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Unformatted text preview: FINAL EXAM  STATISTICS 241f51 July 2004
Time: 1.5 hours Student Name (Please print): .............................................................................. .. Student Number: ................... ................................ .. STAT 241 STAT 251 {check one) Notes: a Total points equal 100. 0 Show the work leading to your solutions in the space provided. Indicate clearly the part of the problem
to which the work relates. a This is a closed book exam. A single 8r1 / 2 x 11 page of notes (2—sided) is allowed.
a Clean statistical tables from the course notes are allowed. a Hand calculators are permitted. Problem 1: Let X be a. continuous random variable with density function f(x)=1—::—$‘ forug 1:39. (1) 1. Determine a .
2. Calculate P(X > 5)
3. Calculate P(X > 3 '5 X < 5). :1. Suppose X1. X2, ..., X 20 are independent random variables with common density function Calcu— late the median of
Y 3: min {.Yh X2, ...‘ X20}. Answer to Problem 1 m
I. Hoodx: 7—995. g.
90 0 g L‘fndx e e
..__’:_. gum; ﬁlm use—Lav '
m Answer to Problem 1 (continued) E13 Problem 2: The temperature reading from a thermocouple placed in a constanttemperature medium
is normally distributed with mean u  the actual temperature of the medium and standard deviation 0. (a) For what value of a. 95% of all the temperature readings are within 0.100f a? (b) Suppose now that a = 0.05”. For what value of n... 95% of all the averages of n independent
temperature readings are within 0.01“ of p? Answer to Problem 2 a. )(Nmyf)
H [II0.19:5 p4+o.)=0.‘15 [a] “'9 Pl $Js§£<£¥Plﬂ<2éﬂ I3] Answer to Problem 2 (continued) [2] C63 is]. Problem 3: 30 lots containing 20 items each are randomly chosen. All the items in the lots are
inspected to estimate the unknown proportion p of defective items. 1. Let X, (t' = 1. 2. 30) be the (random) number of defective items in each of the inspected lots. What
are the expected value and variance. for the total number of defective items 53x¥1+X2+"'L;x'30 I." 2. Suppose that the total number of defective items found in the 30 lots is 150. “that is your estimate
for p ? 3. What is the approximate probability that this estimate differs from the unknown “true” proportion
of defective items p by less than 0.02? Answer to Problem 3
' X; ~ Binor'ﬁal (7.0,?)
EX = hP= 20 P E 2.?
Vary mphpawn?) L z]
30
S = 2 X: 'i‘I
Els) H iXi)‘30EXa= 6oof> cs] YarfS)= Kurt X: ) =._ 3oUMXUa 600,90?) C i] Answer to Problem 3 (continued) “‘— ~N(o.l) My CLT)
E12 3(2) ti] : P("'I.BI €85. N3! ) =Q74'g IQ l.l3l)==o. 97015) L_ Problem 42 A drunkard executes a “random walk" in the following way: each minute he takes a
single step North with probability 2/5 or South with probability 3/ 5. The directions of his successive steps
are independent and their length are random variables with uniform distribution on the interval [20, 60)
cm. The length of each step is independent from the direction he is walking. What is the approximate probability that the drunkard will be at least 1 meter away from his original location after one hour?
Answer to Problem 4 Ai=i+l “I
60
S = 1' MB. 30‘“ Ba ~ unef (20,60) E BIS): L—  "' Bo'E(43)ECB3)
~= 600%)40 :42»
Var“): Vat igAiBi) '=' so VarMiBi) Answer to Problem 4 (continued) 37 0” .I' «v m erg), VarC$»=M(4<PO, {00:50)
1:31
P( lSlzloo)=P( Sawo) +P($gloo) :P( 3'48!) s 'lw'ﬂb)
\l '00'60 J Ivouso
+P( 348" ‘2' [00480) W“ 17.55 =— 0.1!83 [‘1 ...
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 Spring '08
 KANTERS
 Probability theory, Kurt, J Ivouso

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