, we test: :.4 :.4 x o.75.4 The test sttistic is t 1.4 s n.354 The p-vlue is p Pt 1.4. Using Minitb with df n 1 1 1, the p-vlue is p.197. Since the p-vlue is not smll, is not rejected. There is insufficient evidence to indicte the OSA stndrd is violted by plnt 1 for ny vlue of.197. To determine if plnt is violting the OSA stndrd, we test: :.4 :.4 x o.7.4 The test sttistic is t 3.3 s n.83 The p-vlue is p Pt 1.4. Using Minitb with df n 1 1 1, the p-vlue is p.96. Since the Copyright 18 Person Eduction, Inc.

25 356 Chpter 7 p-vlue is not smll, is not rejected. There is insufficient evidence to indicte the OSA stndrd is violted by plnt for ny vlue of.96. If.96, is rejected. There is sufficient evidence to indicte the OSA stndrd is violted by plnt for ny vlue of Since the vlue of ˆp (.63) is much smller thn the hypothesized vlue of p (.7), it is likely tht the null hypothesis is not correct. b. First, check to see if n is lrge enough. np nd nq Since both np 15 nd nq 15, the norml pproximtion will be dequte. : p.7 : p.7 pˆ p pˆ p.63.7 The test sttistic is z 1.53 pˆ pq.7.3 n 1 The rejection region requires.5 in the lower til of the z-distribution. From Tble II, Appendix D, z The rejection region is z Since the observed vlue of the test sttistic does not fll in the rejection region z , is not rejected. There is insufficient evidence to indicte tht the proportion is less thn.7 t.5. c. p p Pz -vlue Since p is not less thn.5, is not rejected z p ˆ p.83.9 pq n.9(.1) 1.33 b. The denomintor in Exercise 7.64 is s compred to denomintor in this problem is smller, the bsolute vlue of z is lrger..3 in prt. Since the c. The rejection region requires.5 in the lower til of the z-distribution. From Tble II, Appendix D, z The rejection region is z Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte the popultion proportion is less thn.9 t.5. d. The p-vlue p Pz (from Tble II, Appendix D). Since the p-vlue is less thn.5, is rejected. Copyright 18 Person Eduction, Inc.

26 Inferences Bsed on Single Smple: Tests of ypothesis No. The p-vlue is the probbility of observing your test sttistic or nything more unusul if is true. For this problem, the p-vlue.33 /.165. Given the true vlue of the popultion proportion, p, is.5, the probbility of observing test sttistic of.44 p.165, there is no evidence to reject z or lrger is.165. Since the p-vlue is not smll. There is no evidence to indicte the popultion proportion is greter thn.5. b. If the lterntive hypothesis were two-tiled, the p-vlue would be times the p-vlue for one-tiled test. For this problem, the p-vlue.33. The probbility of observing your test sttistic or nything more unusul if is true is.33. Since the p-vlue is so lrge, there is no evidence to reject for.1. There is no evidence to indicte tht p.5 for Becuse p is the proportion of consumers who do not like the snck food, ˆp will be: Number of 's in smple 9 pˆ.58 n 5 First, check to see if the norml pproximtion will be dequte: np nq Since both np 15 nd nq 15, the norml distribution will be dequte.. : p.5 : p.5 p ˆ p p ˆ p.58.5 The test sttistic is z pˆ pq.51.5 n 5 The rejection region requires.1 in the upper til of the z-distribution. From Tble II, Appendix D, z The rejection region is z 1.8. Since the observed vlue of the test sttistic does not fll in the rejection region z , is not rejected. There is insufficient evidence to indicte the proportion of customers who do not like the snck food is greter thn.5 t.1. b. p p Pz vlue (using Tble II, Appendix D) 7.68 The smple size is lrge enough to use the norml pproximtion if npo 15 nd nqo 15.. np nd o b. np nd o enough. nq Thus, the smple size is lrge enough. o nq Thus, the smple size is not lrge o Copyright 18 Person Eduction, Inc.

27 358 Chpter 7 c. np nd o d. np nd o e. np nd o nq Thus, the smple size is not lrge enough. o nq Thus, the smple size is not lrge enough. o nq Thus, the smple size is not lrge enough. o pˆ.9 5 b. To determine if more thn 8% of ll customers would prticipte in loylty crd progrm, we test: : p.8 :.8 pˆ p.9.8 c. The test sttistic is z pq.8. n 5 d. The rejection region requires.1in the upper til of the z-distribution. From Tble II, Appendix D, z The rejection region is z.33. e. The p-vlue is p Pz (Using Tble II, Appendix D) f. Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte tht more thn 8% of ll customers would prticipte in loylty crd progrm t.1. g. Since the p-vlue is less thn p.1, is rejected. There is sufficient evidence to indicte tht more thn 8% of ll customers would prticipte in loylty crd progrm t x 68 pˆ.451 n 1, 58 : p.5 : p.5 p ˆ p The test sttistic is z pq.51.5 n 1, 58 The rejection region requires.5 in the lower til of the z-distribution. From Tble II, Appendix D, z The rejection region is z Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte the proportion of compnies tht hd not implemented whistle-blower hotline in 15 is less thn.5 t.5. Copyright 18 Person Eduction, Inc.

