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Unformatted text preview: MAT 221 Signature: Instructions: IﬁnalExani ”” "' ' ”naay9,2005 You should have 12 pages (6 sheets), 12 problems.
Write the answers and show the main steps of your work on this test sheet. Your final answers must include the appropriate units (e. g. dollars, dollars per
week, miles per hour, etc. ) If you use a table, state the table used: for example, 1887 (from table 7?).
If you use a function on the T183 OR (T189), write out the command you entered as well as the result: For example, 0.0668 (normalcdf(10,l.5, O, 1)). DO NOT WRITE ON THE REST OF THIS COVER SHEET!
(Your instructor will use this sheet forrecording your scores.) Problem IE) 1 Problem 4 L5) 1 Problem 9 11.0)
, Problem 2 (9) ﬂLProblem 5 Q) ;Problem 10 (7)
Problem 3 @ _Problem 6 (5) 1 Problem 11 (10y
__ Problem 7 (5L ' :Problem 12 (9
Problem 8 Q) 1
*rARrigﬂ PARTQQQ ;PART3an
L T TOTALUw PART 1: Chapters 1 & 2
Problem 1 (15 points) National League baseball uses the designated hitter rule while the American League baseball does not. Does the NL on average produce more runs? Here
are the average number of runs data for each league in the 2001 season. AL: 11.1 10.8 10.8 10.3 10.3 10.1 10.0 9.5 9.4 97.3 9.2 9.2 9.0 8.3
NL: 14.0 11.6 10.4 10.3 10.2 9.5 9.5 9.5 9.5 91. 8.8 8.4 8.3 8.2 8.1 7.9 a) (4 points) Make a backtoback stemplot of the two data sets. b) (5 points) Compute the fivenumber summaries of each data set. c) (2 points) Determine any suspected outliers using the 1.5 IQR criterion. If you think
there are none, state none as your answer. d) (4 points) Which of the following statements are valid and which are not valid?
Circle all valid statements. NL data is approximately normal. Mean ofNL data is smaller than its median. The 90th percentile of the NL data is at greater than or equal'to 1 1.6. The use ofthe designated hitter in the NL does NOT produce more runs on average. Problem 2 (9 points) The following data comes from a sample of eight college women,
each of whom was asked for her height (to the nearest inch) and her weight (to the nearest pound). 1 2 3 4 5 6 7 8
Weight (x) 130 95 135 140 120 110 125 105
Height (y! 67 61 65 69 67 62 65 65 The following computations have been made for you: 2x = 960, 2.112 =116,900 s, = 15.5839,
2y = 521, .2112": 33979, .3 = 2.6424, r = .7979 a). (3 points) Compute the least squares regression line for this data set (y as a function
of x). b). (3 points) Find the predicted height ofa woman weighing 1 15 pounds. c). (3 points) The variance ofthey data is 6.982. What fraction of this is explained by the
least squares regression on the other variable? Problem 3 (10 points) Suppose that the readings ofa certain blood test are nomlally
distributed with mean 17 and standard deviation 3. a) (5 points) Sketch the normal curve ofthis distribution marking off on the horizontal
axis the points 8, 1 1, l4, l7. . .etc and give the test reading with the property that
approximately 2.5% of the test readings are less than it. b) (2 points) Compute the zscore ofa test result of 18.8. c) (3 points) Compute the zscore that a test result would have so that only 1% of the
test results would have larger zscore than this zscore. PART 2: Chapters 3 & 4 Problem 4 (5 points) Is diet or exercise effective in combating insomnia? Maybe cutting
out desserts can help alleviate the problem; maybe exercise can work also. Forty
volunteers suffering from insomnia agreed to participate in a monthlong regiment. Half
were randomly assigned to a no~desserts diet; the others continued desserts as usual.
