1. (15%) Sampling for defectives £rom large lots of manufactured ...

1. (15%) Sampling for defectives £rom large lots of manufactured product yields anumber of
-
defectives, Y, that follows a binomial probability distribution. A sampling plan consists of
specifying the number of items n to be included in a sample and an acceptance number "a".
The lots is accepted if Y I a and rejected if Y > a. Letp denote the propofiion of
defectives in the lot.
(a) For n = 5 and a = 0, calculate the probability of lot acceptance if (a)p= O,.(b)p = .l,(c)
p= .3,(d) p = .5,(e)p = 1.0.
I
(b) A graph showing the probability of lot acceptace as a function of lot £radon defective is
called the operating characteristic curve for the sample plan. Construct the operating
characteristic curve for the plan n = 5, a = 0.
2. (10%) Let m(t)=(1/6)ef+(~6)ez'+(3/6)df. Find the following:
(a) E(Y)
(b) V(Yr
(c) The distribution of Y
3. (10%) A diagnostic test for a disease is said to be 90% accurate in that if a person has the
disease, the test will detect it with probability 0.9. Also, if a person does not have the disease,
the test will report that he or she does not have it with probability 0.9. Only 1% of the
population has the disease in question. If a person is chosen at random £rom the population
and the diagnostic test indicates'that she has the disease, what is the probability that she does
have the disease? Are you surprised by the answer? Would you call this diagnostic test
reliable?
4. (15%) Let Y, be the amount of pollutant per sample collected above the stack wifhout the
cleaning device and Y, be the amount collected above the stack with the cleaner. The joint
density of Y, and Y, is
osy,s2, O ~ Y , ~ ~ , ~ Y , ~ Y I
elsewhere.
The random variable (YI -Yz) represents the amount by which the weight of pollutant can be
reduce by using the cleaning device.
(a) Find E(Y, -Y,).
(b) FindV(Y, -Y,).

5. (20%) Let Xbe a random variable whose p.d.f.@robability density function) f is either the
U(0,1), Uniform, to be denoted byfo, or the triangular over the interval [0,1], to be denoted by
fi, that is f;(x) = 4x, for0 2 x ~112; f;(x) = 4-4x, for112 2 x 2 1; and 0, otherwise.
(a) Test the hypothesis Ho: f =fo v.s. HI: f =fi at level of significance a = 0.05.
(b) Compute the power of the test.
6. (15%) Answer the following questions:
(a) A random variable Xis said to be memoryless if P{X > s+t I X > t) = P{Y > s) for all
s, t 1.0. Show that Xis memoryless when Xhas an exponential distribution with parameter h.
(b) Let X be the interarrival time of buses at a station. IfX is exponentially distributed,
describe how the memoryless property affects the waiting time of a passenger.
7. (15%) After running a multiple regression analysis with five independent variables, the
following ANOVA table is olitained
Source
Regression
Residual
Total
d f
45
SS
224
270
MS
F
(a) Complete the above ANOVA table.
(b) Calculate R2 and adjusted R2.
(c) Test the hypothesis Ho:PL=P2=P3=P4=P5=0 with a =0.05.

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TABLEIV -
Values of 1. df
-

TABLE VII (cont.)
Valuer of x:

TABLEVIII (cont)
Values of Fa
dfd
u
dfn
0.10
0.05
25 0.025
0.01
0.005
0.10
0.05
26 0.025
0.01
0.005
0.10
0.05
27 0.025
0.01
0.005
0.10
0.05
28 0.025
0.01
0.005
0.10
0.05
29 0.025
0.01
0.005
0.10
0.05
30 0.025
0.01
0.005
0.10
0.05
60 0.025
0.01
0.005
0.10
0.05
120 0.025
0.01
0.005