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Unformatted text preview: IE 4521
' Fall Semester 2008 (Day)  2"d Test Name: _____Hu_ _____
ﬁction 00'
Problem 1 (20 points) The desired percentage of SiOg in a certain type of aluminous cement is 5.5. To test whether the true
average percentage is 5.5 for a particular production facility, 16 independentiy obtained samples are analyzed. Suppose the percentage of Si02 in a sample is normally distributed with o = .3 and that Y =
5.25 (8 points) a) Does this indicate conclusively that the true average percentage differs'from 5.5? Carry out the analysis, clearly stating the test statistic used, its sampling distribution and the sequence of
steps leading to the inference drawn by you. (2 points) b) Compute the p value of the test. (5 points) 0) It the true average percentage is p = 5.6 and a level of or = 0.01 is used in the above test with
n = 16, what is the probability of detecting this departure from HO. (5 points) d) What value of n is required to satisfy 0: = 0.01 and B (5.6) = .01?
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a 5.55.3 ' IE 4521
Fall Semester 2008 (Evening)  2MI Test , Name:
Dedein (7t? 2 Problem 1 ( points)
23: use The yield of a chemical process is being studi . From the previous experience with this process the
standard deviation of the yield is known to b 3 The past five days of plant operation have resulted in the foilowing yields: 91.6%, 88.75%. 90.8%, 89.95% and 91.3%. Use 0t = 0.05 in the following analysis.
(8 points) a) Is there enough evidence that the yield is not 90%? Formulate and define HQ and HA, and then analyze the situation stating clearly the test
statistic being used, its sampling distribution and inference from the data. (4 points) b) What is the P value for the test?
{4 points) 0) What sample size is required to detect a true mean yield of 85% with probability 0.95? (5 points) d) What is the Probability of the type ll error if the true mean yield is 92%? ._. if
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Problem 2 (20 points) Water samples are taken from water used for cooling, as it is discharged from a power plant into a river. It
has been determined that as long as the mean temperature of the discharged water is at most 150°F, there
will be no negative effects on the rivers ecosystem. To investigate whether the plant is in compliance with
regulations that prohibit a mean discharge water temperature above 150°F, 50 water samples were taken at
randomly selected times, and the temperature of each sample recorded. The resulting data will be used to test the hypotheses Hozp 3 150° versus HA = p. > 1500. {10 points) a) in the context of this situation describe Type I and Type II errors. Which type of error would
you consider more serious? Explain. (10 points) b) The above problem could have been formulated as
H0: 1.12150 HA=u<150. Compare the two formulations. Which one do you like? In general, what principle can you state
in defining Ho and HA. I
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Problem 2 (20 points) ; A manufacturer of car batteries guarantees that his batteries will last on the average, three years with a
standard deviation of one year. If five of these batteries have lifetimes of 1.9, 2.4, 3.0, 3.5, and 4.2 years: (10 points) a) Is the manufacturer still convinced that the average of the batteries is 3 or more years? (10 points) b) Is the manufacturer still convinced that the standard deviation of life is not more than 1 year?
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Problem 3 (16 points) 0 A research engineer for a tire manufacturer is investigating tire life for a new rubber compound. He has built
16 tires and tested them to endoflife in a road test. The sample mean and standard deviation are 60. 139.70 and 3,645.94 km.
59% ‘51}? 9. MT M “H,” ,1 619M}! whterfreﬁiﬁu (8 points) a) The engineer ' ' this new tir is in excess of
60,000 km. Formulate and test appropriate hypotheses and draw oonclu ons using 0L: 0.05. i
(8 points) b) Find 95% confidence interval on mean tire file.
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63 =0 é1+(l—a)é2 A new estimator (2 points) a) Show that Q, is an unbiased estimator of 6. (4 points) b) How should the constant a be chosen to minimize the variance of 9}? Assume that 5'1 and ég are independent. (4 points) 0) How should the constant a be chosen to minimize the variance of when 671 and $2 are not independent, and cov ((631, 92 ) = c. A A
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the hole diameters are me 0 the data ives s = .008 mm. (8 points) a) (i) Is there . viation of the hole diameter d .Otm'“? = . 1.
excee s 0 Use on O aéfd 1‘“ ﬁrm {a m (2 points) (ii) State any necessary assu on about the underlying distribution of the data. (4 points) b) Find the PValue for stest. /0 lower confidence interval for 02. 1 {$3 35 n [ ,HA ; {curl I! 07/ " Her 6 504',HA:6?0.01" (6 points) 0) Constructa it}
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Problem 4(12 points)
(2 points) (i) The following three distributions correspond to t sampling distribution with v = 2, v = 5, and v =
m (inﬁnity).
