Question

1. The POM1 company produces lawnmowers in UK. The production level is around 2000 per day. To test for quality, n-40 lawnmowers are randomly selected from the production line, and two attempts are made to see if the mower starts by pulling the cord. The management wishes to use p-chart to monitor this crucial test. Date for the first month, consisting of 22 working days, are shown in the following Starting Failure Data for the First Month with n-40 Sample Number Day Number of Staring Failures Sample Fraction Nonconforming 0.050 0.075 0.025 0.100 0.075 0.050 0.025 0.025 0.000 0.075 0.050 0.100 0.175 0.050 0.075 0.075 0.050 0.200 0.000 0.025 0.075 0.050 p 0.0648 2 3 2 3 2 4 12 13 15 16 17 18 3 2 21 3 57

1. a) Develop a preliminary control chart and plot the sample values on your chart. Explain what you found.

2. b) After a thorough search for the records, it was found that the data recorded on Days 13 and 18 are not reliable. These two data are then eliminated. Redevelop p-chart from the new data set and plot the sample values on your chart again.

3. c) Estimate the probability that the value of a sample is zero.

4. d) A more functional control chart will occur when the probability of having zero no starts is decreased. The appropriate method is to raise the sample size n. Estimate the probability of having zero starts with n = 80 .

5. e) If the management wishes that the probability be not greater than 0.05, find the minimum sample size (you may use Poisson distribution).

Answer 1

Solution

Back-up Theory

p-Chart

Let Xi = number of defectives in the ith sample.

Estimate of the population fraction defective, pbar = (1/kn)Σ(i = 1, k)Xi, where k = number of samples and n = sample size.

Central Line (CL) = pbar ……………………………………………………………………. (1)

Lower Control Limit (LCL) = pbar - 3√{pbar(1 – pbar)/n} …………………………………. (2)

Upper Control Limit (UCL) = pbar + 3√{pbar(1 – pbar)/n} …………………………………. (3)

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and p = probability of one success, then probability mass function (pmf) of X is given by

p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2, ……. , n ……………………………………..(4)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST.(4a)

Now to work out the solution,

Given k = 22, n = 40, Σ(i = 1, k)Xi = 57 and pbar = 0.0648.

Part (a)

CL	0.064773	Vide (1)
LCL	0	Vide (2)
UCL	0.18152	Vide (3)

Answer 1

Observations

Two days, namely D18 and D13, Xi values are outside the UCL. And barring these two aberrations, no specific trend is seen. On other days the process is well under control. Answer 2

Part (b)

After eliminating the data of D18 and D13,

k	20
sum(Xi)	42
pbar	0.0525
CL	0.0525
LCL	0
UCL	0.158294
Control State

All pi's are within limits.

Answer 3

Part (c)

Xi ~ B(40, p), where p is estimated by the pbar obtained under Part (b).

So, probability that the value of a sample is zero

= P(Xi = 0)

= (⁴⁰C₀)(0.0525⁰)(0.9475)⁴⁰ [vide (4)]

= 0.1157 Answer

Part (d)

The probability of having zero starts with n = 80

= = P(Xi = 0)

= (⁸⁰C₀)(0.0525⁰)(0.9475)⁸⁰ [vide (4)]

= 0.0134 Answer

DONE