Final Exam Review Guide Sp 21

IE3302 Final Exam – Review Guide Exam Coverage The Final Exam is comprehensive (any materials covered in the course) but emphasis will be on the latest chapters, particularly chapters 8 & 10. Proportionately, exam coverage will be approximately: Chapters: 4 1-3 5-6 8,10 In this guide, I have only included review problems related to Chapters 8 & 10; you should also do a quick review of the review guides and test solutions from Midterms #1 & #2. Following is a summary of the topics which you should be familiar with from chapters 8 and 10 specifically: Chapter 8 (Sampling Distributions):  Standard error vs. standard deviation  X-bar & difference in means: Z, T statistics & distributions  S 2 : Chi-Square statistic & distribution  Ratio of variances: F statistic & distribution  For each distribution: o Conditions for when to use, key words to identify o How to calculate critical values or find probabilities o Meaning of  o Two-tailed vs. one tailed test o How to use tables and Excel functions Chapter 10 (Hypothesis tests):  Hypothesis Tests: o Formulating/stating o Selecting right test statistic  One sample mean  One sample variance  Two sample means – 4 cases: Known variance, unknown but equal variance, unknown unequal variance, paired observations  Two sample variance o Understanding & calculating  ,  , power, p-value, critical values o Interpreting results & drawing conclusions  Confidence Intervals – calculating, interpreting Operation of the Exam Please review the following carefully:  The final exam is scheduled for 3-5pm Monday April 26 th , 2021 . o If you have requested an extended time accommodation (& provided an ODS form to document), extended time exam will instead run 3-6pm.  The exam itself will be on Moodle during the scheduled exam time . Note the exam will automatically open at 1:30pm and automatically close at the end of the exam period.  You must be on Zoom with video turned on throughout the exam for proctoring purposes . The exam Zoom URL will be the same as the "Scheduled Class Time" zoom link on Moodle. Once you are admitted to Zoom, you will be assigned to a breakout room with a proctor.

o During the exam, your audio should be muted unless a proctor or I need to talk to you. Use Chat to communicate questions to your proctor. The proctor will either relay the question to me or ask you to return to the main session to talk to me if they are not able to answer it directly. o If you need to leave the video frame of view for any reason, notify the proctor first by chat and indicate why. Any break is expected to be short. If you will need to leave for more than a couple of minutes, you must end and submit your exam before leaving or be assigned a zero. If you have not submitted your exam and leave the video frame of view without notifying the proctor first, you will be assigned a zero for the exam. o If your video turns off during the exam, you will be asked once to turn it back on. If that does not occur quickly, you will be assigned a zero for the exam. o For anyone arriving late to Zoom, or leaving early from Zoom, the Zoom attendance logs will be matched against the Moodle exam logs. If the logs do not match (i.e., you were working on the exam while not on Zoom), you will be assigned a zero for the exam.  See rules below as to what you may use during the exam. Rules of the Exam You may use the following during the exam:  Calculator (graphing calculators ok) and/or Excel. Excel may also be used for generating graphs.  Pen / pencil and blank paper for calculations  Formula sheets: Up to four sheets (8.5’ x 11’) double-sided (8 pages total) or eight single-sided pages (same size restriction). You may print and/or hand-write.  Discrete (Binomial & Poisson) and Continuous (Standard Normal) Probability Tables from Appendix A. These may be accessed on your computer, you can but do not have to print. As you will have access to Excel, I strongly advise you use Excel functions instead to avoid the need for interpolation .  Computer, to be used ONLY for zoom proctoring, calculations (calculator app or Excel), probability tables, and entering exam solutions in Moodle. o Any other form of messaging, file sharing, web browsing/search, etc. is not allowed o Use of any materials / files / apps on your computer other than listed here is not allowed NO cell phones may be out during the exam, except to scan handwritten work for essay style questions. For all calculation problems, maintain calculations to at least 4 decimal positions through final answer. Multiple choice, matching, true/false , or Moodle numeric answer problems (the last is a type of auto-graded Moodle problem for which you provide a single numeric answer):  No or limited partial credit will be given for incorrect answers on these problems.  You do NOT show work for these problems. Work must be shown for essay type questions. For these problems, if using Excel or statistical / integration functions on your calculator, formulation must be shown to earn credit. If using Excel, include the Excel formula(s) used. To show integrations, use the Moodle equation editor or a MATLAB-like notation; e.g.,, ∫ x = 1 10 3 x 2 dx can be written as int(3x^2, x=1 to 10) 1

