the problem may look long but the first two paragraphs is just background about gloves.

the problem is:

quality-control monitoring procedures are almost always statistical. often, to check the

quality of a product requires that it be used up or even destroyed, so it's never

possible to do a quality check on every product. an agent must select a sample.

in the case of a product like latex surgical gloves, "quality" is related to consistent latex

thickness, and the relationship can be complex. for instance, a glove that is a little too

thick might slightly reduce the sensitivity of a surgeon's fingers, which could possibly

lead to the surgeon overlooking a clinical fact because he or she couldn't feel it

through the glove. this condition is obviously undesirable, so it's important that gloves

not be too thick. on the other hand (bad pun, sorry), a glove that's too thin might

break, bringing the risk of infection to surgeon and patient alike. this condition is close

to a disaster (much worse than simply undesirable), so it's absolutely imperative that

gloves not be too thin.

imagine a manufacturer of surgical gloves with latex thickness known to be

approximately normally distributed with mean thickness 244 microns (μm) and

standard deviation 13μm. according to the manufacturer's quality-control procedures,

latex in the 95th percentile or higher is defined to be "too thick," capable of reducing

sensation in a surgeon's fingertips. at the other extreme, latex thickness below 200μm

is defined to be "too thin," putting the glove at risk of breakage.

to monitor latex thickness in its product, a quality control agent selects a random

sample of 30 gloves from each production run. if none are found to be at risk of

breakage (zero tolerance for gloves that are too thin), and no more than three exceed

the maximum thickness standard, the agent will certify the quality of the entire

production run.

1. what's the maximum thickness allowed by the manufacturer's standard?

2. is there any reasonable chance that the average thickness of the gloves in the

agent's sample will exceed the maximum thickness allowed? why or why not?

3. what's the probability that a randomly selected glove

a) exceeds the maximum thickness standard?

b) is at risk of breakage?

c) meets both thickness standards?

4. what's the probability that a production run will fail the quality certification test?

Can you show me how to solve this problem including the work you did? It's due tomorrow, I need help.