# Mathematics

**QUALTIY CONTROL PROBLEMS**

1. **System Reliability**

A system is composed of five components in series, and each has a reliability of 0.96.

a) What is reliability of the series system?

If the system can changed to parallel arrangement with all five components,

b) What is reliability of the parallel system?

c) What is the change in the reliability?

2. **Failure Rate**

In a failure rate test of a particular medical device, 10 items are set for a 60-h run. During the test run, 3 items failed at 25, 27, and 35 h.

a) Determine the failure rate?

b) Consider the test continued till 80-h run. Each time a part was failed, it was replaced by another one. In addition to the 3 failure (above described), two more devices failed during the next 20-h run. Determine the failure rate by the end of 80 hours?

c) Determine the mean life of above two failure rate determined in (1.a) and (1.b)

**3. Machine Failure Analysis**

What is the reliability of an engine at 5000 hour if the failure pattern of the machine follows Weibull distribution with shape parameter, β = 1.4. The mean life of machine during the early failure (at debugging) phase is 6260 hours.

Hints:

4. **Probability**

Binomial Distribution

Using the binomial distribution, find the probability of obtaining 2 or more nonconforming units when sampling 5 computers from a batch known to be 6% nonconforming.

Hints: use P(d) = n!/[d!(n-d)!] pd q(n-d) formula

a) Find *d* = 1 and 2 *i.e.*, the value for P(1) and P(2).

b) P(1) = n!/[1!(n-1)!] p1 q(n-1) {also replace n with a total number}

c) P(2 or more) = P(T) – P(0) – P(1), where P(T) = 1

**Design of Experiment**

5. **Design of Experiment**

In a sample of five (piston) cylinders, the diameter found was 15.0, 15.1, 15.8, 15.3 and 15.5 cm. Determine the following:

(i) Sum of squares (),

(ii) Mean square (variance) () and

(iii) Standard deviation () of the data.