Half ofthe people in each ofthese groups were randomly assigned to an exercise
program, while the others did not exercise. The most improvement was achieved by
those who ate no desserts and engaged in exercise. a) (1 points) Was this an experiment or an observational study? Give reason. b) (2 points) Identify the factors and the levels of each. c) (2 points) Identify the treatments and the response variable. Problem 5 (9 points) Suppose E and F are events in a sample space, with P(E) = 1/4,
P(FC) = 3/8, and P(EC and F) = 1/2. Determine the following: a) (2 points) P(F) b) (2 points) P(E or F) c) (3 points) P(F[E°) d) (2 points) Are E and F independent? Give reasoning. Problem 6 (5 points) An urn contains three balls numbered 1, 2 and 3. Balls are drawn
one at a time without replacement until the sum of the numbers drawn is four or more.
Find the probability of stopping after exactly two balls are drawn. (A tree diagram might
help.) Problem 7 (5 points) At a self—service gas station, 40% ofthe customers pump regular
gas, 35% pump midgrade, and 25% pump premium gas. Ofthose who pump regular,
30% pay at least $20. Ofthose who pump midgrade, 50% pay at least $20. And ofthose
who pump premium, 60% pay at least $20. What is the probability that the next customer pays at least $20? Problem 8 (9 points) Kim has a strong serve (in tennis!); whenever it is good (i.e., lands
in) she wins the point 75% ofthe time. Whenever her second serve is good, she wins the
point 50% of the time. Sixty percent of her first serves and 75% of her second serves are good. a) (4 points) What is the probability that Kim wins the point when she serves? b) (5 points) If Kim wins a service point, what is the probability that her first serve was
good? Part 3: Chapters 5 & 6 Problem 9 (10 points) Assume that 13% of people are lefthanded. If we select 5 people
in sequence at random, ﬁnd the probability of each outcome described below. a) (2 points) The ﬁrst person is a lefty. b) (3 points) There are some lefties among the 5 people. c) (2 points) There are exactly 3 lefties in the group. d) (3 points) There are at least 3 lefties in the group. Problem 10 (7 points) a) (3 points) Only about 20% of people who try to quit smoking succeed. Sellers ofa
motivational tape claim that listening to the recorded messages can help people quit.
Write the null and alternative hypotheses to test this situation. H02 Ha: b) (4 points) The seller ofa loaded die claims that it will favor the outcome 6. We don’t
believe that claim, and roll the die 200 times to test an appropriate hypothesis. Our P—
value turns out to be 0.03. Circle the appropriate conclusion: i) There is a 3% chance that the die is fair. ii) There is a 97% chance that the die is fair. iii) There is 3% chance that a loaded die could randomly produce results as extreme as
our observation. Problem 1 l (10 points) A study of the nutrition of dialysis patients measured the level of
phosphorus in the blood of several patients on six occasions. The data for one patient in
mg/dl units are: 5.4 5.2 4.5 4.9 5.7 6.3. Consider these measurements an SRS ofthe
patient’s blood phosphorus level. Assume that this level varies normally with a = 0.9 mg/dl. a) (5' points) Give a 95% conﬁdence interval for the mean blood phosphorus level. b) (5 points) The normal range of phosphorus in the blood is considered to be 2.6 to 4.8
mg/dl. Is there strong evidence that this patient has a mean phosphorus level that exceeds
4.8? In your answer give the zstatistic and the Pvalue. Problem 12 (6 points) A computer has a random number generator designed to produce
random numbers that are uniformly distributed on the interval from 0 to 1. If this is true, the numbers generated come from a population with y = 0.5 and a = 0.2887. A command to generate 100 random numbers gives outcomes with mean f = .4365 . We want to test
H0: ,u = 0.5
Ha: ,u i 0.5 a) (2 points) Calculate the value ofthe 2 test statistic. b) (2 points) Is the result signiﬁcant at the 5% level? c) (2 points) Is the result signiﬁcant at the 1% level? Formulas for MAT 221 Chapter 1 : Looking at DataDistributions 1. Mean: :13 2 I 1:1“ 271?in
_— 2 _— 2 __ 2
2. Variance: 82 = zi—IZM —g‘3)2 : 11 3) “12 I) + +(In I) 11—1 3. Standard deviation: 5 = #7}, 2(12 — i)2 = 5%(25512— mi?) 13—
U 4. z—score: 2 = Chapter 2 : Looking at DataRelationships
5. 7" : 'HITI Z (Eli—j) (9:532) 01‘ 7‘ : (hi—1) (11;:(yaL7) 6. Least—squares regression line : 3,7 = a + bzc, where b : T31 and a = y) — bi".
