# Example of Tolerance Intervals (Nonnormal Distribution)

A manufacturer wants to evaluate the variability of their paper bleaching process. The manufacturer randomly collects 200 samples of pulp and records the brightness level. The manufacturer wants to calculate a range within which the brightness of at least 99% of the batches of pulp are expected to fall.

1. Open the sample data, PulpBrightness.MTW.
2. Choose Stat > Quality Tools > Tolerance Intervals (Nonnormal Distribution).
3. In Variables enter Brightness.
4. In Minimum percentage of population interval, enter 99.
5. Click OK.

## Interpret the results

The probability plot shows that the plotted points fall along the Weibull distribution fitted line, which indicates that the data follow a Weibull distribution. Also, the p-value of the goodness-of-fit test is 0.178, which is greater than the significance level of 0.05. Because you cannot conclude that the data do not follow the Weibull distribution, you can use the interval for the Weibull distribution.

The Weibull interval ranges from approximately 69.1 to 89.7, so the manufacturer can be 95% confident that at least 99% of all batches of pulp will fall in this interval. For all the batches of pulp, the mean brightness level is approximately 82.8.

### Tolerance Interval: Brightness

Method Distribution Weibull Confidence level 95% Percent of population in interval 99%
Statistics Variable N Mean StDev Brightness 200 82.757 3.358
Distribution Parameters Variable Shape Scale Brightness 30.171 84.252
95% Tolerance Interval Nonparametric Achieved Variable Weibull Method Method Confidence Brightness (69.059, 89.684) (70.570, 90.050) 59.54% Achieved confidence level applies only to nonparametric method.

### Tolerance Interval Plot for Brightness

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