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22S:8 MIDT-A 2 April 2007 print NAME___________________ 1) Find the SD of the following data set: 12 18 18 24 28 a) 4.52 b) 5.68 c) 6.16 d) 7.08 e) 8.60 82 2 2 2 2 4 2 82 mean = 20. 6.16 4 (.05) (.03) 2) Find the height of the missing rectangle in the histogram → ? 0 10 25 30 a) .02 b) .03 c) .05 d) .06 e) .08 other rectangles have areas (10)(.03) = .30 & (5)(.05) = .25 for a total = .55 Thus, missing area = 1 - .55 = .45 so that the missing height is .45/15 = .03 SD 3) The coefficient of variation of a data set is defined as CV mean Suppose the mean and SD of a data set are 50 & 12 respectively. The data is transformed as follows: new value = 5(old value) + 6. Find the new CV. a) .23 b) .28 c) .35 d) .40 e) .44 new mean = 256 & new SD = 60 so that new CV = 60/256 = .234 4) Find the correlation between the x & y variables x 3 4 7 7 7 8 SD = 2 mean = 6 y 6 4 12 12 14 12 SD = 4 mean = 10 a) .75 b) .80 c) .84 d) .90 e) .95 (1.5)(1) (1)(1.5) (.5)(.5) (.5)(.5) (.5)(1) (1)(.5) r .90 5 5) Which statement is true regarding data sets A & B ? a) median(A) > median(B) A b) maximum(A) < 3rd quartile(B) c) IQR(A) > IQR(B) d) 1st quartile(A) > median(B) B e) none of the above are true __________________________ 0 10 20 30 40 6) At right is the stemplot indicating 1 6 6 8 9 the daily profits (in thousands of 2 0 0 1 2 2 3 6 6 6 dollars) at XYZ Corporation 3 0 0 2 4 7 over a 22 day period in March. 4 1 5 8 8 Find the IQR of this data. a) 12 b) 14 c) 15 d) 16 e) 18 quartiles are colored purple in the stemplot. 34 – 20 = 14 7) Which one of the following statements is true about the correlation coefficient, r ? a) r is affected by the units of measurement (inches, feet, meters, etc.) naaaa b) r is not affected by the presence of outliers naaaa c) r is always between 0 and 1 naaaa d) a high ( + or - ) r is proof of a cause & effect relationship between x & y naaaa e) none of the above statements are true 8) The daily numbers of visitors to the Tate Modern Gallery is approximately normally distributed with mean = 6320 & SD = 288. What proportion of days does the Tate receive more than 6500 visitors ? a) .27 b) .34 c) .40 d) .46 e) .52 (6500 – 6320) / 288 = .62 table look-up 9) In (8), on the busiest 10% of days, the Tate receives about __?__ or more visitors. a) 6420 b) 6496 c) 6564 d) 6688 e) 6824 1.28 has 90% area to the left (& thus 10% to the right) under the std normal curve. So, 6320 + 1.28(288) = 6688 10) The lifetimes of ACME batteries (when run 24-7) are normally distributed with mean = 62 hours. Suppose 20% of these batteries last more than 70 hours. What is the SD of these lifetimes? a) 7.78 b) 8.26 c) 9.52 d) 10.28 e) 12.50 .84 has 80% area to its left (& thus 20% to its right) under the std normal curve. 70 62 Solve the equation .84 to get x = 9.52 x 11) Suppose that 5% of the suits filed with Superior Court are declared frivolous. Find the probability that at least one of the next five suits filed are frivolous. a) .118 b) .226 c) .284 d) .325 e) .388 1 – (.95)5 = .266 12) In problem (11), find the probability that 2 of the next 10 suits filed are frivolous. a) .075 b) .125 c) .156 d) .180 e) .196 This one’s a binomial: (10C2)(.05)2(.95)8 = .075 13) In problem (11) find the probability that the first frivolous suit filed this month is the 10th suit that is filed. a) .01 b) .03 c) .05 d) .08 e) .10 P(1st nine are not frivolous & the 10th is frivolous) = (.95)9(.05) = .031 14) Two cards are chosen without replacement from the box A A B B B C Find the probability that the cards are DIFFERENT. (hint: think “outside the box” i.e., think “complement”) a) .56 b) .64 c) .68 d) .70 e) .73 1 – P(same) = 1 – P(both AA) – P(both B) = 1 – (2/6)(1/5) – (3/6)(2/5) = .73 15) Suppose 72% of Iowa adults drink tea or