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You're in charge of an operation that manufactures steel dowels with a specification of 100 mm +/- 1 mm in length. You're concerned about the quality in your process and have taken the following samples. What is the probability of exceeding the maximum dimension?

Sample	Measurement
1	100.15
2	99.85
3	99.75
4	99.15
5	100.25
6	101.05
7	98.9
8	100.15
9	100.05
10	99.95
11	98.85
12	100.5
13	100.75
14	100.65
15	99.75
16	100.05
17	99.25
18	101.15
19	98.95
20	99.95
21	100.15
22	100
23	100.15
24	99.95
25	101.05

Answer 1

From the given data of 25 samples :

Mean process data = m =100.016

Sample standard deviation = Sd = 0.6451

Note: we have placed all data in excel and have used the formula AVG ( ) and STDDEV.S() to derive values of process mean and sample standard deviations respectively

Following are given :

Upper specification limit = USL = 101 mm

Lower specification limit = LSL = 99 mm

Thus maximum dimension = 101

Let z value corresponding to probability of process output to be maximum of 101= Z1

Therefore ,

M = Z1 x Sd = 101

Or, 100.016 + 0.6451.Z1 = 101

Or, 0.6451.Z1 = 0.984

Or, Z = 1.5253 ( 1.53 rounded to 2 decimal places )

Value of probability for Z = 1.53 as derived from standard normal distribution table=0.93699

Thus, probability that process output will be maximum 101 = 0.93699

Hence, Probability of exceeding maximum dimension

= 1 – Probability that process output will be maximum 0.93699

= 0.063

PROBABILITY OF EXCEEDDING MAXIMUM DIMENSION = 0.063