Question

Problem 1
(a) A study is undertaken to assess the effect on assembly of the manufacturing tolerance of the outside diameter of a rod and the inside diameter of a mating hole in a wheel. A random sample of ten rods and ten wheels are selected for assembly. Differences (in inches) between the measured hole diameter and shaft diameter are as follows: 0.01, 0.09, 0.15, -0.01, 0.11, 0.06, -0.03, 0.13, 0.8, -0.4
Estimate the proportion of assemblies with interference (can’t assemble) using a point and interval estimate (90% confidence), if the distribution of the difference is
(i) Not known
(ii) Normal
(b) The number of defects per inspected PC-X based on a random sample of 15 from a days’ production is: 1, 3, 1, 0, 2, 0, 0, 1, 1, 1, 0, 1, 2, 1
(i.) Analyze these data and present your results.
(ii.) Estimate the probability that a randomly selected PC will have at least 3 defects.

Problem 2
Two years ago, management of X company initiated a program to reduce employee turn-over. The turn-over rate per month was researched yielding the following results (in % and by month):
2.1, 2.5, 2.7, 3.1, 2.5, 2.3, 1.8, 2.1, 2.2, 2.7, 4.4, 4.3,
1.3, 1.7, 5.3, 4.7, 2.5, 3.8, 4.5, 4.7, 6.2, 5.1, 3.2, 4.1
Do these data indicate that the program is effective? Provide the rationale for your answer.

Problem 3
A robot that shapes metal needs overhauling if it is out of tolerance on 4.5% of the items processed, and it is operating satisfactorily if it is off on only 0.8% of its output. A test is performed involving 50 sample items. If the sample proportion of out-of-tolerance items is greater than 0.02, the robot will be overhauled. Otherwise it will be allowed to continue operating.
(a) What is the probability that a satisfactory robot will be overhauled unnecessarily?
(b) What is the probability that a robot in need of overhauling will be left in operation?

Problem 4
If f(x) = a(5-|5-x|), for 0 ≤ x ≤ 10
             0, elsewhere

find the value of a so that f(x) is a probability density function.
If X=waiting time (in minutes) with probability density function f(x), find:
(i) P(X > 7)
(ii)The probability that an individual will have to wait for at least 7 minutes on at least 3 out of the next 5 days?

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