Chapter 4 : Probability: The Study of Randomness 7. Probability Rules (i) P(Ac) : 1 — P(A).
(ii) If events A and B are disjoint, P(A or B) : P(A) + P(B).
(iii) For any events A and B, P(A or B): PM) + P(B) — P(A and B).
(iv) If events A and B are independent, P(A and B) : P(A) P(B).
(v) For any events A and B, PM and B) : P(A)  P(BA).
(vi) When P(A) > 0, P(B)A) = ———_”<A and ’3). PM)
(vii ) Bayes’s Rule: Suppose that A1,A2, . . . ,Ak are disjoint events whose probabilities are not
0 and add to exactly 1. If C is any other event whose probability is not 0 or 1,
P C A, P A,
PWC): (I >( > P(CA1)P(A1) + P(CIA2)P(A2) +    + P(ClAk)P(/4k) 8. Probability distribution (i) Mean: #X : $11)] + Igpg + ‘   +Ikpk : Eggpp (ii) Variance: 0% =($1— #X)2P1+($2 — #X)2P2 + ' ' ' + (3% — #X)2Pk = 2(131' — Hxlzpz (iii) If a and b are ﬁxed numbers, then 2 2 2
Ma+bX : a + bltx, Ua+bx : b 0X l If X and Y are random variables, then ,IL_\/+y 2 ﬁx + [Ly
and if X and Y are independent, then 2 2 2
0X+Y = 0X + 0y 2 2 2
UX—Y I 0X + UY Chapter 5 : From Probability to Inference 9. Binomial distribution: X ~ B(n,p) (i) Binomial coefﬁcient: (2) = Emu—IT)” where n! = n X (n — 1) X (n — 2) X X 3 X 2 x 1.
(ii) Binomial probability: P(X = k) = (Dpkﬂ — p)("_k) for k : 0,1,... ,n
(iii) ux : M)
(W) 0X = m (V) For the sample proportion, 13, 11,5 2 p, 01, : ﬂip) n 10. Let i be the mean of an SRS of size n from a population having mean a and standard deviation 0.
Then Mi = u, 02 = U/x/E Chapter 6 : Introduction to Inference 11. A level C conﬁdence interval for n ( a known, SRS from a normal population): a? :l: 2* 2* from N(O,1) table a
ﬁ’
(C : 90%, z“ = 1.645 ; C : 95%, 2* : 1.96 ;C = 99%, 2* : 2.576)
12. Sample size for desired margin of error m :
* 2
Z 0'
m 13. Test statistic for H0 : u : #0 (0 known, SRS from a normal population): f—uo
2— _ U/x/ﬁ l0 Table entry for z is the area under the
standard normal curve
to the left of z . TABLE A I. Standard normal probabilities A. 's‘ Prbbésﬂify w ' HProlbabi‘lity ; Table entry for z is
the area under the .
