PAPER NO.: 444, 445 PAGE NO.: Page 1 of 1 INSTRUCTIONS I. You have bee provided with: a) the examiatio paper i two parts (PART A ad PART B), b) a multiple choice aswer sheet (for PART A), c) selected formulae (at the ed of PART A), d) a booklet of tables. II. The total umber of marks possible is 50. a) PART A is worth 35 marks ad cosists of multiple choice questios. It is suggested that you first complete the questios o the examiatio paper by choosig the BEST aswer out of five i each case; the trasfer your aswers to the multiple choice aswer sheet by blackeig the appropriate space with a "HB" or "F" pecil. Oly oe space should be blackeed; otherwise, the questio will be marked wrog. The questios are of equal value. There is o pealty is made for guessig; therefore, all questios should be attempted. III. PART B is worth 15 marks ad cosists of log aswer questios that are to be aswered i the spaces provided o the examiatio paper. SHOW ALL YOUR WORK. IV. At the ed of the examiatio period, tur i (i) your multiple choice aswer sheet for PART A ad (ii) your examiatio paper for PART B. Be sure to write your NAME, STUDENT NUMBER, ad INSTRUCTOR s NAME o your MULTIPLE CHOICE ANSWER SHEET ad your EXAMINATION PAPER for PART B. C. No-graphig calculators are permitted. ----------------------------------------------------------------------------------------------------------------- 1. Customer complaits about tissues tearig led the compay to look at the stregth of the tissues. The data were collected over 3 days ad the compay would like to kow if the average stregth is the same for all three days. A aalysis of variace was coducted o a data set, which cotais 5 observatios for each day. The group sum of squares is 89.0 ad the total sum of squares is 5,97.40. The calculated value of the F statistic for testig the equality of mea stregth of tissues is: (A) 1. (B) 0.74 (C) 4.93 (D) 1.18 (E).43. Suppose we wat to perform a two-tailed of test the hypothesis H o :µ = 64 agaist the alterative µ = 60 where σ is kow to be = 10. If we select a sample of size 16 ad decide to reject H o if the sample mea x < 61 or if x > 67, the the probability of a Type II error is: (A).6554 (B).3446 (C).30 (D).340 (E).1151
PAPER NO.: 444, 445 PAGE NO.: Page of 1 Questios 3 to 6 refer to the followig. A marketig maager coducted a study to determie whether there is a liear relatioship betwee moey spet o advertisig expeses (1000s of $) ad compay sales (1000s of $). The data are preseted i the followig table. Advertisig (X).4 1.8.0.6 1.4 1.6.0. Sales (Y) 5 184 0 40 178 184 186 15 The followig iformatio is also give. ΣX = 16, ΣY = 163, ΣXY= 333.8, ΣX = 33.1, ΣY = 336,84 S x = 0.4, Sy = 3.6, r = 0.90, ad ( y yˆ) = 71. 1 3. To see whether the data provide sufficiet evidece to idicate a ozero populatio correlatio betwee advertisig expeditures ad compay sales, the value of the test statistic is: (A) 0.90 (B) 6.97 (C) 5.46 (D) 11.6 (E) 5.06 4. The least squares regressio lie to predict compay sales from the advertisig expeses is: (A) yˆ = 310. + 53. 1x (B) yˆ = 97.8 + 53. 1x (C) yˆ = 1.11+ 0. 015x (D) yˆ = 5.11+ 0. 015x (E) yˆ = 0. + 0. 