A lot of 1000 parts is shipped by a supplier to a receiving company. A sampling plan agreed to by the supplier and receiver dictates that 100 parts are to be sampled at random and without replacement and the lot is accepted if no more than 2 of these 100 sampled parts are defective.
- Let p=K/N be the fraction defective in the lot. The operating characteristics (OC) curve is a function of this proportion, p. Describe it using the hypergeometric distribution.
- The supplier recommends approximating the OC curve with a binomial distribution. Why or why not is this okay? Write down the OC curve using the binomial approximation.
- The customer recommends that they use a Poisson distribution. Why or why not is this okay? Write down the OC curve using the Poisson distribution.
Draw (approximately) the OC curve from
0£p£0.1.
Hint: You cannot evaluate these distributions because you don’t know
p. The answers are written as OC curves versus
p, the probabilities of accepting the lot given
p.
Problem #2Suppose that iron plates in a production process are required to have a certain thickness, and the machine work is done by a shaper. These plates will differ slightly due to variation in the material properties, machine behavior, tool usage, and other uncontrollable sources. Let the thickness of the plates be a random variable,
X, in
mm. For a certain production setting, the variable
X is normal with a mean of
10 mm and standard deviation
0.02 mm. Determine the percentage of defective plates to be expected, assuming that the plates are considered defective when they are
(a) thinner than
9.97 mm, or
(b) thicker than
10.05 mm, or
(c) the plates deviate by more than
0.03 mm from the mean.
(d) What value of
c shall we choose for the interval
(10-cs)mm to
(10+cs)mm ensures that the expected percentage of defectives will not be greater than 5%? Use the fact that the probability distribution is symmetric about
X=10 mm, that is,
Pr(X³10+cs)=Pr(X£10-cs).
(e) Suppose you want to determine what happens when a 1% bias (e.g. prior to recalibration) is made to the tool bit. How does the percentage of defective plates determined in part (d) change if the mean is shifted from
10 mm to
10.01 mm? Discuss.
Hint: To solve all 5 parts, use Appendix A Table III in Montgomery & Runger.
Problem #3Let
X and
Y be random variables of the discrete type having the following joint probability mass function defined for the integers specified.
- Prove that the denominator constant, c=18.
- Compute the marginal probability mass functions of X and Y.
- Are the two random variables X and Y dependent or independent? Why?
- Compute the means mX and mY.
- Compute the variances sX2 and sY2.
- Compute the covariance, sXY=cov(xi,yj).
- Compute the correlation, rXY.
- Determine the equation of the least squares regression line Y versus X that would best fit random data generated with this distribution.