Question

 

A domestic manufacturer of watches purchases quartz crystals from a Swiss firm. The crystals are shipped in lots of 1,000. The acceptance sampling procedure uses 15 randomly selected crystals

If

p₀ = 0.01

and

p₁ = 0.08,

what are the producer's and consumer's risks for each sampling plan in part (a)? (Round your answers to four decimal places.)

c	Producer's Risk	Consumer's Risk
0
1
2

Answer 1

Producers risk

The producers risk is the risk of rejecting a sample when the quality is acceptable

When c=0 the lot will be accepted if there are zero defects. n=15 and p = 0.01. The probability can be found out using the Binomial theorom. The probability of acceptance is given by

P ( X=0) = 15C0* (0.01)⁰ (0.99)¹⁵ = 1* 1*0.86 = 0.86

Hence probability of rejection when the lot is actual good is 1-0.86=0.14 or 14% This is the producers risk

When c=1 the probability of acceptance is

P(X=0) + P (X=1)

= 0.86 + 15C1* 0.01*0.99¹⁴ = 0.86 +15*0.01*0.868 = 0.86 + 0.13 = 0.99

probability of rejection = 1-0.99 =0.01 or 1%

When c=2

Required probability = P(X=0) + P(X=1) + P(X=2)

P(X=2) = 15C2(0.01)²*0.99¹³= 105*0.0001*0.877 = 0.0092

Prob = 0.86+ 0.13+ 0.009 = 0.999 =99.9%

Probability of rejection or producers risk = 1-0.999 = 0.001 or 0.1%

The producers risk =1-0.99 = 0.01 or 1%

Consumers Risk

Here the probability is that of a type II error or accepting a lot when the quality if bad or p=0.08

When c=0

Probability = 15C0(0.08)⁰*0.92¹⁵= 0.2862

When c=1

Probability = P(X=0) + P (X=1)

P(X=1) = 15*0.08*0.31=0.372

Total probability = 0.2862+ 0.372 =0.6582

When c=2

P(X=0) + P(X=1) + P(X=2)

P(X=2) = 0.2272

Total probability = 0.6582+ 0.2272= 0.8854

The table is as follows

	Producer Risk	Consumer Risk
c=0	0.14	0.2862
c=1	0.01	0.6582
c=2	0.001	0.8854