NOTE: Some students may have reversed methods 1 and 2 in calculations, so that delta-bar is positive. Still correct so long as correctly state H1 and do upper tail test instead of lower tail. 3

2. [20pts total] A medical equipment company supplies rechargeable batteries for use in AEDs (automatic external defibrillators). The batteries have historically provided N(200, 5) milliamps (mA) of current. 12 AEDs are selected at random from the most recently produced batteries, and the mean amperage and standard deviation are calculated to be 196 mA and 4 mA, respectively. a. [10pts] Does this sample appear to come from the historical distribution? State the appropriate hypothesis test and assuming a 0.05 significance level, calculate the critical value(s) for the test. Also state your test conclusion. H0:  = 200 H1:  1 <> 200  =0.05 n=12 Comparing sample mean against known mean with known variance. Because of small sample size, use t-distribution. t= (196 – 200) / (4/sqrt(12)) = -3.4641 Two sided test, 0.025 in each tail Critical values +/- t 0.025; 11  +/- 2.201 Since test statistic is outside critical values, reject H0. The batteries do not appear to come from the same distribution. b. [10pts] Find the 90% confidence interval on the sampled mean for the average amperage. = 196 ± t 0.05 , 11 ( 4 √ 12 ) ¿ 196 ±( 1.796 ) ( 4 √ 12 ) ¿ 196 ± 2.0738 ¿ 193.9262 < μ < 198.0738 3. [15pts] A drug manufacturer will face serious sanctions if its variability exceeds  =.001mg per pill in the key ingredient of one of its medicines. A sample of 30 pills is taken at random by an FDA inspector and S is found to be 0.0013mg. At a 5% significance level, what conclusion should the inspector draw? Full state the specifics of your hypothesis test in answering this question. H0:  = 0.001 H1:  > 0.001  =0.05 n=30 Comparing sample variability against assumed variance. Test Statistic:  2 = (30-1)*0.0013 2 / 0.001 2 = 49.01 Upper tail test, Critical value:  2 .05;29 = 42.557 Since 49.01 > 42.557, reject H0 – pills exceed maximum variability allowed in key ingredient. 4

the two months preceding putting the process in place is collected for each facility, and then again for two months after implementing the new process. Perform an appropriate hypothesis test to test whether your improvement was effective. State your hypothesis and significance level, calculate your test statistic and critical value(s), and state both your statistical and practical conclusion for the test. Facility Before After  Houston 230 231 -1 Mobile 180 140 40 Lafayette 250 194 56 New Orleans 130 67 63 Vicksburg 145 148 -3 Sample Mean: 187 156 31 Sample S: 52.153 6 61.866 8 31.265 Difference in means - Paired observation (delta column added for solution) H 0 :   =0 H 1 :   >0 (since before – after, positive means after is smaller)  =0.05 Test Statistic: t = (31 - 0)/(31.265/sqrt(5)) = 2.2171 Critical value: t .05,4 = 2.132 Since 2.2171 > 2.132, Reject H 0 . The new maintenance process is effective in reducing production downtime. 6