28 Inferences Bsed on Single Smple: Tests of ypothesis 359 b. x 436 pˆ.64 n 681 : p.5 : p.5 The test sttistic is z The rejection region requires.5 in the upper til of the z-distribution. From Tble II, Appendix D, z The rejection region is z Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte the proportion of compnies tht hd not implemented whistle-blower hotline in 13 is greter thn.5 t x 785 pˆ.3513 n,4 To determine if the true frction of dult TV viewers who re subscription stremers differs from 1/3, we test: : p 1/3 : p 1/3 The test sttistic is z , The rejection region requires /.1/.5in ech til of the z-distribution. From Tble II, Appendix D, z The rejection region is z or z Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte the true frction of dult TV viewers who re subscription stremers differs from 1/3 t If there is no reltionship between color nd gummy ber flvor, then.5 of the popultion of students will correctly identify the color. b. To determine if color nd flvor re relted, we test: : p.5 : p.5 Copyright 18 Person Eduction, Inc.

29 36 Chpter 7 c. From the printout, the p-vlue is. p..1, is rejected. There is sufficient evidence to indicte tht color nd flvor re relted t.1. p. Since the p-vlue is less thn To determine whether the true proportion of toothpste brnds with the ADA sel verifying effective decy prevention is less thn.5, we test: : p.5 : p.5 b. From the printout, the p-vlue is p.188. c. Since the observed p-vlue is greter thn p.188.1, is not rejected. There is insufficient evidence to indicte the true proportion of toothpste brnds with the ADA sel verifying effective decy prevention is less thn.5 t x 414 pˆ.1 n 1, 971 To determine if the percentge of ll residentil properties purchsed for vction homes is greter thn 1%, we test: : p.1 : p.1 p ˆ p.1.1 The test sttistic is z pq.11.1 n 1,971 The rejection region requires.1in the upper til of the z-distribution. From Tble II, Appendix D, z The rejection region is z.33. Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte the percentge of ll residentil properties purchsed for vction homes is greter thn 1% t.1. b. The return rte is only1,98 / 45,.44. This is very low return rte. Since this is selfselected smple (only those who wnted to respond returned their questionnire), it is very likely tht the smple ws not representtive. Therefore, the results re very suspect x pˆ.585 n 845 To determine if fewer thn 6% of the coffee growers in southern Mexico re either certified or trnsitioning to become certified, we test: : p.6 : p.6 Copyright 18 Person Eduction, Inc.

30 p ˆ p p ˆ p The test sttistic is z.89. pˆ pq.61.6 n 845 Inferences Bsed on Single Smple: Tests of ypothesis 361 The rejection region requires.5 in the lower til of the z-distribution. From Tble II, Appendix D, z The rejection region is z Since the observed vlue of the test sttistic does not fll in the rejection region z , is not rejected. There is insufficient evidence to indicte tht fewer thn 6% of the coffee growers in southern Mexico re either certified or trnsitioning to become certified t x 515 pˆ.86 n 639 To determine if the percentge of Europen diry frms tht crry out clf dehorning differs from 8%, we test: : p.8 : p.8 p ˆ p.86.8 The test sttistic is z.38. pq.81.8 n 639 Since no ws given, we will use.5. The rejection region requires /.5/.5in ech til of the z-distribution. From Tble II, Appendix D, z The rejection region is z 1.96 or z Since the observed vlue of the test sttistic does not fll in the rejection region z , is not rejected. There is insufficient evidence to indicte the percentge of Europen diry frms tht crry out clf dehorning differs from 8% t.5. This supports the figure reported by SANKO Le p proportion of middle-ged women who exhibit skin improvement fter using the crem. For x 4 this problem, pˆ.77. n 33 First we check to see if the norml pproximtion is dequte: np , nq Since nq 13. is less thn 15, the ssumption of normlity my not be vlid. We will go hed nd perform the test. To determine if the crem will improve the skin of more thn 6% of middle-ged women, we test: : p.6 : p.6 Copyright 18 Person Eduction, Inc.

31 36 Chpter 7 pˆ p.77.6 The test sttistic is z 1.49 pq.6.4 n 33 The rejection region requires.5 in the upper til of the z-distribution. From Tble II, Appendix D, z The rejection region is z Since the observed vlue of the test sttistic does not fll in the rejection region z , is not rejected. There is insufficient evidence to indicte the crem will improve the skin of more thn 6% of middle-ged women t (Using Tble II, Appendix D.) Since the p-, is not rejected. There is insufficient evidence to indicte b. The p-vlue is p Pz vlue is greter thn p the crem will improve the skin of more thn 6% of middle-ged women t First, check to see if n is lrge enough. np nd, ,663. nq, Since both np 15 nd nq 15, the norml pproximtion will be dequte. x 1, 554 pˆ.654 n,376 To determine if the true detection rte for pictures of PTW is different from.7, we test: : p.7 : p.7 pˆ p pˆ p The test sttistic is z 4.89 pˆ pq.7.3 n,376 The rejection region requires /.1/.5in ech til of the z-distribution. From Tble II, Appendix D, z The rejection region is z or z Since the observed vlue of the test sttistic flls in the rejection region z , is rejected. There is sufficient evidence to indicte the true detection rte for pictures of PTW is different from.7 t Answers will vry. The trget popultion will be ll households in the \$75, income brcket in the United Sttes. The experimentl unit will be n individul household in the United Sttes in the \$75, income brcket. The vrible to be mesured is whether or not the household hs Smrt TV with internet nd video streming ccess. Let p = proportion of be ll households the \$75, income brcket in the United Sttes tht hve Smrt TVs with internet nd video streming ccess. The hypotheses of interest re: Copyright 18 Person Eduction, Inc.

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