HW 6 Hypothesis Testing

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Apr 3, 2024

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8 problems Note: Problems 6-8 can be solved in a separate sheet by hand, which is then scanned Problem 1 Can you conclude that the mean of this data set is greater than 11? Use the appropriate hypothesis test. Use ** Do this "manually" by calculating the test statistic and comparing to the critical value. You may then use bu 16.11383372191 12.21679044783 12.54830755537 11.77001117073 11.08819673558 6.59612465448 11.40694937948 10.49983290549 14.88865669544 11.43743987738 6.885663846119 5.838637434005 13.36174883304 16.6140443214 9.41367463704 11.33574021723 8.85360171204 15.14525451822 11.15134391717 13.58139078593 7.371202535497 6.876353718236 9.073732817152 11.62291349108 Problem 2 Can you conclude that the mean of this data set is greater than 11? Use the appropriate hypothesis test. Use ** Do this "manually" by calculating the test statistic and comparing to the critical value. You may then use bu 8.85360171204 15.14525451822 11.15134391717 13.58139078593 7.371202535497 6.876353718236 9.073732817152 11.62291349108 Problem 3 Can you conclude that there is a real difference between X and Y? Use the appropriate hypothesis test. Use a ** Do this "manually" by calculating the test statistic and comparing to the critical value. You may then use bu X 15.99364611826 15.51945828924 15.71114067382 14.77223780929 13.05462663185 15.50060909779 13.76289521519 18.03904797461 13.74588046638 16.92595761523 9.51867556951 11.85891717113 17.10458553943 14.30250059684 15.93867423987 12.20338089847 20.29761059598 9.357604378982 12.65151937363 14.82175113868 15.63177243932 11.93870945935 11.4812328082 10.50210030545 Y 16.11383372191 12.21679044783 12.54830755537 11.77001117073 11.08819673558 6.59612465448 11.40694937948 10.49983290549 14.88865669544 11.43743987738 6.885663846119 5.838637434005 13.36174883304 16.6140443214 9.41367463704 11.33574021723 8.85360171204 15.14525451822 11.15134391717 13.58139078593

7.371202535497 6.876353718236 9.073732817152 11.62291349108 Problem 4 Can you conclude that there is a real difference between X and Y? Use the appropriate hypothesis test. Use a ** Do this "manually" by calculating the test statistic and comparing to the critical value. You may then use bu X 2.757994699905 12.57883885676 11.93286612342 15.47061018231 Y 16.11383372191 12.21679044783 12.54830755537 11.77001117073 Problem 5 Repeat P1 -P4 in Mintab. Copy the Minitab output from the session window for each of the problems below. **By doing this, you have a way of checking your work. The Minitab results should match your own calculatio Problem 6

n Mean StDev H0 H1 50 84.8 0.5 85 85 Z Z value P -2.82842712475 0.0024 0.0048 Since the P value is very small, the sample mean cann Problem 7 H0(NULL) =23 𝜇 H1(ALT) . =/=23 𝜇 n DOF 10 9 Avg StDev 23.3 0.2 t-test P-Value 368.4053474096 0.001 p<0.05 Significant result Reject H0(NULL) Since P_value is small and mean averages differ from the percentage, the process should be recalibrated. Problem 8 In a process that manufactures tungsten-coated silicon wafers, the target resistance for a wafer is 85 m  . random sample of 50 wafers, the sample mean resistance was 84.8 m  , and the standard deviation was 0 represent the mean resistance of the wafers manufactured by this process. A quality engineer tests _ : 𝑯 𝟎 _ : ≠ . 𝑯 𝟏 𝝁 𝟖𝟓 a) Find the P-value. b) Do you believe it is plausible that the mean is on target, or are you convinced that the mean is not on t your reasoning. A certain manufactured product is supposed to contain 23% potassium by weight. A sample of 10 specim product had an average percentage of 23.2 with a standard deviation of 0.2. If the mean percentage is fo from 23, the manufacturing process will be recalibrated. a) State the appropriate null and alternate hypotheses. b) Compute the P-value. c) Should the process be recalibrated? Explain. A process that manufactures glass sheets is supposed to be calibrated so that the mean thickness  of the than 4 mm. The standard deviation of the sheet thicknesses is known to be well approximated by = 0.2 𝝈

