April 6, 2020
 Quality Applications SPC Guide First Word Last Word

## A Colorful Solution, But Is It Correct? Hypothesis testing entails more than guessing.

Michael J. Cleary, Ph.D.
mcleary@qualitydigest.com

Walker Runn is quality manager for Color In A Can, a paint processing company with facilities in three states. He’s pleased to be assigned to the plant located farthest from company headquarters, hoping that an “out of sight, out of mind” sensibility will keep the home office from pestering him about things like using statistical process control, getting more training to do his work or a host of other things that are pebbles in his shoe.

The company sends a vice president to audit his quality system each year, but Runn has been able to sustain a façade of competence because the vice presidents that the home office sends have generally known little about statistical process control. In the past, he simply threw around a few impressive terms and charts to satisfy them. This time, however, the company has chosen to send Dan Druff, a highly competent (though somewhat flaky) statistician.

Thumbing through an old statistics textbook, Runn decides that the only way to convince Druff that the plant is producing consistently high-quality paint is to wow him with hypothesis testing. Because Runn has never done hypothesis testing himself, he finds that he must actually review the chapter to learn some terminology. A section entitled “Differences Between Means” draws his interest. In the associated case study, a plant has two identical production lines that produce identical products--not unlike the way Color In A Can is set up, with two lines that produce the same product.

As part of the corporate quality effort, each production line is rated on a quality index each day. Data for the two lines

follows:

Quality Index

 Line 1 Line 2 14 10 17 7 9 11 16 9 15 5 11 12

Line 1      Line 2

Runn collects the data and asks one of his employees to enter the data into a statistical software program so it can be presented to Druff in a major PowerPoint presentation that he has planned.

As he shares the presentation, he points to the high t value of 2.83.

“Aha,” he says. This demonstrates how different the lines are from each other, he points out to Druff, who nods and then asks Runn what alpha value might be used.

Oh-oh. “Selecting the Alpha” was in the part of the chapter that Runn hadn’t skimmed. The only connection with “alpha” that came to his mind derived from the sports car his neighbor just bought, an Alfa Romeo GTV6. Fishing for a response, he blurts out, “12,” because the sum of n1 and n2 is 12. Not fancy, but a fast calculation, he thinks to himself.

Was Druff impressed? Should he have been?

Runn was flat-out wrong. He had confused the sum of the two sample sizes with a type 1 error.

Data from the production lines were as follows:

Line 1      Line 2

The production on line 2 is clearly less than that of line 1. The question remains whether the difference is due to natural variation or whether it can be ascribed to the two lines actually operating differently. Using traditional hypothesis testing, one can apply the “t-test”:

Step 1:

Interpretation: The null hypothesis (H0) is that line 1 and line 2 are not significantly different.

Step 2:

Alpha value (a) = 0.01

Interpretation: An alpha value of 1 percent suggests a willingness to accept a 1 percent chance of rejecting the null when it’s actually true. This is known as a type 1 error.

Step 3: Calculate statistical t value:

Step 4: Make a decision:

a) Look up tabular t value in a statistics textbook. In this case, it’s equal to 2.2281. (Note: You have 10 degrees of freedom.)

b) Compare this to the value from step 3 of 2.83. If it’s greater, reject; if not, accept.

Interpretation: In this case, the mean values are different enough from each other that one would conclude that lines 1 and 2 are indeed different from one another.

How would X-MR charts created for each line compare?

If this exercise brings back dark memories of a statistics course and its innumerable calculations, welcome to the new technology.