The Six Sigma standard has been used for quality control in

The Six Sigma standard has been used for quality control in business and manufacturing.   While it is implemented differently in different situations, it uses the standard normal distribution to arrive at an expectation as to how many or what percentage of items will fall outside of 3 standard deviations on either side of the mean on some important dimension such as size, temperature, pressure tolerance, etc. Simply put, even when a manufacturing process is working properly, there is variation in the product produced. We have learned that for a normal distribution, we expect 99.7% of all values to fall within 3 standard deviations on either side of the mean (within the middle 6 standard deviations). Thus we expect .3% to fall more than 3 standard deviations from the mean. If more than .3% of a product falls outside of 3 standard deviations on either side of the mean on some important measure, assume it is likely that the manufacturing process needs to be inspected and perhaps adjusted.

You are a fan of Levi’s 501 jeans and that you always buy a size 30 inch waist. Levi’s has published data saying that the average waist size for their size 30 inch jeans is 30 inches with a standard deviation of .2 inches. They say the distribution is normally distributed.

Lately you have noticed that the size 30s do not always fit you well even though you have not gained or lost weight. Then you read an article in a popular consumer magazine that describes a study done by Levi Strauss Company after hearing concerns about large variation in the sizing of their jeans. Below is a summary of what they found.

Levi Strauss randomly sampled 6000 pairs of their size 30 inch 501 jeans as they came off the assembly line. Assume they used the expectation derived from the standard normal distribution to define the number of jeans that should have waist measurements less than 29.4 inches or greater than 30.6 inches. They found that 32 pairs of the size 30 inch jeans in the sample had waist measurements less than 29.4 inches and 35 pairs of the size 30 inch jeans in the sample had waist measurements greater than 30.6 inches.

According to the six sigma standard, if the manufacturing process is working properly, how many pairs of jeans in the sample of 6000 would you expect to find with waist measurements less than 29.4 inches? How many would you expect to find with waist measurements greater than 30.6 inches?

What would you conclude about the manufacturing process based upon the sample of 6000 pairs reported in the study? Is it working properly or does it need to be calibrated? Explain using the expectations of the normal distribution.

Solution

Levi Strauss randomly sampled 6000 pairs of their size 30 inch 501 jeans as they came off the assembly line. Assume they used the expectation derived from the standard normal distribution to define the number of jeans that should have waist measurements less than 29.4 inches or greater than 30.6 inches. They found that 32 pairs of the size 30 inch jeans in the sample had waist measurements less than 29.4 inches and 35 pairs of the size 30 inch jeans in the sample had waist measurements greater than 30.6 inches.

We expect .3% to fall more than 3 standard deviations from the mean. If more than .3% of a product falls outside of 3 standard deviations on either side of the mean on some important measure, assume it is likely that the manufacturing process needs to be inspected and perhaps adjusted.

We expect 0.15% to fall more than 3 standard deviations from the mean and 0.15% to fall less than 3 standard deviation

0.15% of 6000 is 9

( 0.15/100)*6000 =9

We expect to find with waist measurements less than 29.4 inches is 9 and we expect to find with waist measurements greater than 30.6 inches is 9.

We found that 32 pairs of the size 30 inch jeans in the sample had waist measurements less than 29.4 inches and 35 pairs of the size 30 inch jeans in the sample had waist measurements greater than 30.6 inches.

We conclude about the manufacturing process based upon the sample of 6000 pairs reported in the study is not working properly and it need to be calibrated.