9x 5. The stadard error for the regressio coefficiet b is closest to: (A) 10.96 (B) 3.63 (C) 9.59 (D) 10.36 (E) 8.34 6. The percetage of variatio i compay sales explaied by the regressio of compay sales o advertisig expeses is: (A) 90 (B) 81 (C) 8 (D) 95 (E) 0.90 Questios 7 ad 8 refer to the followig: A experimet was desiged to test the effectiveess of Vitami B1 i stimulatig mushroom growth. Eleve similar mushrooms were used. Six were radomly selected to be treated with Vitami B1 while the other five were left utreated. The weight gais (i grams) of the mushrooms after a week were as follows: Vitami B1 treated 18 1.1 13.5 14.6 4 1 Utreated 1.4 10. 13.5 9.1 11.3 7. The calculated value of the Rak- Sum test statistic for the utreated group is: (A) 15.0 (B) 10.5 (C) 17.5 (D) 48.5 (E) 51.0 8. The 5% critical regio for the test statistic is: (A) T U 6 (B) T U 0 (C) T U 40 (D) T U 18 or (E) T U 0 or T U 4 T U 40
PAPER NO.: 444, 445 PAGE NO.: Page 3 of 1 Questios 9 to 1 refer to the followig: The followig cotigecy table shows the results of a radom sample of 550 compay CEO s classified by age ad size of compay. The researcher is iterested i determiig if the CEO s ages are depedet o the compay size. NOTE: (1) Expected values for some cells are give i brackets i the secod row. () Cell χ values are give i the third row of some cells. (3) The sum of the give χ values i the table is 45.1 Age of CEO s Compay size 39 ad 40-49 50-59 60-69 70 ad Total uder over Small/Midsize 4 69 108 60 1 300 (?) 10.44 (?) 9.78 (105.3) 0.07 (98.) (3.5) 0.6 Large 5 18 85 10 50 (?) 1.53 (?) 11.74 (87.73) 0.08 (81.8) (19.5) 0.31 Total 47 87 193 180 43 550 9. Uder the appropriate ull hypothesis, the expected frequecy for the cell correspodig to age 40-49 ad large size of compay is: (A) 105.7 (B) 3.45 (C) 81.8 (D)39.5 (E) 47.5 10. The ull hypothesis will be rejected at the 0.01 level of sigificace if the test statistic exceeds: (A) 0.09 (B) 15.086 (C) 13.77 (D) 11.143 (E) 1.838 11. The calculated value of the test statistic is: (A) 65.1 (B) 81.69 (C) 45.1 (D) 77.90 (E) 67.71 1. The ull hypothesis to be tested is : (A) The CEO s ages are idepedet of the compay size. (B) The CEO s ages are depedet of the compay size. (C) The probability of selectig CEO s ages is related with the compay size. (D) The proportio of CEO s ages associatig with a particular compay size is the same for all ages of the employees. (E) The umber of employees for every cell is depedet o the compay s size. 13. I a study of various fast foods, you fid that the mea calorie cotet of 15 grilled chicke sadwiches from Arby s is 430 calories with a stadard deviatio of 6. calories. You also fid that the mea calorie cotet of 1 similar chicke sadwiches from McDoald s is 440 calories with a stadard deviatio 8.1 calories. A 95% cofidece iterval for the differece i mea calorie cotet of grilled chicke sadwiches is: (A) 15.66 µ A -µ M 4.34 (B) 16.654 µ A -µ M 5.336 (C) 16.4 µ A -µ M 3.76 (D) 14.564 µ A -µ M 4.336 (E) 15.39 µ A -µ M 4.61
PAPER NO.: 444, 445 PAGE NO.: Page 4 of 1 Questios 14 to 16 refer to the followig situatio: A public opiio poll idicates that 31 out of 400 members of the Alliace party support the proposed merger with the Progressive Coservative party compared to 7% out of a sample of 300 members of the Progressive Coservative party that support the merger. Let p A represet the populatio proportio of members of the Alliace party that support the proposed merger with the Progressive Coservative party ad p P represet the proportio of members of the Progressive Coservative party that support the proposed merger. 