MAT221-2004Spring - MAT 221 Signature: Instructions: 0 You...

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Unformatted text preview: MAT 221 Signature: Instructions: 0 You should have 12 pages (6 sheets), 12 problems. 0 Write the answers and show the main steps of your work on this test sheet. 0 Your final answers must include the appropriate units (6. g. dollars, dollars per F inal Exam week, miles per hour, etc.) o If you use a table, state the table used: for example, 1.887 (from table W). 0 If you use a function on the T183 or (T189), write out the command you entered as well as the result: Dday3,2004 For example, 0.0668 (normalcdf (-10, -1.5, 0, 1)). DO NOT WRITE ON THE REST OF THIS COVER SHEET! (Your instructor will use this sheet for recording your scores.) Problem 1(8) Problem 200) Problem 3(15) PARTIO” Problem 4 (4) Problem 5(3) Problem 6(9) Problem 7(10) P 2(34) Problem 8(8) Problem 9(9) Problem 1001) Problem 1100) Problem 1200) PART 3(33) 1 00) Part 1: Chapters 1 & 2 Problem 1 (8 points) Babe Ruth’s yearly home run totals for the 22 years 1914-1935 are: 0,4,3,2,1 1,29,54,59,35,41,46,25,47,60,54,46,49,46,41,34,22,6. Barry Bonds’ yearly home run totals for the 18 years 1986-2003 are: l6,25,24,19,33,25,34,46,37,33,42,40,37,34,49,73,46,39. a) (3 points) Make a back-to-back stem plot of these data. b) (3 points) Give the five-number summaries of these data. c) (2 points) List any suspected outliers of each data set using the 1.5 x IQR rule. If you think that there are none, say so. Problem 2 (10 points) In a certain population the birth weight X of infants in pounds is normally distributed with mean [1X = 7.75 and ox = 1.25. a) (3 points) Find the probability that an infant will have a birth weight of less than or equal to 6 pounds and 10 ounces ( recall there are 16 ounces in one pound). b) (2 points) Draw a labeled normal distribution curve forX and shade the area that represents the percentage of infants whose weight fall between 7 pounds and 9.75 pounds. 0) (3 points) What is the weight of the infant who is in the 90th percentile? d) (2 points) Professor Judy has invented a charm index C which she asserts is related to birth wei ghtX by the formula C=2X + 10. Find the mean p c and standard deviation 0' c of C. Problem 3 (15 points) Consider the following data on heights (in inches) of dating couples: Observation: # l 2 3 4 5 6 7 Women (x- data) 60 66 64 66 65 70 65 Men (y- data) 64 72 68 70 68 71 65 a) (5 points) Make a scatter plot of the data y-axis 75 70 65 60 55 50 55 60 65 70 75 x—axis b) (4 points) Given the following information, i = 65.14, s. = 2.97, 2 xyi = 31179, y = 68.29, sy =2.98, compute the correlation coefficient. Circle your answer below. (If you cannot compute the correlation coefficient, circle the right-most option and use that value to complete this problem.) .77 .27 0 -.27 -.77 .5 c) (4 points) Circle the equation of the linear regression line for predicting y from x: A A A A A y =35.72+5x, y =68.29, y=18.ll+.77x, y =51.35+.27x, y=85.22-.27x d) (2 points) Using this regression line, predict the height of the date of a 58 inch woman: Part 2: Chapters 3 & 4 Problem 4 (4 points) A research team is interested in the possibility that aspirin or extra vitamin A will decrease the number of heart attacks in adult males. They plan a multiyear study involving 21,996 male physicians. Each will take an aspirin tablet or a placebo each day and a vitamin A tablet or a placebo each day. a) (1 point) Who are the subjects? b) (2 points) List the treatment(s): c) (1 Point) What are the response variable(s): Problem 5 (3 points) We wish to take a sample of MAT 