# Example of Sample Size for Tolerance Intervals

A quality engineer at an automotive parts plant wants to assess the variability in the thickness of round metal washers. The engineer plans to measure a sample of washers and calculate a tolerance interval that includes 95% of the population. If the sample size is small, then the maximum acceptable percentage of population in the interval may be too large and the tolerance interval may greatly overestimate the variability in the thickness of the washers. From historical data, the engineer assumes that the data are normally distributed.

The engineer wants to determine the sample size of washers that is necessary to measure to achieve maximum acceptable percentages of population in the interval of 96% and 97% for the tolerance interval. The engineer also wants to know the maximum acceptable percentages for sample sizes of 50 or 100 washers. The engineer can assume that the data are normally distributed.

## To calculate sample sizes

1. Choose Stat > Power and Sample Size > Sample Size for Tolerance Intervals.
2. Select Calculate sample sizes.
3. In Minimum percentage of population in interval, enter 95.
4. In Maximum acceptable percentages of population in interval (p*), enter 96 97.
5. Click OK.

## To calculate margins of error

1. Choose Stat > Power and Sample Size > Sample Size for Tolerance Intervals.
2. Select Calculate maximum acceptable percentages of population in interval (p*).
3. In Minimum percentage of population in interval, enter 95.
4. In Sample sizes, enter 50 100.
5. Click OK.

## Interpret the results when calculating sample sizes

With the normal method, to achieve a maximum acceptable percentage of population in interval of 96%, the engineer needs to collect 2480 observations. With 2480 observations, the probability that a tolerance interval coverage exceeds 96% of the population is only 0.05.

If the engineer is willing to accept a maximum acceptable percentage of population in interval of 97%, the sample size can be reduced to 525 observations.
###### Note

If the engineer cannot assume normality, the sample sizes will be much higher with the nonparametric method.

### Sample Size for Tolerance Intervals

Method Confidence level 95% Minimum percentage of population in interval 95% Probability the population coverage exceeds p* 0.05
Sample size for 95% Tolerance Interval Normal Nonparametric Achieved Achieved Error P* Method Method Confidence Probability 96.000% 2480 4654 95.0% 0.049 97.000% 525 1036 95.1% 0.048 P* = Maximum acceptable percentage of population in interval Achieved confidence and achieved error probability apply only to nonparametric method.

## Interpret the results when calculating maximum acceptable percentage of population in interval

When the engineer specifies the target sample sizes, Minitab calculates the maximum acceptable percentages of population in interval. With the probability the population coverage exceeds p* equal to 0.05 (5%), the maximum acceptable percentage for the normal method is 99.4015% when the sample size is 50. When the sample size is 100, the maximum acceptable percentage is 98.6914%.
###### Note

If the engineer cannot assume normality, the maximum acceptable percentages of population will be higher with the nonparametric method.

The engineer might decide that the maximum acceptable percentage is too high and might rerun the analysis using larger sample sizes to decrease the maximum acceptable percentage. For example, the engineer could try 250 washers or 400 washers. However, the engineer knows from the first analysis that at least 525 washers are required to have a 5% probability that the tolerance interval contains no more than 97% of the population, assuming a normal distribution.

### Sample Size for Tolerance Intervals

Method Confidence level 95% Minimum percentage of population in interval 95% Probability the population coverage exceeds p* 0.05
Maximum Acceptable Percentages of Population for 95% Tolerance Interval Sample Normal Nonparametric Achieved Achieved Error Size Method Method Confidence Probability 50 99.4015% 99.2846% 72.1% 0.050 100 98.6914% 99.6435% 96.3% 0.050 Achieved confidence and achieved error probability apply only to nonparametric method.
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