Math 130 Fall 2011 Test 2

Math 130 Fall 201 1 November 15, 2011 Name: Test#2 Please complete the folloing problems. Be sure to ask me if you ha\e any questions or anything is unclear, Partial credit i11 be given, so please be sure to sho all of your work. Please use complete sentences to answer ALL questions. Lnless specifically asked to check or comment, ‘you ma\ assume that an\ necessar\ assumptions are reasonably vel1 satisfied. The Z table and T table are included at the end of the test. General Theory These first 7 questions relate to hypothesis testing and confidence intervals in general. 1. (5 pts) Which is true about a 98% confidence interval for a proportion based on a given sample? Circle the correct response. I. II, Ill. We are 98% confident that the sample proportion is in our interval. There is a 98% chance that our interval contains the population proportion. The interval is wider than a 95% confidence interval would be. A) I only B) I and II C) II only only E) None 2. (5 pts) We have calculated a confidence interval based on a sample of size n = 100. Now we want to get a better estimate with a margin of error that is only one-fourth as large. How large does our new sample need to be? ?1’ pAE 7 3. (5 pts) We have calculated a 95% confidence interval and would prefer that our next confidence interval has a smaller margin of error without losing any confidence. How can we achieve this? € e .L:ee rc Ly LIe £;e 4fie vErns; 4. (5 pts) Botanists observed 30 bristiecone pines and estimated their ages. A 95% confidence interval for the mean age of bristlecone pines was calculated to be (1775 4225) years. , In addition, the botanists wanted to do a hypothesis test. Let p be the mean age of bristlecone pines. The botanists want to test H 0 p = 4000 versus H 0 p 4000. They plan to use a significance level of a = 0.05. Based on the information given, what can you say about the p-value for such a hypothesis test? qao, i’f 4 /c.e# 1 +L A! C*L4. t05 _ 5. (10 pts) A diagnostic test for Alzheiiner’s disease tests the hypotheses H The patient does not have Alzheimer’s disease : 0 : The patient does have Alzheimer’s disease 4 H Suppose researchers set a a. = 0.01, and have calculated the power of the test to be 92%. Describe a Type I and Type II error in the context of this hypothesis test. Which is worse? 0LLS Z Lr9 I J A Ii. So 0 . t, • Sot: (o. b. beet Find the probability of both the Type I and Type Ii elTors. +pe P(Tpt t) ( Tp ): c. C I- P’ i Define what is meant by the “power” of a hypothesis test. poi Yke # pr.(1y 4- s Cvre cI Y vt fevi 6. (8 pts) Many people have trouble setting up all the features of their cell phones, so a company has developed what it hopes will be easier instructions. The goal is to have at least 96% of customers succeed. The company tests the new system on 200 people. of whom 188 were successful. Is this strong evidence that the new system fails to meet the company’s goal? The work was given to a summer intern. Evaluate the student’s test, pointing out any errors you find. H : 0 ô=0.96 3 *0.96 HA:? b I4 Leefr SRS. 0.96(200)> 10 zfl( 094 SD(?3)=J (09 ‘1 =0017 4 t— 0.94 ‘ ! P = P(z> 1.18) Ø() fl = 0,12 p( L4 a —i 3 -t.) There is strong evidence that the new instructions do not work. L f of _thz 7. (5 pts) For the following scenarios, state which hypothesis test procedure should be used. a. You are comparing the job placement success of LJMASS Business School graduates with those of Yale. You randomly sample 25 LJMASS graduates and 15 Yale graduates, and record the starting salary of each graduate. What test could you use to determine whether the starting salary of UMASS Business School graduates is less than that of Yale graduates? b. To report a mileage estimate to the EPA for a new sedan, a car manufacturer randomly selects I cars from their production line. They test each car under identical conditions, and record the mileage per gallon for each car. They wish to test the claim that the mean mileage for this sedan is over 30 mpg. What test would they use to do this? c. Drug companies do a lot of clinical trials while researching their products. Early in drug development, these companies conduct trials of their drug on normal, healthy individuals. In a “crossover design,” each sample person receives both the drug and a placebo. Measurements are made each time to record information such as blood pressure when using the placebo and blood pressure when using the drug. The company wishes to claim that their drug lowers blood pressure. What test could be used to test such a statement? I PI d. Are Idaho’s “famous potatoes” really better? The Idaho Potato Growers Association wants to find out. They randomly sample 50 potatoes from each state. The potatoes are baked, and a trained tester tastes each one, ranking their satisfaction from I to 10. The Association would like to claim that the mean satisfaction rating of Idaho potatoes is higher than that of Maine potatoes. Which test could be used to test this claim? posbf, Ldp’s DfI e. The state of Massachusetts is interested in comparing the proportion of households with broadband internet access for households west of the Quabbin reservoir versus households east of the reservoir. We sample 250 households