1.
A chemical engineer monitors a dyeing process for polyester yarn used in clothing by comparing a sample of the yarn against a standard color chart. He accepts or rejects the entire dyeing batch based on the sample results. Historically, this process averages 10% rejected batches. The process dyes 20 batches in each shift. Assume that the next shift forms a random sample.
(a) Find the probability that 3 or 4 of the batches will be rejected in the next shift.
(b) Find the expected number and the standard deviation of batches rejected.
(c) If 4 batches were rejected for this shift, would this be unusual? Justify your answer by calculating the relevant probability.
2.
The number of trac accidents at a certain intersection can be modeled by a Poisson process with mean 2 accidents a year.
(a) Sketch the Poisson distribution for number of accidents in a particular year.
(b) What is the probability that there are at least 4 accidents in a particular year?
(c) What is the probability that are between 4 and 6 accidents (inclusive) in 2 consecutive years?
3.
A large rm wants to experiment with "telecommuting" i.e. allowing employees to work at home with their computers. Among other things, telecommuting is supposed to reduce the number of sick days taken.
Over the last few years employees at this rm have taken a mean of 4.2 sick days. This year the rm introduces telecommuting with the view of reducing the mean number of sick days taken. Management chooses a simple random sample of 50 employees to follow in detail, and at the end of the year these employees average 4.8 sick days with a standard deviation of 1.8 days.
(a) Set up the null and alternative hypotheses. Why would a 1-sided alternative be appropriate in this situation?
(b) Find the P-value for the test of the null hypothesis in (a).
(c) Explain what the P-value means in this context and draw the appropriate conclusion at 5% level of signi cance.
4.
The weight of potato chips in a small-size bag is stated to be 125 gm. The amount that the packing machine puts in these bags is believed to be normally distributed with mean 127 gm and standard deviation 2.0 gm.
(a) What proportion of all bags sold are underweight i.e. less than 125 gm?
(b) What proportion of bags will be within 2 standard deviation of the mean?
(c) Some chips are sold in \bargain packs" of 6 bags. What is the probability that at least four bags are underweight?
(d) If the mean remains at 127, what must the standard deviation be so that only 10% of bags will be underweight?
5.
In a process for the chemical etching of silicon wafers used in integrated circuits, this process etches the layer of silicon dioxide until the layer of metal beneath is reached. The thickness of the silicon oxide layer is monitored because thicker layers require longer etching times. Assume that layer thicknesses are normally distributed and, historically, mean thickness is 1.100 micron with a standard deviation of 0.06 micron.
(a) A recent sample of six wafers yielded a mean thickness of 1.150 micron. Find the probability of observing such a mean or larger.
(b) To monitor layer thickness, wafers are sampled in subgroups of ve.
i. Calculate 3-sigma control limits for a x chart based on the historical values?
ii. Suppose a new batch of wafers being processed has mean thickness 1.2 micron but the same standard deviation. Find the probability that the shift in the mean for this batch will be detected on the rst sample taken.
iii. Calculate the ARL. ARL equals 1/p where p is the probability calculated in ii and it is the number of samples, on the average, we have to observe before detecting the shift in the mean.