# Food Protection Trends - July/August 2017 - 285

```APPENDIX B. DEVELOPMENT OF SAMPLE SIZES FOR MEDIAN AND RANGE/d2 BY MINITAB
SIMULATION

Sample Size Calculations
* Goal: determine the core temperature sample size for a variable sampling plan with a lower specification limit (LSL) and
the following specifications:
* RQL = 1% and β = 0.05. Based on these values, we can be 95% confident that at least 99% of the population is good when
we accept the lot.
* Core temperature follows a normal distribution.
* Instead of estimating μ with the sample mean, it is estimated with the sample median.
* Instead of estimating σ with the sample standard deviation, it is estimated with range/d2.
* The k factor equals 3. In other words, the decision rule has the form
Accept the lot if median -3 range > LSL
d2
Simulation Steps
1. Simulate 10,000 samples of size n from a normal distribution with a mean 2.32635 μ above the LSL. This results in
exactly 1% of the population out-of-spec. When the true defect rate is equal to the RQL (1%) and β = 0.05, we should
accept the lot 5% of the time and reject the lot 95% of the time.
2. Store the median and range for each of the 10,000 samples.
3. Estimate σ by dividing the range by d2.
4. Calculate the percent of times the simulation accepts the lot and repeat steps 1-4 for different sample sizes (n) until the
range
percent of times the lot is accepted (median -3
d2 > LSL) equals 5%. In other words, when the true defect rate is
equal to the RQL, as is the case here, we want to find the n so that we accept the lot 100 β% of the time and reject the
lot 100 (1-β)% of the time.
To do this for an RQL = 5% and β = 0.05, replace step 1 with:
Simulate 10,000 samples of size n from a normal distribution with a mean 1.64485σ above the LSL. This results in exactly
5% of the population out-of-spec. When the true defect rate is equal to the RQL (5%) and β = 0.05, we should accept the lot
5% of the time and reject the lot 95% of the time.
Comparison of n = 15 to n = 60 for Variables Plan RQL = 1% and Beta = 0.05

Median - 3*Range/d2

Example of Simulation
The graph below shows a histogram of the simulated median-3

range
values for n = 15 and n = 60. The distribution for
d2

core temperatures in the simulation had a defect rate of 1%. When n = 60, the lot is accepted approximately 5% of the time and
rejected approximately 95% of the time. Therefore, 60 is the correct sample size for an RQL = 1% and β = 0.05. When n = 15,
the lot is incorrectly accepted much more than 5% of the time. This high β (or Type II error) indicates that a sample size of 15
is too small.
July/August Food Protection Trends

285

```

Contents
Food Protection Trends - July/August 2017 - Cover1
Food Protection Trends - July/August 2017 - Cover2
Food Protection Trends - July/August 2017 - Contents
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Food Protection Trends - July/August 2017 - Cover3
Food Protection Trends - July/August 2017 - Cover4
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