coffee, 48% drink coffee, and 28% drink tea and coffee. Find the probability that a randomly chosen Iowa adult drinks tea. a) .14 b) .30 c) .40 d) .52 e) .64 P(coffee or tea) = P(coffee) + P(tea) – P(coffee and tea) .72 = .48 + P(tea) - .28 So, P(tea) = .52 16) In problem (15), find the probability that an Iowa adult drinks coffee, given that s/he drinks tea. a) .538 b) .625 c) .688 d0 .725 e) .745 Draw a diagram with 100 Iowans, 28 of them drinking coffee and tea, etc. From the diagram we have P(coffee | tea) = 28/52 = .538 17) Three independent observations are made from a standard normal distribution. Find the probability that ALL THREE are less than 1.5 a) .64 b) .70 c) .76 d) .81 e) .87 area to the left of 1.5 under the std normal curve = .9332 .93323 = .81 18) Suppose P(L > 5) = .75, and P(L > 6) = .60 Find P(L > 6 | L > 5). a) .50 b) .70 c) .80 d) .90 e) .95 P(both) / P(given) = P(L > 6 and L > 5) / P(L > 5) = P(L > 6 ) / P(L > 5) = .80 19) A gambler is equally likely to select a card from box A or B: A 1 2 3 4 5 B 1 2 3 4 5 6 Use Bayes’ Theorem to find the probability that the gambler selected from box A, given that s/he chose a number less than 4. a) .545 b) .612 c) .688 d) .741 e) .768 P ( 4 | A) P ( A) (3 / 5)(1 / 2) P ( A | 4) .545 P ( 4 | A) P ( A) P ( 4 | B) P ( B ) (3 / 5)(1 / 2) (3 / 6)(1 / 2) 20) Use the following information to calculate the regression equation that predicts the monthly number of TV sets sold at Crazy Eddie’s Electronics Emporium from the monthly number of DVD players sold there. Avr. number of TV’s sold = 288 SD = 28 Avr. number of DVD’s sold = 200 SD = 25 r = .80 Now use your equation to predict the number of TV’s sold during a month when 240 DVD players were sold. a) 312 b) 320 c) 324 d) 330 e) 336 28(.8) slope .896 y .896x b 288 .896(200) b 25 b = 108.8 y .896x 108.8 predicted value: y .896(240) 108.8 323.84 21) In (20), find the residual associated with June 2007 when 380 TV’s and 290 DVD players were sold. a) 6.8 b) 8.5 c) 11.4 d) 15.8 e) 18.2 y .896(290) 108.8 368.64 380 – 368.64 = 11.36 22) In problem (21), Regression SS = 840 and Residual SS = 290. Find R2. a) 67% b) 70% c) 74% d) 80% e) 86% 840 / (840 + 290) = 74.3% 23) A computer fits the equation y = 32 + 2.4x1 + 3.5x2 to salary data at XYZ corporation. y = salary, x1 = years of service and x2 = 1 for male employees or 0 for female employees. Last Tuesday I caught up to Molly Stevenson. She told me she’s been with XYZ for 10 years and earns a salary of 60. Find her residual. a) - 2 b) -1 c) 0 d) +2 e) +4 y = 32 + 2.4(10) + 3.5(0) = 56 residual = 60 – 56 = 4 24) Four couples got married over the weekend: husband’s age 22 23 25 26 wife’s age 18 23 23 28 predicted husband's age ← fill in this line first The regression equation is husband’s age = .4 wife’s age + 14.8 Find the Regression SS. a) 4 b) 5 c) 6 d) 8 e) 10 25) A regression model relates salary to years of service, age, gender (male or female), education level (HS grad, college grad, graduate degree), and job performance (excellent, good, fair & poor). Including the “loose constant” term, how many terms are in this model ? (hint: this problem’s a no-brainer) a) 157 b) π c) 7.13 d) 9 e) Detroit, Michigan bonus ☺: In problem (11), find the probability that the 3rd fraudulent claim filed this month is the 10th claim filed. Put your answer above your name on the front page. Find the p-value of the appropriate test . Also find 90% & 95% CI's for the parameter. 