The number of contaminant particles (?aws) found on each standard size silicon wafer produced at a certain manufacturing site is a random variable, X, that a quality control engineer wishes to characterize. A sample of 30 silicon wafers was selected and examined for ?aws; the result (the number of ?aws found on each wafer) is displayed in the Table below.
|
4
|
1
|
2
|
3
|
2
|
1
|
2
|
4
|
0
|
1
|
|
3
|
0
|
0
|
2
|
3
|
0
|
3
|
2
|
1
|
2
|
|
3
|
4
|
1
|
1
|
2
|
2
|
5
|
3
|
1
|
1
|
(i) From this data set, obtain an empirical frequency distribution function, fE (x), and compute E(X), the expected value of the number of ?aws per wafer.
(ii) Justifying your choice adequately but succinctly, postulate an appropriate theoretical probability model for this random variable. Using the result obtained in
(i) Above for E(X), rounded to the nearest integer, compute from your theoretical model, the probability that X = 0, 1, 2, 3, 4, 5 and 6, and compare these theoretical probabilities to the empirical ones from (i).
(iii) Wafers with more than 2 ?aws cannot be sold to customers, resulting in lost rev- enue; and the manufacturing process ceases to be economically viable if more than 30% of the produced wafers fall into this category. From your theoretical model, determine whether or not the particular process giving rise to this data set is still economically viable.