standard normal curve , A z
to the left of z . ‘ ' i ' TABLE A}? Standard ﬁbrinal ptbbébilitiés (continued) 9997 .9998 :TABLE C ’Binbnﬁal'probabilities (continued) '1 ’ 4. Entry is P(X = k) = (n)pk(l ~ k
p
n k .10 .15 .20 .25 .30 .35
2 0 .8100 .7225 .6400 .5625 .4900 .4225
1 .1800 .2550 .3200 .3750 .4200 .4550
2 .0100 .0225 .0400 .0625 .0900 .1225
3 0 .7290 .6141 .5120 .4219 .3430 .2746
1 .2430 .3251 .3840 .4219 .4410 .4436
2 .0270 .0574 .0960 .1406 .1890 .2389
3 .0010 .0034 .0080 .0156 .0270 .0429
4 0 .6561 .5220 .4096 .3164 .2401 .1785
1 .2916 .3685 .4096 .4219 .4116 .3845
2' .0486 .0975 .1536 .2109 .2646 .3105
3 .0036 .0115 .0256 .0469 .0756 .1115
4 .0001 .0005 .0016 .0039 .0081 .0150
5 O .5905 .4437 .3277 .2373 .1681 .1160
1 .3280 .3915 .4096 .3955 .3602 .3124
2 .0729 .1382 .2048 .2637 .3087 .3364
3 .0081 .0244 .0512 .0879 .1323 .1811
4 .0004 .0022 .0064 .0146 .0284 .0488
5 .0001 .0003 .0010 ..0024 .0053
6 O .5314 .3771 .2621 .1780 .1176 .0754
1 .3543 .3993 .3932 .3560 .3025 .2437
2 .0984 .1762 .2458 .2966 .3241 .3280
3 .0146 .0415 .0819 .1318 .1852 .2355
4 .0012 .0055 .0154 .0330 .0595 .0951
5 .0001 .0004 .0015 .0044 .0102 .0205
6 .0001 .0002 .0007 ' .0018
7 ' O .4783 .3206 .2097 .1335 .0824 .0490
1 .3720 .3960 .3670 .3115 .2471 .1848
2 .1240 8 .2097 .2753 .3115 .3177 .2985
3 .0230 .0617 .1147 .1730 .2269 .2679
4 .0026 .0109 .0287 .0577 .0972 .1442
5 .0002 .0012 .0043 .0115 .0250 .0466
6 .0001 .0004 .0013 .0036 .0084
7 .0001 .0002 .0006
8 0 .4305 .2725 .1678 .1001 .0576 .0319
1 .3826 .3847 .3355 .2670 .1977 .1373
2 ‘ .1488 .2376 .2936 .3115 .2965 .2587
3 .0331 .0839 .1468 .2076 .2541 .2786
4 .0046 .0185 .0459 .0865 .1361 .1875
5 .0004 .0026 .0092 .0231 .0467 .0808
6 .0002 .0011 .0038 .0100 .0217
7 .0001 . .0004 .0012 .0033
8 .0001 .0002 p )n k .40 .3600
.4800
.1600 .2160
.4320
.2880
.0640 .1296
.3456
.3456
.1536
.0256 .0778
.2592
.3456
.2304
.0768
.0102 .0467
.1866
.3110
.2765
.1382
.0369
.0041 .0280
.1306
.2613
.2903
.1935
.0774
.0172
.0016 .0168
.0896
.2090
.2787
.2322
.1239
.0413
.0079
.0007 .45 .3025
.4950
.2025 .1664
.4084
.3341
.0911 .0915
.2995
.3675
.2005
.0410 .0503
.2059
.3369
.2757
.1128
.0185 .0277
.1359
.2780
.3032
.1861
.0609
.0083 .0152
.0872
.2140
.2918
.2388
.1172
.0320
.0037 .0084
.0548
.1569
.2568
.2627
.1719
.0703
.0164
.0017 .50 .2500
.5000
.2500 .1250
.3750
.3750
.1250 .0625
.2500
.3750
.2500
.0625 .0313
.1563
.3125
.3125
.1562
.0312 .0156
.0938
.2344
.3125
.2344
.0937
.0156 .0078
.0547
.1641
.2734
.2734
.1641
.0547
.0078 .0039
.0313
.1094
.2188
.2734
.2188
.1094
.0312
.0039 ...
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