1. Researchers are interested in the effectiveness of a new allergy drug. In particular, they are interested in knowing whether most allergy patients would benefit from taking it. 432 of 800 randomly chosen allergy patients benefited by taking the new drug. hyp: 50% of all allergy patients would benefit from taking this drug vs. alt: more than 50% would benefit p 432 p ˆ .54 800 .54 .50 T 2.26 (.50)(.50) 800 p value .012 (.54)(.46) 95% conf int .54 1.96 800 replace 1.96 by 1.65 to get 90% confidence ________________________________________________________________________ 2. Average life of 400 ACME car batteries is 2004 days with SD = 35.5 days. hyp: average life of ALL bateries = 2000 days vs. alt: average life of ALL bateries > 2000 days µ (large sample) 2004 2000 T 2.25 35.5 / 400 p value .012 35.5 95% 2004 1.96 400 _______________________________________________________________________ 3. 54 of 225 randomly chosen boron absorbing rods have cracks. hyp: percentage of ALL rods with cracks = 20% vs. alt: percentage of ALL rods with cracks > 20% p 54 p ˆ .24 225 .24 .20 T 1.50 (.20)(.80) 225 p value .0668 (.24)(.76) 95% conf int .24 1.96 225 ________________________________________________________________________ 4. Average caloric content of 6 fast food burgers at a local chain = 768, SD = 171.4 hyp: average caloric content of ALL burgers = 900 vs. alt: average caloric content of ALL burgers < 900 µ (small sample) This is a small sample problem (use t distribution) 768 .900 T 1.89 p-value = area to the LEFT of -1.89 under the t5 curve 171 .4 / 6 ALWAYS CHECK THAT THE DATA IS IN THE DIRECTION OF THE ALTERNATIVE, & THEN TAKE THE SMALLER AREA (in this case the area to the left - - not to the right - - of -1.89) FOR YOUR p-value 768 < 900 IS IN THE DIRECTION OF THE ALTERNATIVE. AREA LEFT OF -1.89 = AREA RIGHT OF 1.89 .05 < p-val < .10 ___________________________________________________________________ 5. 15 of 60 randomly chosen Iowans are worried about the economy. 24 of 80 randomly chosen Californians are worried about the economy. hyp: same percentage of worry in each state vs. alt: higher percentage in CA p1 – p2 39 p1 .25 ˆ p 2 .30 ˆ pooled .278 140 .30 .25 T (.278)(.722) (.278)(.722) 60 80 ________________________________________________________________________ 6. 50 Friday customers at Backwater Mike's: avr. tip = $2.40 SD = .60 80 Saturday customers at Backwater Mike's: avr. tip = $2.82 SD = .80 hyp: Friday & Saturday tips the same., on average vs. hyp: average tip higher on Saturday µ1 - µ2 (large sample) 2.82 2.40 T 3.41 p-val = .0003 .60 2 .80 2 50 80 ________________________________________________________________________ 8. 108 of 144 randomly chosen cars in Iowa are found to be in violation of U.S. emission standards. Let p = the proportion of cars in Iowa in violation. DOT wishes to test Ho: p = .8 vs. H1: p < .8 p 108 p ˆ .75 144 .54 .50 T 1.20 (.80)(.20) 144 p value .1151 (.75)(.26) 95% conf int .75 1.96 144 ________________________________________________________________________ 9. In pblm (8), let µ denote the average odometer reading of observed cars. DOT also wishes to test Ho: = 45,000 vs H1 > 45,000 Avr reading of the cars is 47,186 miles, SD = 12,024 miles µ (large sample) 47,186 45,000 T 2.18 12,024 / 144 p value .0146 12,024 95% 47,186 1.96 144 ________________________________________________________________________ 10. The 5 readings (in meters) given by a rangefinder in repeat measurements of the distance between an observer and a bridge are independent n(, 4) random variables. Suppose average reading is 1802.6 m. and SD = 1.88 (note: this is a different SD than the one I used in class). We wish to test Ho: µ = 1800 vs. H1: µ > 1800 where µ denotes the true distance. µ (small sample) 1802 .6 1800 T 3.09 From t4 curve .01 < p-val < .02 1.88 / 5 ________________________________________________________________________ 11. Continuing (8), suppose 82 of 100 randomly chosen Minnesota cars are in violation of the U.S. standard. DOT wishes to test Ho: The proportions of Iowa and Minn. are the same vs. H1: the proportion in Minn. is higher. p1 – p2 p1 .82 ˆ p 2 .75 ˆ pooled .779 .30 .25 T .925 p-val = .177 (.779)(.221) (.779)(.221) 144 100 ________________________________________________________________________ 12. The mean breaking strength of 120 ACME steel rods is 1026 lbs. with SD = 28 lbs. The mean breaking strength of 200 GlaxCo steel rods is 1018 lbs. with SD = 16 lbs. Are ACME rods better, on average, that GlaxCo? µ1 - µ2 (large sample) 1026 1018 T p-val = area to the right of this number under the std. normal curve 28 2 16 2 120 200 ________________________________________________________________________ 13. The number of particles of a pollutant in a sample of one cubic meter of air collected near Schaeffer Hall was measured. Results : 126 154 163 133 118 125 158 143 The observations are assumed to be from a normal distribution whose mean is the actual number of particles present in the air sample. We wish to test Ho: μ = 150 vs. H1: μ < 150 µ (small sample) In this problem, YOU have to figure out the sample mean & SD. On the Final, I will always tell you these numbers, or (at least) tekll you the SD. 140 150 T 1.67 p-val = area to the RIGHT of +1.57 under the t7 curve 16 .9 / 8 .05 < p-val < .10 ________________________________________________________________________ 16. Is "heads" more likely when a coin is spun on a table than when it is flipped in the air ? 100 spins on the table, 62 heads 100 flips in the air, 54 heads p1 – p2 116 p1 .62 ˆ p 2 .54 ˆ pooled .58 200 .62 .54 p-val = .1250 T 1.15 (.58)(.42) (.58)(.42) 100 100 note: We could have subtracted .54 - .62 in the numerator of T, & then too the area to the left of -1.15 but I prefer to keep things on the positive side of zero (less to worry about) ________________________________________________________________________ 17. Are ACME steel rods stronger (on average) than Ming rods ? 100 ACME rods: avr. breaking strength = 1006 lbs. SD = 28 lbs. 80 Ming rods: avr. breaking strength = 1002 lbs. SD = 18 lbs. µ1 - µ2 (large sample) 1021 1006 T 4.35 p-val = 0 (appx.) 28 2 18 2 100 80 ________________________________________________________________________ 18. Is drug A the same as drug B ? or is A better ? 400 patients take drug A, 224 improve 250 take drug B, 118 improve. p1 – p2 224 118 p1 .56.0 ˆ p 2 .472 ˆ pooled .526 650 .560 .472 p-val = .014 T 2.19 (.526)(.474) (.526)(.474) 400 250 19. Is gas the same price in Iowa (on average) as it is in Minnesota ? Or is it more expensive in Minnesota ? A random sample of 40 gas prices in Iowa & 50 gas prices in Minnesota resulted in the observations: Iowa avr. = 1.04, SD = .02 Minn. avr. = 1.06, SD = .025 µ1 - µ2 (large sample) 1.06 1.04 T 4.21 p-val = 0 (appx) .02 2 .025 2 40 50 ________________________________________________________________________ 20. The scores on an aptitude test are assumed to be normally distributed. Eight prospective employees are randomly chosen from a large group of applicants. The sample mean = 76 and the sample SD = 4.2 . Ho: μ = 80 vs. Ha: μ < 80. µ (small sample) 76 80 T 2.69 from t7 curve .01 < p-val < .02 4.2 / 8 ________________________________________________________________________ 21. Two types of tire treads are compared. 6 cars are each equipped with a "type A" & a "type B" tire (randomly placed in the front). The differences in wear between the tires is noted for each car. We wish to test Ho: A same as B vs. H1: A better than B. CAR 1 2 3 4 5 6 A 48 55 49 51 56 50 B 45 53 50 48 52 46 µ (small sample) diff YOU WORK WITH THE DIFFERENCES 3 2 -1 3 4 4 sample mean = 2.5 sample SD = 1.87 2.5 0 T 3.27 from t5 curve .01 < p-val < .02 1.87 / 6 __________________________________________________________________ 22. A large company believes that more than 20% of its employees arrive late for work. On a Monday morning in November, 256 randomly chosen employees are observed. Of these, 60 are late. p p .23475 ˆ .234 .20 T 1.36 (.20)(.80) 256 p value .0869 (.234(.766) 95% conf int .234 1.96 256 _______________________________________________________________________ 23. A cereal company claims that 40% of its boxes contain a prize. A consumer group doubts this claim. 