14. A 97% cofidece iterval for the true differece, p A p P, is: (A) (B) (C) (D) (E) 31 400 16 58 ±1.96 300 700 17 700 1 400 + 1 31 400 16 58 ±.17 300 700 17 700 1 400 + 1 31 400 16 31 ±1.96 300 400 88 400 1 400 + 16 300 84 300 1 300 31 400 16 31 ±.17 300 400 88 400 1 400 + 16 300 84 300 1 300 31 400 16 31 ±1.88 300 400 88 400 1 400 + 16 300 84 300 1 300 15. The value of the test statistic ad 8% critical regio used to test the hypothesis H 0 : p A p P = 0 versus H a : p A p P 0 is: (A) (B) (C) (D) (E) 31 400 133 31 400 16 31 400 16 31 400 133 31 400 7 31 400 88 400 1 400 + 16 300 84 300 1, z > 1.75 300 58 700 17 1 700 400 + 1, z > 1.75 58 700 17 1 700 400 + 1, z > 1.96 31 400 88 400 1 400 + 16 300 84 300 1, z > 1.96 300 384 700 316 1 700 400 + 1, z > 1.75 16. I testig the hypotheses i questio 15, which of the followig relatioships hold for the observed values of the statistics? (A) t = F (B) z = F (C) χ = F (D) t = χ (E) z = χ
PAPER NO.: 444, 445 PAGE NO.: Page 5 of 1 Questios 17 to 0 refer to the followig: It is difficult to determie body fat without immersig a idividual i water. A researcher hopig to fid a way to make a good estimate immersed 13 adult males i water, the measured their Body Fat (%) ad their Waist (cm). Waist (x): 81 97 84 101 104 89 97 95 10 91 11 86 109 Body Fat (y): 7 15 7 31 3 1 5 30 0 38 10 8 x = 96 y = (x x ) =1096 Bivariate Fit of Body Fat (%) By Waist (cm) 40 Body Fat (%) 30 0 10 0 80 90 100 110 10 Waist (cm) Liear Fit Body Fat (%) = -6.7 + 0.88 Waist (cm) Aalysis of Variace Source DF Sum of Squares Mea Square F Ratio Model Error 31 Prob > F C. Total 1194 Parameter Estimates Term Estimate Std Error t Ratio Prob> t Itercept -6.700 16.1-3.87 0.006 Waist (cm) 0.88 17. The value of the t statistic used to test the hypothesis that the slope parameter is zero, i.e. H o :β = 0 is: (A).16 (B) 7.5 (C) 5.4 (D) 31.18 (E) 3.87 18. The 1% critical regio for the t test agaist the alterative H a :β 0 is: (A) t >.0 (B) t > 3.11 (C) t > 3.05 (D) t >.7 (E) t >.68 19. The residual for the ma with a waist measuremet of 84 cm is: (A) 4.4 (B) -1. (C) -15 (D) -4.4 (E) 15
PAPER NO.: 444, 445 PAGE NO.: Page 6 of 1 0. A 95% predictio iterval for the body fat of a ma with a waist measuremet of 100 cm is closest to: (A) (15.1, 35.9) (B) (1.7, 38.3) (C) (19.7, 31.3) (D) (1.6, 9.4) (E) (-45.7, 96.7) Questios 1 ad refer to the followig A maufacturer wats to test the stregth of plastic produced by a ew process to determie if it produces the same quality as that produced by the old process, which required a force of 19.5 pouds per square ich to break the plastic. A sample was tested with the followig breakig stregths: 19.4 18. 18.9 18.9 0.8 19.5 0.1 19.5 0.8 18.6 17.3 19.5 1. The P-value of the Sig test to test the appropriate hypotheses is: (A) 0.618 (B) 0.0730 (C) 0.1460 (D) 0.539 (E) 0.5078. The value of the test statistic based o raks is: (A) 8 (B) 17 (C) 36 (D) 33 (E) < 10
PAPER NO.: 444, 445 PAGE NO.: Page 7 of 1 Selected Formulae for 005.00 1. (y i y ) = y i 1 i=1 i=1 i=1 y i = y i y i=1. s y = 3. r = 1 1 i =1 (y i y ) 1 x i x -1 s y i y x s = y x i y i 1 ( )( y i ) x i ( 1) s x s y ( )( y i ) ( ) 4. b = r s y = x y i i x i s x x i x i a = y b x 5. If X has a biomial distributio with parameters ad p, the the mea of X is p ad the variace of X is p(1-p). 6. The samplig distributio of x has a mea of µ ad a stadard deviatio of σ. 7. Z = x µ 0 σ x ± z * σ = z * σ m 8. Z = ˆ p p 0 p 0 (1 p 0 ) p ˆ ± z p ˆ (1 p ˆ ) = z m p (1- p ) 9. t = x µ 0 s x ± t * s 10. t = x 1 x (µ 1 µ ) s p 1 1 + 1 s p = ( 1 1)s 1 + ( 1)s 1 + 1 ( x 1 x ) ± t * s p + 1 1 with df = 1 + if σ 1 = σ 11. t = x 1 x (µ 1 µ ) s 1 1 + s with df = smaller of 1 1 ad 1 ( x 1 x ) ± t * s 1 + s 1