IE3302 Sample Final Exam For problems 1-5, you are given X~NORM(  =40,  =20). n=36,  =0.05 (significance level). 1. [5pts] An ´ X = 42.4 is drawn. For a one-sided test (H 1 :  >40), what is the p-value? a. 0.7200 b. 0.7751 c. 0.2358 d. 0.3156 e. 72.8971 f. 0.4522 2. [5pts] Pr( 27 ≤ X ≤ 40) (to 2 decimal places): a. 0.50 b. 1.00 c. 0.24 d. 0.76 3. [5pts] For a one-sided test on the mean (H 1 :  >40), the critical value for ´ X is: a. 45.4828 b. 34.5172 c. 0.95 d. 7.1029 e. 0.05 f. None of the above 4. [4pts] X 0.25 ? Choose closest from following: a. 4.04 b. 37.75 c. 0.25 d. 26.51 e. 24.30 5. [5pts] Pr( ´ X > 40 | ´ X > 39 )? a. 0.9617 b. 0.5000 c. 1.3086 d. 0.8092 e. None of the above 7  known, n>= 30, use Z = Pr( ´ X  42.4) = 1 -  ( (42.4-40)/(20/sqrt(36)) ) = 1 -  (0.72) = 1 – 0.7642 = 0 .2358 = F(40) – F(27) = 0.5 -  ( (27-40)/20=-0.65) = 0.5 – 0.2578 = 0.24 = ´ X 1 − 0.05 = 0.95 = Z 0.95 *(20/sqrt(36))+40 = 1.645*(20/sqrt(36))+40 = 45.48 Closest Z in table is -1.67 = -0.67*20+40 = 26.6 Interpolating or using norm.inv yields 26.51 = Pr( ´ X >40) / Pr( ´ X >39) = 0.5 / (1 -  ( (39-40)/(20/sqrt(36))=-0.3) = 0.5 / (1 – 0.3821) = 0.8092

10. [4pts] B(3; 9, 0.4)? a. 0.7334 b. 0.2706 c. Pr(X<3) d. 0.4128 e. None of the above 11. [5pts] You are studying a maintenance operation. You have made significant changes in how the process is performed (new tools and methods) in an effort to reduce repair time. You have taken a sample of 40 service times both before and again after the improvement and calculated ´ X and S for each. Which hypothesis test would be most appropriate? a. T test on mean of a single population b. Comparison of means, unknown but equal variances (t test) c. F-test d. Chi-Square test e. Paired observations (Paired-t) f. Comparison of means, unknown and possibly unequal variances (t test) g. Z test on mean of a single population h. Comparison of means, known variance (t test) 12. [5pts] Same problem as Q#11. Write the H 0 and H 1 hypothesis for this problem H 0 :  b =  a , or  b -  a = 0 , or d 0 = 0 H 1 :  b >  a , or  b -  a > 0 , or d 0 > 0 13. [5pts] Given degrees of freedom v=21, write the Excel formula for finding Pr( T ≤ - 0.12 ). = t.dist(-0.12,21,true) 14. [5pts] 10 th percentile for t for a sample size of n=30? t 0.9,29 = -t 0.1,29 = -1.311 (from student-t table, or t.inv(0.1,29)) 9 =Pr(X<=3; n=9,p=0.4) From binomial table or using Binom.Dist(3,9,0.4,true), = 0.4826 You've changed the process, so two populations. Variance unknown, and since how the process is performed changed, you can't assume variances equal. Comparison of means, unknown and possibly unequal variances. Also accepted if you reversed before and after, as long as sign changed direction for H1

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18. [10pts] You have two coordinate measurement machines (CMM), which are used to take quality control measurements of part dimensions. Due to measurement variability, you will get differences between the two machines, but you suspect there is an average difference between the two. You use the two machines to measure the same small dimension (in mm) on seven different parts; i.e., each part is measured by both machines (CMM1 and CMM2). Results for each of the parts is shown in the table below. Perform the appropriate hypothesis test using a significance level of 0.05. State H 0 , H 1 ,  , your test statistic, the test critical value, and conclusion (both statistically and in practical terms). Part CMM1 CMM2  1 7.30 7.01 0.29 2 6.29 6.19 0.1 3 7.19 6.82 0.37 4 6.94 6.79 0.15 5 7.72 8.17 -0.45 6 6.60 6.43 0.17 7 7.15 7.13 0.02 ´ X d : 0.0929 S d : 0.2661 Comparison of means: paired observation test. Calculate deltas, then sample mean and standard deviation of the deltas (see above) H 0 :  1 =  2 (d 0 = 0) H 1 :  1 <>  2 (d 0 <> 0)[two-sided]  =0.05 Test statistic: t = ( 0.0929 - 0) / ( 0.2661 / sqrt(7)) = 0.9237 Critical values: t 0.025,7-1=6 = 2.447 (upper critical value) t 0.975,6 = -2.447 (lower critical value Since the test statistic falls between these two critical values, fail to reject H 0 There is insufficient evidence that CMM1 and CMM2 have different measurement means. 12