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X StDev H0 H1 4.04 mm 0.2 mm ≤4 𝜇 >4 𝜇 A) n StDev a 𝜇 𝜎 100 0.2 0.05 4 0.02 Z 1.644853626951 H0* X= 4.032897072539 Reject H0 𝜇 B 4.03 -0.35514637305 0.355146373049 B) a Con.Lvl Z N sheets 0.05 0.95 1.644853626951 270.5543454095 C) P B TRUE X StDev 𝜎 0.9 0.1 0.04 0.02 a 0.281551565545 D) X TRUE X StDev n 4.02 4.04 0.2 100 Z P -1 0.841344746069 than 4 mm. The standard deviation of the sheet thicknesses is known to be well approximated by = 0.2 𝝈 Thicknesses of each sheet in a sample of sheets will be measured, and a test of the hypothesis _ : ≤ v 𝑯 𝟎 𝝁 𝟒 will be performed. Assume that, in fact, the true mean thickness is 4.04 mm. a) If 100 sheets are sampled, what is the power of a test made at the 5% level? b) How many sheets must be sampled so that a 5% level test has power 0.95? c) If 100 sheets are sampled, at what level must the test be made so that the power is 0.90? d) If 100 sheets are sampled, and the rejection region is ̅≥ . , what is the power ofthe test? 𝑿 𝟒 𝟎𝟐

d and included with the excel solution. e an alpha of 0.05. Be sure to clearly state the null and alternative hypotheses. uilt in functions or the XLminer Analysis Toolpak to check your results. 9.905112874983 10.55478919675 10.12129959497 12.29959070077 14.47058870139 12.59209546216 15.93683420792 16.71547625808 13.93784335446 14.11894486981 11.51012867527 11.76756142708 7.786938152729 7.317429767277 10.16603396904 11.10582317914 10.67265639445 12.85222422499 12.40920100579 11.16221069687 8.657948387273 12.66292432636 9.662952893734 15.05779942965 12.27691304608 13.56349149299 12.85249293475 14.86431774325 14.68896570014 16.51114987741 e an alpha of 0.05. Be sure to clearly state the null and alternative hypotheses. uilt in functions or the XLminer Analysis Toolpak to check your results. 8.657948387273 12.66292432636 9.662952893734 15.05779942965 12.27691304608 13.56349149299 12.85249293475 14.86431774325 14.68896570014 16.51114987741 an alpha of 0.05. Be sure to clearly state the null and alternative hypotheses. uilt in functions or the XLminer Analysis Toolpak to check your results. 14.67945749709 15.08320314828 14.39105636841 10.80907985862 13.8646398223 15.79938972146 10.3419483579 12.25422636148 15.12415876845 21.39558771695 15.89072992254 10.78407177439 19.5680100139 12.02397485191 14.75107523852 10.22932622245 13.99488038586 17.26279227063 14.6576821811 15.83615013024 20.48136916335 15.58844434816 12.12441189495 12.18624074398 18.77542585705 16.76237554579 12.58014608878 19.35945990397 18.34314565336 16.65182186562 9.905112874983 10.55478919675 10.12129959497 12.29959070077 14.47058870139 12.59209546216 15.93683420792 16.71547625808 13.93784335446 14.11894486981 11.51012867527 11.76756142708 7.786938152729 7.317429767277 10.16603396904 11.10582317914 10.67265639445 12.85222422499 12.40920100579 11.16221069687 8.657948387273 12.66292432636 9.662952893734 15.05779942965 12.27691304608

13.56349149299 12.85249293475 14.86431774325 14.68896570014 16.51114987741 an alpha of 0.05. Be sure to clearly state the null and alternative hypotheses. uilt in functions or the XLminer Analysis Toolpak to check your results. 10.41451090135 11.20388660744 7.103587903996 14.17643997891 9.705847207397 9.905112874983 10.55478919675 10.12129959497 12.29959070077 14.47058870139 ons.

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not be on target and must be rejected. . In a simple 0.5 m  . Let 𝝁 : = versus 𝝁 𝟖𝟓 target? Explain mens of this ound to differ e sheets is more 20 mm.