PAPER NO.: 444, 445 PAGE NO.: Page 8 of 1 p 1. Z = ˆ 1 p ˆ s p = p ˆ (1 p ˆ ) s p 1 + 1 1 p ˆ = x 1 + x 1 + ( p ˆ 1 p ˆ ) ± z *s D s D = p ˆ 1 (1 p ˆ 1 ) p + ˆ (1 p ˆ ) 1 13. Poisso Probability Distributio: P(X = k) = e λ λ k k! k = 0, 1,, 14. Test statistic for zero Correlatio Coefficiet: t = r 1 r 15. s b = s e (x i x ) = MSE (x i x ) 1 16. SE µ ˆ = s e + ( x * x ) ( x x ) ( ) ( ) 17. SE y ˆ = s e 1+ 1 + x * x x i x i 18. χ = over all cells (Observed cout Expected cout) Expected cout
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PAPER NO.: 444, 445 PAGE NO.: Page 10 of 1 Part B Part A Part B Total Istructio: (1) Show all work. () Test statistics should be to decimal places. (3) Probabilities should be to 4 decimal places. Sectio : L0 Your Istructor : Name : Studet # :
PAPER NO.: 444, 445 PAGE NO.: Page 11 of 1 1. Volume 16, No. 1 issue of Chace cotaied a article o the scorig prowess of The Great Oe, Waye Gretzky. Icluded i that article was the followig distributio of the umber of poits Waye scored i each game of the five seasos from 198 to 1986. The article begis with a examiatio of how adequate the Poisso distributio is i modelig the poits per game for Gretzky. The followig table also gives a partial aalysis of a test of the hypothesis that Gretzky s poit distributio has a Poisso distributio. NOTE:Durig those five years he scored a total of 1036 poits. NOTE: For coveiece you may roud ay itermediate results to oe decimal place. NOTE: The sum of the values of the test statistic give i the table is 3.39. Poits Number of Games Expected Frequecy Test Statistic 0 3 9.3 1.35 1 83 76.1 0.63 96 98.9 0.09 3 88 85.7 0.06 4 48 55.7 1.06 5 39 6 11 0.0 7 4 4.6 8.1 Total 394 (a) Defie the coditios a radom variable must satisfy to have a Poisso distributio. (b) Complete the test at the 5% level of sigificace. Clearly state your hypotheses ad coclusios. (c) Darryl Sittler of Toroto holds the NHL record for the most poits i a sigle game, amely 10. What is the probability of Gretzky scorig 10 or more poits i a sigle game durig the five seasos from 198 to 1986 (assumig the Poisso distributio referred to above)?
PAPER NO.: 444, 445 PAGE NO.: Page 1 of 1. A group of researchers studied hemodyamic chages i patiets with acute pulmoary thromboembolism. The mea pulmoary artery pressure of ie of these patiets before ad 4 hours after urokiase therapy is preseted i the followig table. We wish to kow whether these data provide sufficiet evidece to idicate that urokiase therapy lowers pulmoary artery pressure. Mea pulmoary artery pressure, millimetres of mercury Patiet 1 3 4 5 6 7 8 9 0 hour (X) 33 17 30 5 36 5 31 0 18 4 hours (Y) 1 17 13 33 0 19 13 9 (a) State your hypotheses, clearly defiig ay otatio that you itroduce. (b) Aalyze the data usig the best oparametric procedure. (Use α=0.05). State why you chose the test you used. (c) Determie the P-value of the test ad state clearly your coclusio based o the cotext of the problem. (d) Suppose you ca assume that the data above is ormally distributed, what is the best parametric test would you use? No calculatios are ecessary for this part.