221 students. For each of the following, circle the correct description of the sampling technique. a) Select all of the students in section 1. Circle one: Simple random sample Multistage sample Stratified random sample None of these b) Write all student names on slips of paper and randomly select 30. Circle one: Simple random sample Multistage sample Stratified random sample None of these c) Randomly select 3 sections and randomly select 10 students from each of these sections. Circle one: Simple random sample Multistage sample Stratified random sample None of these Problem 6 ( 9 points) Let A, B, and C denote events in a sample space with P(A) = .4, P(B) = .75, and P(C) = .3 a) (3 points) Given that A and B are independent, compute 1'. PM and B) = ii. PM or B) = iii. PM I B) = b) (3 points) Given that PM and C) = .18, compute i. PM”) = ii. PM or C) = c) (3 points) Using the information above and that PM and B and C) = .1 and P(B and C) = .15 draw the Venn diagram for A, B and C and label the eight separate areas with their associated probabilities. Problem 7 (10 points) On 8- sided die is rolled. Two of its sides are numbered 1, three sides are numbered 5 and the remaining sides are numbered 8. If X is the random variable that gives the number rolled, then a) (2 points) Fill in the probability distribution for X X , l 3 5 8 Pr0b_ ' b) Compute [ show computations] i. (2 points) ,uX [ show computations] ii. (2 points) 0X c) Say we also have two additional standard 4 sided dice, each numbered from 1 to 4. Take as given that the random variable Y representing the sum of numbers rolled on these two dice has the probability distribution. Y 2 3 4 5 6 7 8 Prob 1/16 1/8 3/16 1/4 3/16 1/8 1/16 For which px = 5 and 0y =\/—:'—. Now let 2: X+ Y, i.e., the sum ofthe three numbers rolled on the three dice. Compute i. (2 points) ,1: z = [ show computations] ii. (2 points) a Z = [ show computations] Problem 8 (8 points) A hat contains 3 coins, a standard quarter (with a head and a tail) and two fake quarters each with both sides head! A coin is selected at random and is flipped with the result that head shows on top. Was this coin a fake??? a) (4 points) Give a tree diagram for this question, with labeled branches. b) (4 points) Compute P (fake I head) = probability that the selected coin was fake, given that head showed on top. Part 3: Chapters 5 & 6 Problem 9 (9 points) 75% of the members in Congress support a bill about gun control. A sample of 20 members of Congress is taken. Let X count the number who are against the bill. a) (2 points) How many members of the sample would you expect to be against this bill? ( the answer need not be a whole number!) b) (2 points) Find the standard deviation 0' X forX. c) (3 points) Find the probability that exactly 2 members in the sample are against the hill. (1) (3 points) Use the normal approximation with continuity correction to estimate the probability that at least 10 members of the sample are against the bill. Problem 10 (1 1 points) A study of the writing ability of 125 4th graders yielded a mean score 75. The population standard deviation was assumed to be 8. a) (3 points) What is the margin of error for a 95% confidence interval for this study? b) (4 points) Give a 95% confidence interval for the mean score derived from this sample of 125. c) (4 points) How many 4lh graders should be sampled in order to estimate the mean score within 2 points at a 95% confidence? Problem 11 (10 points) A manager for a plant that makes lawn gnomes has been receiving complaints his products are shorter than they claim to be. They are advertised as being 18.00 inches