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[size=12.0000pt]Question 1: (32)
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A manufacturer receives a shipment of 100 parts from a vendor.
#The shipment will be unacceptable if more than five of the parts are defective.
#The manufacturer is going to randomly select K parts from the shipment for inspection,
#and the shipment will be accepted if no defective parts are found in the sample.
#How large does K have to be to ensure that the probability that the manufacturer
#accepts an unacceptable shipment is less than 0.1?
#Hint: Use R to plug in different values of K.
# k <- unknown
[size=10.0000pt]m <- [size=10.0000pt]6
[size=10.0000pt]n <- [size=10.0000pt]94
[size=10.0000pt]x <- [size=10.0000pt]0
[size=10.0000pt]k <- [size=10.0000pt]0
[size=10.0000pt]prob <- dhyper(x, m, n, k)
while[size=10.0000pt] (prob > [size=10.0000pt]0.1[size=10.0000pt]){
[size=10.0000pt] k <- k +[size=10.0000pt]1
[size=10.0000pt] prob <- dhyper(x, m, n, k)
[size=10.0000pt]}
[size=10.0000pt]k
[size=10.0000pt]## [1] 32
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[size=12.0000pt]Question 2: (51)
Now suppose that the manufacturer decides to
#accept the shipment if there is at most one defective part in the sample.
#How large does K have to be to ensure that the probability that
#the manufacturer accepts an unacceptable shipment is less than 0.1?
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[size=10.0000pt]m <- 6
[size=10.0000pt]n <- 94
[size=10.0000pt]x <- 1
[size=10.0000pt]k <- 1
[size=10.0000pt]prob <- phyper(x, m, n, k)
while[size=10.0000pt] (prob > 0.1[size=10.0000pt]){
[size=10.0000pt] k <- k + 1
[size=10.0000pt] prob <- phyper(x, m, n, k)
[size=10.0000pt]}
[size=10.0000pt]k
[size=10.0000pt]## [1] 51
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[size=12.0000pt]Question 3: (n)
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[size=12.0000pt]Answer Let X be the number of trials needed to open the door. Let ‘S 0 denote success i.e. the door is opened and ‘F 0 denote failure for each trial. (a) The event X = x is equivalent to the event F F F . . . F | {z } x−1 times S Also P(S) = 1 n and P(F) = n − 1 n . X has the geometric distribution with p = P(S) = 1 n . Therefore E(X) = 1 p = 1 1 n = n
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[size=12.0000pt]Question 4: (7)
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CHOCOLATE CHIPS IN A COOKIE
#Let the number of chocolate chips in a certain type of cookie have a
#Poisson distribution. We want the probability that a randomly chosen
#cookie has at least two chocolate chips to be greater than 0.99.
#For which of the following values of the mean of the distribution
#is this condition assured? (Please select all that apply!)
# Number of choco chips X ~ Pois(lambda)
# Pr(X > = 2) > 0.99
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[size=10.0000pt]lambda = 0
[size=10.0000pt]prob = 1[size=10.0000pt] - ppois(1[size=10.0000pt], lambda)
while[size=10.0000pt] (prob <= 0.99[size=10.0000pt]){
[size=10.0000pt] lambda = lambda + 0.001
[size=10.0000pt] prob = 1[size=10.0000pt] - ppois(1[size=10.0000pt], lambda)
[size=10.0000pt]}
[size=10.0000pt]lambda
[size=10.0000pt]## [1] 6.639
[size=10.0000pt]prob
[size=10.0000pt]## [1] 0.9900056
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[size=12.0000pt]Question 5: (400)
[size=12.0000pt]Question 6: (E(300 + 10M – 100B + 50*1/4)
[size=12.0000pt]Question 7: (342.5)
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[size=12.0000pt]Workings for both q5. & q6 & q7.
file:///C:/Users/F/AppData/Local/Temp/ksohtml16876/wps1.png
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[size=12.0000pt]Question 8: (2.5)
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[size=12.0000pt]Question 9: (0.0833)
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[size=10.5000pt]Random sample of size 100 is used..
[size=10.5000pt]V(2*sample mean) = 4* V(sample mean)/n {since random sample is taken}
[size=10.5000pt]Variance of estimator put forward by Researcher A is,
[size=10.5000pt]= 4 * (25/12)/100 = 4 * 25/(12*100) = 0.08333
[size=12.0000pt]Question 10: (Researcher A)
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[size=12.0000pt]Question 11: (Theta B)
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[size=12.0000pt]Question 12: ([size=10.5000pt]0.008333)
[size=10.5000pt]Random sample of size 1000 is used..(Variance comes down when sample size goes up)
[size=10.5000pt]V(2*sample mean) = 4* V(sample mean)/n {since random sample is taken}
[size=10.5000pt]Variance of estimator put forward by Researcher A is,
[size=10.5000pt]= 4 * (25/12)/1000 = 4 * 25/(12*1000) = 0.008333
[size=10.5000pt]Question 13: (n=1000)
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