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Unformatted text preview: Ql. In a study of water pollution in a river , the level of pollution is compared at two river stations. Random
samples of sizes n1 2 12 and n2 2 10 are taken and the pollution degrees are measured . Following data are obtained in proper units : £1 = 2%:1($11i— 51)2 = 5'2 = 2.04, 31M“ — :53)2 = 1.806. (Here 531,1 , = 1,2,    , 12) are 12 measurements at Station 1 and xzij, (j : 1, 2, ~   ,10) are 10 measurements
at Station 2 a) Find a 90% two—sided confidence interval for the difference ul—ug , between the mean pollution levels at the two
locations. Assume that the populations are approximately normally distributed with equal variances. 11pts b) Do the data support the assumption that a? = 0% ? Answer this question by using another confidence interval, this time related to the variances. Fix a degree of
confidence 1 — a at your choice. 11pts I ’L lo
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0. 7/ 3110 5' 6'2); o‘wcq ( 90’) a 1. .i‘U/Hhﬁ C C'ch,gLLJ/f L“ Q2. In a machine shop we decide to take a random sample of n = 75 axle shafts to estimate the proportion 9 of
the shafts having surface finish not meeting the specifications. We shall estimate this proportion of defective shafts
by means of a two—sided confidence interval. a) Before the random sample of 75 shafts is taken, with what level (degree) of confidence we can claim that the
maximum error in our confidence interval estimation will be 0.09 ? 10pts b) Suppose now that we have taken the random sample of 75 shafts and tested them. If 12 were found not meeting
the requirements of the specifications, find a two—sided 90% confidence interval for 9. 8pts O ‘l6 “l: l‘é'd’i/T/J my * one t 1645610429)
.4;  9' 0‘16 _+_ 0.07 Q3. A machine produces metal rods used in an automobile suspension system . A random sample of 10 rods is
selected and their diameters are measured . The resulting data (in mm.) are shown below 8.24 8.28
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8.26 8.24 Assuming the rod diameter normally distributed a) Construct a 99% twosided confidence interval for the standard deviation 0. 1111753 b) Constructa 99 upper—sided confidence interval for the variance 02. 10pts _ 'L
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Q4. A population random variable X has an exponential density f(a:;9) : (ED—e 9, a: > 0, with an unknown parameter (9. Since the experiments are very expensive we have only a single observation of X, (i.e. n = 1). a) Consider the hyptotheses
H0 2 G) = 90
H1 1 9 : 90 + 4
Rejection Rule (RR) : Reject H0 if the single measurement X > 90 + 2. 1) Find oz, the probability of Type I error, 5pts
2) Find ﬂ, the probability of Type II error. 5pts b) Suppose now that the hypotheses are as follows
H0 : E) = 2
H1 2 e > 2
Rejection Rule (RR) ; Reject H0 ifX > 4 1) Find an expression for 5(9) (i.e. the Type II probability as function of a general 9 > 2), 6pts 2) Sketch the graph of 5(9). 6pts
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,_j_éx G>Z i re) a) Q5. A study was conducted to study the true reaction times of men and women to a stimulus. Independent random
samples with large sample sizes n1 2 50 and n2 2 50 were used in the experiment. The results are shown below Men Women n1 2 50 n2 2 50 :51 = 3.6 sec. :52 = 3.9 sec.
521’ = 0.18 a; = 0.14 a) Does the data present sufficient evidence to suggest a difference between the true mean reaction times of men
and women ?
State proper hypotheses H0 and H1 and then test at the a = 0.05 level of significance. 11pts b) Does the data present sufficient evidence that the mean reaction time of women is 0.2 seconds longer than the
mean reaction time of men ? Again state the proper hypotheses and test this time at the a = 0.01 level of
significance. 11pts (j) H0 ll lllﬁNLf/O TC :3 ~ 3.6“ .3 9 Ho
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