tall. The manager takes a simple random sample of 100 gnomes, in order to test the consumers complaints, and finds the average height to 17.78 inches. He knows that his machines cause the heights of the gnomes to have a normal distribution with a standard deviation of 0' = 1 inch. a) (2 points) State the hypotheses: H01 Ha: b) (3 points) Find the value of the z-statistic for testing the hypotheses in part (a). c) (3 points) Compute the P-value of the z-statistic you found in part (b). d) (2 points) Is this statistic significant (circle your answer) 'at the 5% level? Yes No at the 1% level? Yes No Problem 12 (3 points) Consider the following test of significance: H0: ,u = 20 Ha, p > 20 At the a = 5% level of significance this test accepts the null hypothesis if f S 22 and rejects the null hypothesis if I > 22. Find the power of this test to detect a true mean of 21 if the sample standard deviation is ( = 5-1) is 1.216. (Note 22 is the n — 22 — 20 value of )7 such that x 0’“ 0 = 1.216 = 1.645 = the 2 -value corresponding to J; the 5% level of significance.) Formulas for lVIAT 221 Chapter 1 : Looking at Data-Distributions . Mean; i: w=%211 Tl (I; —I')2+(:rg—:i')7+ +(In -J’:)7 o Variance: s2 = "—1—, 2(1, — :2)? = n_1 0 Standard deviation: 5 : n—h 2(11 — i)2 = 112 — mi?) - z-score: z = 1—;E (I = p + zo) Chapter 2 : Looking at Data-Relationships - an?) (m) or r = «1-) 5,, 71-1 5:5,, 0 Least-squares regression line : g = a + bx, where b = rat and a = 3} * bi. Chapter 4 : Probability: The Study of Randomness 0 Probability Rules — P(AC):1— PM). — If events A and B are disjoint, PM or B) : PM) + P(B). For any events A and B, PM or B) 2 PM) + P(B) — PM and B). If events A and B are independent, P(A and B) = P(A) - P(B). — For any events A and B, P(.4 and B) 2 PM) - P(BIA). — When P(A) > (rpm/1): INA—[3mg_ - Bayes’s Rule: P(BlA)P(A) PMIB) = WW 0 Probability distribution - Mean: ux = Iip1+ $2112 +- + rm '= 211m ‘ Variance 0% : (11 ~ #Xlzpi + "'+ (1k — #XlZPk = ZOE ~IIX)2P1 — Standard deviation: (7X : «2(1, — pxfip, : MQ: 112p,) — pfi. — If a and b are fixed numbers, then l—‘(a+bX) = a + b/lxi 0(a+bX) = 00x — If X and Y are random variables, then p(X+y) = px + ,uy — and if X and Y are independent, then 0(X+y) 2 V0; + 03, and 0(X-y) 2 H0; +03, Chapter 5 : From Probability to Inference o Binomial distribution: X ~ B(n,p) — Binomial coefficient: = k,(:_'k),, where 71!: n x (n — 1) x (n —- 2) x -x 3 x 2 x 1. — Binomial probability: P(X = k) (2);)“(1 —p)("'kl for k = 0,1,. .,n ~Forthecount,X, pxznp 0x=\/np(l—p) ~ For the sample proportion, 75, ,ufi = p, 015 = NIH—p) ll a Let i: be the mean of an SRS of size n from a population having mean p and standard deviation 0. Then #i=#i Cir—(WV; Chapter 6 : Introduction to Inference o A level C confidence interval for p ( 0 known, SRS from a normal popula— tion): fiz' z'li'omN(0,1) at (C = 90%, z' 21645; C = 95%, z' = 1.96 ,C = 99%, z‘ =25?6)' 0 Sample size for desired margin of error 771 ' (2‘0)? n = —— _ m 0 Test statistic for Ho : 2 pg (0 known, SRS from a normal population) v.30... .258... 50.. o. 0000.. 0.0 0...!- o... .00.m M w .0". . [If .0000. 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R 000...... 20.0.0 0000.0 0.0.0.0 0000 0000.0 0.80.0 0.00.0 00.0.0 0200 . 00 0.000 «.000 0.00.0 0.000 0.000 :000 :80 :000 0.00.0 0.8.0 0.0: Mnu 00.00 00.0.0 00000 .0000 000.0 0.0000 0000.0 00000 0.000 .0000 . 0.0 0.000 0000.0 00000 0800 .0000 0000.0 00000 00000 800.0 0000.0 2.: F. 00.000 0.0000 00:0 0020 3.000 00:0 «.0000 300.0 :03 002.0 _ 00 0000.0 00000 00000 00000 00000 00000 00000 0000.0 0000.0 080.? 3.: I... 0.0.0.0 0.0.0.0 003.0 0.9.0.0 3000 00.0.00 00.0.00 «~20 .300 RN00 0.0 . 