from each area, and compare the percentage of households that have broadband internet access. Which test could be used to make this comparison? Scenario I — The following scenario will be used to answer questions 8 - 10 A group of scientists is interested in studying air pollution. One component of air pollution is airhorne particulate matter such as dust and smoke, To measure particulate pollution, a vacuum motor dras air through a filter for 24 hours. The filter is weighed at the beginning and at the end of the peiod. The weight gained over the 24 hour period is a measure of the concentration of particles in the air. This study made measurements in the center of a small city and at a rural location 10 miles southwest of the city. The data are shown below: Particulate Level (grams) Location Rural 67.42.33,46,43,54,38,88. 108.57,70,42.43.39,52,4X,56,44,51,2l,74. 38, 84. 51.43, 45, 41,47.35 39, 68. 42, 34. 48, 82, 45. 60. 57. 39. 123, 59, 71, 41, 42, 38. 57. 50, 58. 45, 69, 23, 72, 49, 86. 51, 42, 46. 44, 42 City The alternative hypothesis used in this analysis is the 2-sided (not equal) hypothesis. Some summary statistics are given below: Two-Sample T-Test and CI: Rural, City Two-sample T for Rural vs City N 29 30 Rural City Mean 52.1 54.1 StDev 18.2 19.4 SE Mean 3.4 3.5 /2 You may assume that the degrees of freedom are df 2\2 tl-, = ,=57. - i fs i —I—I +—l—= fll_1fllJ Th_1\fl 8. (8 pts) Is there a difference in particulate pollution between the rural and city locations? Carry out an appropriate inference procedure. I’ It 3f L4, s1,i i ,-. +. L -2. -v V 1 1/ ,- ‘- c-4? • 5 t F L — - . VC4 . pJfee > o2.° 9. (6 pts) Name the assumptions the data must in satisf order for the conclusions based your inference procedure to be valid. A i eh -L L/ + &(s & ot’s o 4 4? given below. Discuss of the graphical output. 10. (5 pts) Additional eraphical output is you listed in *‘-I evi/ ?D0? Question 9 on the basis the validity of the assumptions Remember to cite the graph number you are referring to. c s4&V (1 “ .rOt b4. 0 4tie#c we 4Lc,e A rnA prOLtiit2. Graph #2: Graph #1: Particulate Pollution Data Particulate Pollution at Two Locations 40 5 120 , o 00 100 110 Rur 10 C 100 & t H 4 20 c:ty 40 00 1 — Graph #4: Graph #3: Probability Plot of Rwal Probability Plot olCity -___ / / / / / /V 7 -‘ / -z / 7/4 40 . aurM // H a V Scenario 2 — The following scenario will be used to answer questions 11 - 14 A study on crew teams analyzed the weiehts of randomh selected rowers from the Oxford and Cambridge crew teams. From data collected over past years, 8 Oxford and 8 Cambridge rowers were randomly selected, and their weight their senior year on the team as recorded. A curious crew fan wants to know if Oxford rowers weigh more on average than Cambridge rowers. 11. (5 pts) State the hypotheses you would test (be sure to define your order of subtraction) to address the fan’s question. kt urat L ‘€ L4 o 12. (4 pts) You can assume the 16 rowers selected are a representative sample of rowers from the schools and that the weights are independent. What other assumption needs to be valid to perform your hypothesis test, and what graphs (specifically) would you make to check that assumption? (i.e. if you say bar chart bar chart of what?) - f s 4 4L f. 4,,. f1— I 0 J : fl g 13. (6 pts) We will assume the conditions hold. Complete your test procedure, using the appropriate R output below. Provide the numeric values of the test statistic and p-value for j’our hypotheses. Paired t-test (Cambridge-Oxford) t = 0.7501, df = 7, p-value = 0.2388 alternative hypothesis: true difference in means is greater than 0 Welch Two Sample t-test (Cambridge- Oxford) t = 0.4259, df = 13.775, p-value = 0.3384 alternative hypothesis: true difference in means is greater than 0 Test statistic= p-value C. 33 Interpret your p-value in terms of its definition (not in terms of whether we reject the null hypothesis). pS ‘; L.s4 ( .ç I 4. (5 ptsj What tYpe of error might you have made? Tie I eII) No Error Scenario 3 — The following scenario will be used to answer questions 15 - 17 Bullying has become a hot topic’ due to recent media attention and events. A study on secondary school bullying in England found that out of 92 short students, 42 had experienced bullying. Out of 117 not-short students, 30 had experienced bullying. The determination of short vs. not-short was made based on an unknown but likely reasonable height cutoff Researchers want to know what the difference in bullying rates is for short vs. not-short students. 15. (5 pts) If the percentages of bullying experiences for short and not-short students were equal, what would your best estimate of that percentage be? 2. Ld L4 3% ç .t’ic.( 16. (8 pts) Address the researcher’s question by creating a 98% confidence interval, showing your work, and checking conditions. - qç) b f ‘ P/ 22L 4 j. II o2’tt ± f53 ( ‘i7 o.3c) - c 17. (5 pts) Based on the Cl, is it reasonable to conclude that a higher percentage of short students have experienced bullying compared to not-short students? Explain in one sentence. At what significance level can you state that conclusion? s c ifyi, 4 4 se s& fr;Ft SLt 4 Ct L- t s ‘L’r *- 4- 1 9. + t. 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