Question

The part diameters are normally distributed. The lower tolerance limit corresponds to -2 z (minus two standard deviations below the mean). The upper tolerance limit corresponds to 3 z. What percent of parts will be out of tolerance? Describe the method you used to find the answers.

Answer 1

It is given that the part diameters are normally distributed, this means that we can apply the empirical formula in this case.

By empirical formula, we know that 99.7% of the data fall within 3 standard deviation below the mean and 3 standard deviation above the mean.

Now, tolerance range given in the question are -2z and 3z, i.e. 2 standard deviation below the mean and 3 standard deviation above the mean.

If it was -3z to +3z, then 99.7% of the parts will be within tolerance limit, but the tolerance limit given is -2z to +3z

So, we need to subtract the % of values that falls within -3z and -2z area on the normal distribution.

By empricial rule, we know that only 2.35% of the values fall within -2z and -3z. So, we can say that only 2.35% of parts will be out of tolerance as the range -2z to 3z contains remaining part diameter or 97.35% of part diameter which are within tolerance range.

We have used empirical formula or rule for normal distribution to find the % of parts that will be out of tolerance.