Label each curve appropriately. Which of the curves represents Z distribution. 3 .3 .1 D i 2 The r—distribution curves for v = 2, 5, and 00. (2 points) (ii) The following two sampling distributions correspond to the distribution of Y for
,u = go under H0 and ,u = #0 + 6 , and one of position HA . Let a = 0.05 Shade the area denoting ,6. where 5 represents the probability of making type II error in a
testing of hypotheses problem. f 5mm 00; (2 points} (iii) The following diagram shows several 95% confidence intervals for ,u . Explain in terms of
probability what does 95% confidence interval mean? ,
ff Wild Lt 5“ 5:0}:4hmgaf {n Jplﬂwe oldEWVJS 1 1:6 #‘fﬁ Ma! “we 1"? day{Wm Flu/«(Ell «41‘ ‘5"? CE
ﬂlﬂwly l——————_l 1" .\ Interval estimates of a for different samples. (2 points) (iv) The following diagram gives sampling distributions of 92, 93, which are three alternative
estimates of the unknown parameter (9 of the population. A A
a. Which estimators are unbiased? {9,66% by Which estimator would you select? (9/: 3131' as at ‘ L 9. ae' m5 +5
, :5 wt 1' 5w! any! 8 AIM mﬂ
‘1 f 6f Val/1W Sampling distributions of different estimators of 9. . 2 *—
(2 points) (v) How are Z and 25., related? 7ft! $42231 hrg: aim 2,, 2, . .33, (we 3M4”. uWMJcI 94’wa: EJ‘tini» . . 2
(2 pomts) (VI) How are 11, and Fvlow2 O x {dark/i . ___._ .___.
Wig/2. related? F 7f" it, Seem dolﬁaal
Problem 5 (25 points) a) In the context of 'Testing of hypothesis’ where HD is the null hypothesis and HA is the alternative
hypothesis. Name the following events in their terms. (1 point) (i) Ho/HA Twat IL error (1 point) (ii) HA/HD (We I WW Give the traditional symbols for the following Probabilities.
(1 point) (iii) WHO/HA) ’9 (1 point) (iv) Prob (HA/Ho) C4 (1 point) (v) Prob (HA/HA) ]. f3 DOW ﬁst
b) (4 points) II J 6"], and 62 are two point estimates of the Parameter 6 A We know then [gr 15 a“ “£95560, Ei‘h‘rm‘bv! Jimt (£339, yawMa, ,.
Suppose A A {213 a gamer. [nit [4.9 My, mew = 6 V3r(61) = A A g 9 81% (52] = Mania: 5 19;” 2_ A g A .
5192):; Mag): 4 MW W Lnw (ma).
‘ Mew wants: 2
I ] : 0 ’_" A i. 1'“ A l
_ Which estimator is better? MSE { (an ) : wrﬁgﬂfﬁﬂ: 2 / 92/0 ﬂ) t9,isLefter=9MfEf9J<ﬂ$Er6J=3lu<§~lfyg
1:. z A) : 4 + E ‘ A n n {K
In what sense is it better? + 3.3912.th 79mﬂ9ﬂww. 195:4999 (4; c) (2 points) (i) If Y is the mean of a random sample of size n taken from a population with mean ﬁe and
"I?  #5 ark/E finite variance 02, then the limiting quantity of follows what sampling distributes? r7
2.: 5W dol Z002— Problem 5 can’t (2 points) (ii) What is 56) = (ﬂ ?
.r’
_ 2
and Var (X)= 5/ ? (1 point) (iii) What does (i) say additional to (ii)? HM SMEWsamdm mmdﬂéébl/Mrwlﬁwbhﬁzﬁvf d) (2 points) i) Let 2 be a standard normal ran om variable e a chisquare random var able with
V degrees of freedom. lf 2 and V are independent, then what is the name of the distribution of the
. 2
quantity — t (2 points) (ii) If If is a mean of a random sample from a normal population with mean ,u, and population variance unknown and s is the sample standard deviation of the data then what distribution
does the following quantity follow? X—ﬂ Six/E Tit—l e) (2 points) Both sample mean (denoted by E) and sample median (denoted by X” ) are unbiased
estimators of the population mean y , which of the two estimators is more efficient, and why? SSElo Mead 1‘5 at Letfer 65$~wpﬁﬂ if 45 WWW :4 ﬁve . . .
f) ( pomts) For each of the fol owmg assertions, state whet er or not it is a legitimate statistical hypotheSIs
and why. .2 Ifg‘wtﬂ'fp 4L! 9;! [‘5 (i) H: or>100 V/ [Mn MM”!
(ii) H: E :45 X
(iii)H: ? ? =5;( 9) (2 points) Write a note not exceeding 2 sentences on ‘why is it important for decision makers to learn
about inferential Statistics.‘ ' i‘ Wig wﬂﬁt "(Lo RULE 51594131336 9h FonMgm M Lakdf 9“ {Why i. u Den‘a‘ms Lib“V6! 0% $9M?!“ (5m ygﬁﬂ an “10%)” .i £13455th {g Veg a“???pr 4‘”. ’[LﬂLclttlj hark] ts “Ml/LE Wad igvﬁg'jﬁyms" ".\ ...
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 Spring '08
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 Standard Deviation, Type I and type II errors

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