Question: Consider a variant of problem where any package with at least one imperfection (a crease, a hole, a smudge, or a broken seal) is considered to be nonconforming. Reinterpret the values X in the table of problem 1 as counts of nonconforming packages in samples of size 30. Suppose that in the past .05 (5 %) of packages have been nonconforming.

(a) Does this variant of problem involve variables data or attributes data? Why?

(b) What probability distribution (fully specify it, giving the value of any parameter(s)) can be used to model the number of nonconforming packages in a sample of 30?

(c) What is the mean number of nonconforming packages in a sample of 30?

(d) What is the standard deviation of the number of nonconforming packages in a sample of 30?

(e) Find the standards given control limits and center line for monitoring the proportion of nonconforming packages in samples of size 30.

(f) Repeat (e) for monitoring the number of nonconforming packages in samples of size 30.

(g) Suppose no standard is given for the fraction of nonconforming packages. Based on the data in the table above, find appropriate retrospective control limits and center line for an chart

Problem: In a packaging department of a food processor, types of packaging "imperfections" are carefully defined and include creases, holes, printing smudges, and broken seals. 30 packages each hour are sampled, and X = the total number of imperfections identified on the 30 packages is recorded. On average about .05 imperfections per package have been seen in the past. Below are data from 7 hours one day in this department.

Hour        1        2        3         4         5        6         7

X            1        0        2         0         1        1         3

(a) Are the data above variables or attributes data? Why?

(b) What distribution (fully specify it, giving the value of any parameter(s)) can be used to model the number of imperfections observed on a single package?

(c) What is the expected total number of imperfections observed on a set of 30 boxes? What probability distribution can be used to model this variable?

(d) What is the standard deviation of the total number of imperfections on 30 boxes?

(e) Find the standards given control limits and center line for a chart based on the data above, where the plotted statistic will be X/30. Do any values of X/30 plot outside your control limits?

(f) What is the name of the type of chart you made in (e)?

(g) Suppose no standard is given for the rate of imperfections. Using values above, find appropriate retrospective control limits and center line for plotting X/30.