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Unformatted text preview: Problem I (14 points) IE 4521 Fall Semester 2009 — 2"“ MidTerm Name In a random sample of 100 batteries, the average lifetime was 150 hours and the standard
deviation was 25 hours. " (4 points) (4 points) (4 points) (2 points) (:9 "‘ {QEJMSI a) Find a 95% confidence interval for the mean life time of the batteries. b) An engineer claims that the mean life time is between 147 and 150 hours.
With What level of confidence can this statement be made? ﬁe c) Approximately how many batteries must be sampled so that a 95% confidence
interval will specify the mean within 1 2 hours? cl) State the test statistic you are using while solving parts a. b, and 0 above and
also what assumptions, if any, you are making. W i : m I “3’03; 3:13 ) 01:001— ._ 1c S“ fetvkjmf 3 if??? “a e (“H—.95; E) Y—Joi s/m A W M‘T‘JM C/fc) 2 : PHI? < w «z r Ff?) 1' i?< "MSW 56w <2 em) :F('T>f/;mﬂ42 4% m raftdam saw is BMW: C, . @314ch I 'ﬂe Mfume?! Samplr 3329 (“cm be (gram cad 150
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paw(c) T— lﬁmeﬂ’ﬁ' Problem 2 (13 points)
Fit! in the blanks in 1 & 2. (1 point) . 1. The value measures the plausibility of H0. (Based on the given
data). (1 point) 2. The smaller the Pvalue the stronger is the evidence against H0" . 3. A certain type of automobile engine emits a mean of 100 mg of oxides of
nitrogen (NOX) per second at 100 horsepower. A modification to the engine design has been proposed to reduce NOx emissions.
The new design will be put into production if it can be demonstrated that _theimean
__ernission rate is less than 100 mg/s. L Hal“: .1 diff! L" hit: J4” 55;! L LU! ié “1'
A sampie of 50 modified engines are built and tested. The sample mean NOx
emission is 92 mg/s and a standard deviation is 21 mgls. The manufacturers are concerned that the modified engines might not reduce
emissions at all. ____.— in $315475“ NJ“: ; ,a (loo; HAﬂaﬁ’
(2 points) a) Formulate the problem as a testing of hypotheses (Define Ho an'd HA) ” 3 .19 tie in?» regime [‘5 "’0, nab S9" Ha“ M3!“ . Hi; : ﬂ d“? We” ,wéci‘ mm; Ho
(2 points) b) Give the test statistic you would use. (g Hawaiian , (Last. 990% evident of H.
(2 points) 4. Draw the sampling distribution of E, when H0 is true. _ {93° "' Ha {aging ) Hﬂﬁqq Y (a sample size of 50 is large enough to assume normality of X ) {WC ,mm { weaker W 9'? oint' F ‘ on HUI»? evrtfence
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(Ht/Q Oman/er (1 point) 5. Show the observed vaiue of 92 on the graph .in 4.
(2 points) 6. Shade the region which represents the P value in 2. above. (2 points) 7. If in your decision making a = 0.01, what inference would you draw in the
above case. I Problem 3 (8 points) (2 points) (4 points) (2 points) a) X is a normal distribution with mean u and variance oz. Both parameters are
unknown. A small random sample of size n, with values X1, X2, , Xn is taken from the above population. it and s are computed from the data. Then the statistic X135 followsa afar/)5szin
s n _ Write the sampling distribution. b) Let X be the octane ratings for a particular type of gasoline in (a) above. The
results (in %) for a sample of size 5 are 87.0, 86.0, 86.5, 88.0, 85.3 Find a 99% confidence interval for the mean octane rating for this type of gasoline. 0) Given below are 4 curves (density functions) representing tm, t4, t1, and Z. Labeieachcurve. 4'5) {1 87i86+%~§t 9'5Jr3’5~5
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1 MED/l «(Moi Wit?!) 33”. 5:3) Problem 4 (10 points) Fifi in the blanks with the words provided out of the list given below. I ~'\ \ "I. ‘ 'H—E’uiir". I «(if ‘ «I ‘i
(1 Point) 1. A $1; 5126 Ha random vanale that depends only on the observed
values of a random sample. ' (1 Point) 2. A Fﬁmlﬁiﬂconsists of the totality of the observations with which we are
conce ned. . (1 Point} 3. The probability distribution of a statistic is called a ﬁancﬁ'bt‘ru? distribution. (1 Point) 4. The standard deviation of the sampling distribution of a statistic is called them gaggng of the statistic. (2 points) 5.  Statistical inference may be divided into two major areas:
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(2 points) 6. in estimation if e is the unknown parameter of the population. and 6 is its estimator. then unbiasedness of 8 is a very desirable property oi a good
point estimator. Write the missing left hand side in the following equation
which makes 9 unbiased. , 1R = 3 (1 point) '7. If we consider all possible unbiased estimators of so ‘parameter s, the
estimator with the smallest variance is called the most (2%?(5MT estimator of e. (1 point) ‘ 8. Both mean (denoted by 11' ) and median (denoted by 3?) are unbiased estimators of the population mean p, which of the two estimators is more
efficient1 and why? lea it
List of words D: ’ 1+5 ’V‘JlHHEerCQ, emit/lid 'bQ % WWW/l” (mean, E(6' )lefficient. population, statisticsI standard error, sampling.'estimation, testing
of hypothesis) Problem 5 (12 points) The mean sitting height of adult males may be assumed to be normaliy distributed, with mean
36" and standard deviation 1.3”. For a sample size of n = 100 men, determine .the. foliowing probabilities.
(i). P (S s 1.00”) (ii) P (S .>. 1.4") (iii) P (1.25” s S s 1.35") (HINT? *There is a weii known distribution connected with 32 * Also use Central Limit Theorem {2310 9% 6" a}; glw )w 36 L
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: P(')(Wz4 "‘ p()(oql7t2§:.z.ﬂig Problem 6 (8 points) A good estimator is unbiased and has minimum variable. Suppose X1, X2, X3 denote a random
sample from an exponential distribution with density function ie—J‘m, x > 0
6‘ for) = elsewhere
0 Consider the following four estimators of 6' ' él=XI
6A1 : X1+X2
" 2
53 = X1+X2
3
634:} (4 points) a) Which of the above estimators are unbiased estimators of 9. (4 points) b) Among the unbiased estimators of 9., which has the smallest variance. 00. Bahama 9 WWW: a} :1) bum/aimed ix ; 2 3 to) 3 MR Moved
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 Spring '08
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