A certain production process produces an average of 5 defect

A certain production process produces an average of 5% defective items. All items produced are inspected and on average an inspector misclassifies an item 5% of the time. What proportion of items will be classified as bad? Hint: Assume there are 1000 items produced. Construct a table with rows labelled Good and Bad and Columns labelled Inspection: Correct and incorrect. (Please show work) Thanks!

Solution

To simplify the calculations we will undertake the assumption that there are a total of 2000 items produced by the production process.

Now it has been given that on an average 5% defective items are produced. So in a lot of 2000 items number of bad (defective) items would be 5% of 2000 or 100. Hence the number of good(non-defective) items will be 2000-100 = 1900.

Now the next information that is given in the problem is that on an average an inspector misclassifies the labels 5% of the time. So among 100 bad(defective) items he is going to label 5% of them as good(which is incorrect). In numbers this will represent 5% of 100 or 5. Thus there will be 5 items which are labelled as good but they are not good. In proportions this will represent 5/2000 or 0.0025.

Since 5 out of 100 bad items are labelled good, the rest 95 items which are labelled bad must have been labelled correctly. Thus the proportions of bad items labelled correctly is 95/2000 or 0.0475

Now moving towards good items. There are a total of 1900 good items. 5% of them are mislabelled as bad. In numbers 5% of 1900 or 95 items are labelled bad but they are good. In proportions it represents 95/2000 or 0.0475. Thus there are 0.0475 proportion of items which are labelled bad but the labelling is incorrect.

Now this leaves a total of 1900-95 or 1805 items which are good and are labelled correctly. In proportion it represents 1805/2000 or 0.9025. Now since we have all proportions, we can create the required table.