Power 0.644853626951 20 mm. versus _ : > 𝑯 𝟏 𝝁 𝟒

Problem 4 Can you conclude that there is a real difference between X and Y? Use the appropriate hypothesis test. Us ** Do this "manually" by calculating the test statistic and comparing to the critical value. You may then use X Y X Mean X STDev 2.7579947 16.1138337 10.5938425 3.8263282 12.5788389 12.2167904 11.9328661 12.5483076 15.4706102 11.7700112 Y Mean Y STDev 10.4145109 9.90511287 12.2222582 2.03346497 11.2038866 10.5547892 7.1035879 10.1212996 14.17644 12.2995907 (X-Y) t 9.70584721 14.4705887 -1.62841572 -1.12742509 v DoF H0 No significant diff in meanson trial 12.1850368 H1 There may be a sig diff on means

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se an alpha of 0.05. Be sure to clearly state the null and alternative hypotheses. built in functions or the XLminer Analysis Toolpak to check your results. t-Test: Two-Sample Assuming Unequal Variances Variable 1 Variable 2 Mean 10.5938425 12.2222582 Variance 14.6407875 4.13497978 Observations 9 9 Hypothesized 0 df 12 Round Down t Stat -1.12742509 12 P(T<=t) one-t 0.14080086 t Critical one- 1.78228756 P(T<=t) two-t 0.28160172 t Critical two-t 2.17881283 t<t* Reject H0 p>0.05

Problem 3 Can you conclude that there is a real difference between X and Y? Use the appropriate hypothesis test. Use an alp ** Do this "manually" by calculating the test statistic and comparing to the critical value. You may then use built in f X 15.99364612 15.51945829 15.711140674 14.77223781 14.679457497 15.08320315 13.05462663 15.5006091 13.762895215 18.03904797 15.799389721 10.34194836 13.74588047 16.92595762 9.5186755695 11.85891717 15.890729923 10.78407177 17.10458554 14.3025006 15.93867424 12.2033809 10.229326222 13.99488039 20.2976106 9.357604379 12.651519374 14.82175114 20.481369163 15.58844435 15.63177244 11.93870946 11.481232808 10.50210031 16.762375546 12.58014609 Y 16.11383372 12.21679045 12.548307555 11.77001117 9.905112875 10.5547892 11.08819674 6.596124654 11.406949379 10.49983291 12.592095462 15.93683421 14.8886567 11.43743988 6.8856638461 5.838637434 11.510128675 11.76756143 13.36174883 16.61404432 9.413674637 11.33574022 11.105823179 10.67265639 8.853601712 15.14525452 11.151343917 13.58139079 8.6579483873 12.66292433 7.371202535 6.876353718 9.0737328172 11.62291349 13.563491493 12.85249293 a=0.05 (X-Y) Z Xl Z hand P Xl 2.932141326 5.460160099 5.37244867 1.9999999612 0.999999961 P table P 2T H0 on trial (X-Y)>0 Right Z>H0 reject null hyp 0.0233 0.0466 H1 right Favored hypothesis z-Test: Two S X 15.99364612 Y 16.11383372 13.05462663 11.08819674 13.74588047 14.8886567 Mean 17.10458554 13.36174883 Known Varian 20.2976106 8.853601712 Observations 15.63177244 7.371202535 Hypothesized 15.51945829 12.21679045 z 15.5006091 6.596124654 P(Z<=z) one-t 16.92595762 11.43743988 z Critical one- 14.3025006 16.61404432 P(Z<=z) two-ta 9.357604379 15.14525452 z Critical two-t 11.93870946 6.876353718 15.71114067 12.54830756 13.76289522 11.40694938 9.51867557 6.885663846 15.93867424 9.413674637 12.65151937 11.15134392 11.48123281 9.073732817

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14.77223781 11.77001117 18.03904797 10.49983291 11.85891717 5.838637434 12.2033809 11.33574022 14.82175114 13.58139079 10.50210031 11.62291349 14.6794575 9.905112875 15.79938972 12.59209546 15.89072992 11.51012868 10.22932622 11.10582318 20.48136916 8.657948387 16.76237555 13.56349149 15.08320315 10.5547892 10.34194836 15.93683421 10.78407177 11.76756143 13.99488039 10.67265639 15.58844435 12.66292433 12.58014609 12.85249293 14.39105637 10.12129959 12.25422636 16.71547626 19.56801001 7.786938153 17.26279227 12.85222422 12.12441189 9.662952894 19.3594599 14.86431774 10.80907986 12.2995907 15.12415877 13.93784335 12.02397485 7.317429767 14.65768218 12.40920101 12.18624074 15.05779943 18.34314565 14.6889657 13.86463982 14.4705887 21.39558772 14.11894487 14.75107524 10.16603397 15.83615013 11.1622107 18.77542586 12.27691305 16.65182187 16.51114988