0.0000 3000 8000 0000.0 0.0000 300.0 «000.0 3000 0000.0 0000.0 2...: Wm 0200 8:0 2:0 0.0.0.0 00000 3.000 0.000 00000 00.000 0.000 _ 00 .0000 ".0000 00000 00000 00000 «.0000 00000 0.080 00000 008.0 .00.: 7/: 00000 3.0.0.0 0000 0 “20.0 02.00 00000 0000.0 0000.0 .0000 0.0.0.00 3.. 8000 300.0 00000 “0000 0000.0 0000.0 8000 N080 0000.0 «000.0 0...: Mn. :20 00000 03.00 00100 000.00 .200 0000.0 0.0.000. 0.000 0:00 0.0 00000 00000 .0000 .0000..000.0 .0000 .0000 .0000 .0000 .0000 0...: U .. .200 0.0.0.0 .0000 00000 5.00.0 3.00.0 0.0.0.0 .0000 ~20... 0.0000 N0 . .0000 .0000 .0000. .0000 .0000 .0000 .0000 .0000 .0000 .0000 0.0... M 5.00 15.0 0000.0 00000 000.00 .530 2.00.0 0000.0 09.0.0 0000.0 .0 .0000 .0000 .0000 .0000 .0000 .800 .0000 .0000 .0000 .080 0...: m .. 009:0 0.00.0 000...... 0n~00 00.00....830 0200 0000.0 00.3.0 800.0 0.0 . .0080 . 0.0: r... . 00.0 00... 8.0 8.0 3.0 3.0 8.... 00 0 3.0 m00 0.0.0 0.0.0 0.0.0 00.0 00.0 06.0 N 0. 000.0 .2580 003$ ~ 0. 80.0 .2500 0:83 . 0:00 .9500 90000.0 00. .80.. 000:. . .000. ._ 39: Tables 'J‘_,7 .‘ . . 3 TABLE C Binomial probabilities (continued) r i P 7. 6 'r, 1. .50 k 0 .8100 .7225 .6400 . . .4900 . .3600 .3025 .2500 1 .1800 .2550 .3200 . 750 .4200 .4550 .4800 .4950 .5000 2 .0100 .0225 .0400 .0625 .0900 .1225 .1600 .2025 .2500 3 0 .7290 .6141 .5120 .4219 .3430 .2746 .2160 .1664 .1250 1 .2430 .3251 .3840 .4219 .4410 .4436 .4320 .4084 .3750 2 .0270 .0574 .0960 .1406 .1890 .2389 .2880 .3341 .3750 3 .0010 .0034 .0080 .0156 .0270 .0429 .0640 .0911 .1250 4 0 .6561 .5220 .4096 .3164 .2401 .1785 .1296 .0915 .0625 1 .2916 . .3685 .4096 .4219 .4116 .3845 .3456 .2995 .2500 2 .0486 .0975 .1536 .2109 .2646 .3105 .3456 .3675 .3750 3 .0036 .0115 .0256 .0469 .0756 .1115 .1536 .2005 .2500 4 .0001 .0005 .0016 .0039 .0081 .0150 .0256 .0410 .0625 5 0 .5905 .4437 .3277 .2373 .1681 .1160 .0778 .0503 .0313 1 .3280 .3915 .4096 .3955 .3602 .3124 .2592 .2059 .1563 2 .0729 .1382 .2048 .2637 .3087 .3364 .3456 .3369 .3125 3 .0081 .0244 .0512 .0879 .1323 .1811 .2304 1 .2757 .3125 4 .0004 .0022 .0064 .0146 .0284 .0488 .0768 .1128 .1562 5 .0001 .0003 .0010 .0024 .0053 .0102 .0185 .0312 6 0 .5314 .3771 .2621 .1780 .1176 .0754 .0467 .0277 0156 1 .3543 .3993 .3932 .3560 .3025 .2437 .1866 .1359 .0938 2 .0984 .1762 .2458 ..2966 .3241 .3280 .3110 .2780 .2344 3 .0146 .0415 .0819 .1318 .1852 .2355 .2765 .3032 .3125 4 .0012 .0055 .0154 .0330 .0595 .0951 .1382 1861 .2344 54 .0001 .0004 .0015 .0044' .0102 .0205 0369 .0609 .0937 6 0001 .0002 .0007 .0018 .0041 .0083 .0156 7 0 5 4783 .3206 .2097 .1335 0324 .0490 .0280 .0152 .0078 . 1 3720 .3960 .3670 .3115 .2471 .1848 .1306 .0872 .0547 g 2 1240 .2097 2753 .3115 .3177 2985 .2611 .2140 .1641 a 3 .0230 .0617 .1147 .1730 .2269 2679 .2903 .2918 .2734 E 1 1 .0026 .0109 0287 0577 .0972 144: 1935 .2388 .2734 5 5 , .0002 .0012 .0043 .0115 .0250 .0466 .0774 .1172 .1641 g 61 .0001 .0004 0013 .0036 .0054 0172 0320 .0547 1 7 0001 .0002 .0006 1016 0037 .0078 I 6' (I 1 430% 272% 1076’ 1001 .0570 (1319 .0165- .(1084 .0039 1 1 ' .3826 .3847 .3355 .2670 .1977 .1373 .0896 .0548 .0313 1 2 .1488 .2376 .2936 .3115 .2965 .2587 2090 .1569 .1094 3 .0331 .0839 .1468 .2076 2541 .2786 .2787 .2568 .2188 . 4 .0046 .0185 0459 .0865 .1361 .1875 .2322 .2627 .2734 1 5 .0004 .0026 .0092 .0231 .0467 .0808 .1219 .1719 .2188 . .0002 .0011 .0038 .0100 .0217 .0413 .0703 .1094 E .000! .0004 .0012 .0033 .0079 .0164 .0312 .0001 00(12 0007 .0017 .0039 m._._____.__.__.._____.....____% 1 f7 7 8 ...
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MAT221-2004Spring - MAT 221 Signature: Instructions: 0 You...

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