625 randomly chosen boxes are opened. 228 contain a prize. p p .365 ˆ .365 .40 T 1.79 (.40)(.60) 625 p value .0367 (.365)(.635) 95% conf int .365 1.96 625 Note in this example & similar examples involving p that the hypothesized value is used in the square root (denominator) of T, whereas the observed sample proportion is used in the square root of the confidence interval. ________________________________________________________________________ 24. The consumer group in (23) measured the weights of the sampled boxes of cereal. They found avr. weight = 15.88 ounces, SD = .36 . Ho: µ = 16 oz vs. H1: µ < 16 oz µ (large sample) ________________________________________________________________________ 25. A quick check of the records shows that the 256 employees of problem (22) averaged 12.4 days absent in 1996, with SD = 2.8 . Ho: µ = 12 vs. H1: µ > 12. µ (large sample) 12.4 12 T 2.28 2.8 / 256 p value .011 2.8 95% 12.4 1.96 256 ________________________________________________________________________ 26. Five 1999 BMW's are crashed into a brick wall at 10 mph. Average damage = $2,162 SD = $268. Ho: = $2000 vs. H1: > $2000. µ (small sample) 2162 2000 T 1.35 268 / 5 .10 p value .15 from t 4 table ___________________________________________________________________ 26. Five 1999 BMW's are crashed into a brick wall at 10 mph. Average damage = $2,162 SD = $268. Ho: = $2000 vs. H1: > $2000. µ (small sample) 2162 2000 T 1.35 268 / 5 .10 p value .15 from t 4 table ___________________________________________________________________ 27. 52 of 100 randomly chosen college graduates support O'Brien for Congress. 92 0f 150 randomly chosen non-college graduates support O'Brien for Congress. Does a higher percentage of non-college grads support O'Brien ? p1 – p2 pooled extimate of p = (52+92) / 100 + 150) = .576 (. 52 .613 ) 0 Area to the right of T under the std normal curve (. 576 )(. 424 ) (. 576 )(. 424 ) 100 150 _____________________________________________________________________ 28. 50 Crinkly french fries: Avr calories = 12.4 SD = 2.6 80 Home Style french fries: Avr. calories = 13.2 SD = 1.8 µ1 - µ2 (large sample) Does the average Home Style FF have more calories than the average Crinkly ? 13.2 12.4 Area to the right of T under the std normal curve 2 2 2.6 1.8 50 80 29. 8 patients take a drug. Their before & After blood pressures are measured. before: 68 62 78 56 90 88 71 70 after 72 60 86 62 88 90 78 78 What can you conclude ? Does the drug increase BP ? µ (small sample) Work with the differences 4 -2 8 6 -2 2 7 8 mean = 3.875 SD = 4.155 3.875 0 T 2.638 4.155 / 8 .01 p value .02 from t 7 curve ________________________________________________________________________ 30. A simple random sample of 4 "one-pound" cans of ACME coffee were weighed. The measurements (in ounces) were 15.8 15.7 15.3 16.1 Does this data indicate that the true average weight of all "one-pound" cans of ACME coffee is less than 16 ounces? µ (small sample) ________________________________________________________________________ 31. Eight deluxe burgers purchased at Sky Burger were measured for their caloric content. The following results were obtained: average caloric content = 1032, SD = 171.4 Is this significant evidence that the average caloric content of Sky Burger deluxe burgers exceeds 950? (You may assume that caloric content of Sky Burger deluxe burgers is normally distributed.) µ (small sample) 1032 950 T 1.35 171.4 / 8 .10 p value .15 from t 7 curve

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