pha of 0.05. Be sure to clearly state the null and alternative hypotheses. functions or the XLminer Analysis Toolpak to check your results. 14.391056368 10.80907986 13.86463982 X Mean X STDev X,n 12.254226361 15.12415877 21.39558772 14.670904 2.9083532449 54 19.568010014 12.02397485 14.75107524 17.262792271 14.65768218 15.83615013 12.124411895 12.18624074 18.77542586 19.359459904 18.34314565 16.65182187 10.121299595 12.2995907 14.4705887 Y Mean Y STDev Y,n 16.715476258 13.93784335 14.11894487 11.73876268 2.76159470407 54 7.7869381527 7.317429767 10.16603397 12.852224225 12.40920101 11.1622107 9.6629528937 15.05779943 12.27691305 14.864317743 14.6889657 16.51114988 .<0.05 Reject H0 Sample for Means Variable 1 Variable 2 14.670904002 11.73876268 8.4585186 7.62640531 54 54 2.93214133 -7.104828E-09 0.4999999972 1.644853627 0.9999999943 1.9599639845 Z>Z* reject null hyp

Variance 8.458518597 Variance 7.62640531

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8.85360171 Column1 H0 11≤ 𝜇 7.37120254 H1 11> 𝜇 15.1452545 Mean 11.9152639 6.87635372 Standard Erro 0.6877094 11.1513439 Median 12.4699187 2 tail 9.07373282 Mode #N/A t 2.1098155778 13.5813908 Standard Dev 2.91770387 p 0.025 11.6229135 Sample Varia 8.51299589 8.65794839 Kurtosis -1.10686461 13.5634915 Skewness -0.26824846 12.6629243 Range 9.63479616 12.8524929 Minimum 6.87635372 T 1.3308875149 9.66295289 Maximum 16.5111499 P 0.1 14.8643177 Sum 214.474749 15.0577994 Count 18 14.6889657 12.276913 16.5111499

on trial 0 t-Test: Two-Sample Assuming Equal Variances right side 0 Variable 1 Mean 11.9152639 1 tail Variance 8.51299589 -1.73960673 excel Observations 18 table Pooled Varia 8.04005167 Hypothesized 11 df 18 t Stat 0.43306506 P(T<=t) one-t 0.33505616 p>0.05 hand t Critical one- 1.73406361 T<T* Do not rejet H hand p>0.05 P(T<=t) two-t 0.67011231 t Critical two-t 2.10092204

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16.1138337 Column1 11.0881967 H0 11 𝜇⦤ 14.8886567 Mean 11.7387627 H1 _xDF07_ >1 𝜇 13.3617488 Standard Erro0.37580544 8.85360171 Median 11.6952375 7.37120254 Mode #N/A Z Score 1.96581156 12.2167904 Standard Dev 2.7615947 P-table 0.25 6.59612465 Sample Varia 7.62640531 P-excel 0.02466019 11.4374399 Kurtosis -0.50782225 16.6140443 Skewness -0.16770761 P<0.05 Reject H0 15.1452545 Range 10.8768388 6.87635372 Minimum 5.83863743 12.5483076 Maximum 16.7154763 11.4069494 Sum 633.893184 6.88566385 Count 54 9.41367464 11.1513439 9.07373282 11.7700112 10.4998329 5.83863743 11.3357402 13.5813908 11.6229135 9.90511287 12.5920955 11.5101287 11.1058232 8.65794839 13.5634915 10.5547892 15.9368342 11.7675614 10.6726564 12.6629243 12.8524929 10.1212996 16.7154763 7.78693815 12.8522242 9.66295289 14.8643177 12.2995907 13.9378434 7.31742977 12.409201 15.0577994 14.6889657 14.4705887 14.1189449 10.166034 11.1622107

12.276913 16.5111499

0 z-Test: Two Sample for Means on trial 0 right side Variable 1 Mean 11.7387627 Known Varia 7.62640531 Observations 54 Hypothesized 11 z 1.96577676 P(Z<=z) one- 0.0246622 0.05> Reject H0 z Critical one- 1.64485363 P(Z<=z) two-t 0.0493244 z Critical two- 1.95996398

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HW 6 Hypothesis Testing

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