7. Product and Process Comparisons
7.2. Comparisons based on data vrom one process
7.2.7. How can we determine whether the proportion of defectives produced by a process has changed from the "nominal" value?

7.2.7.1.

Confidence limits

Confidence intervals are are another way of testing hypotheses

The results from an hypothesis test for proportions described in the previous section can be expressed in the form of a 100(1-a)% confidence interval as follows: 

where 

  = the sample proportion
p₀ = the population proportion
z_a/2 = the critical value from the normal distribution at a/2
n = sample size. 

Note: For a one sided hypothesis, we use z_a and a one sided confidence interval. 

Example

We construct a 90% one sided confidence interval for p₀ for the example from a previous page.

Since the hypothesized value of .10 is included in the confidence band we accept the null hypothesis of no difference.

Exact Binomial Confidence Bounds

When n is small, say less than 30, the normal distribution approximate confidence bounds used above may not be accurate enough for many applications. Fortunately, there is an exact confidence bound method that is not difficult to compute when n is small. 

As above, assume we want a 100(1-a)% confidence interval for the true p when x defects are found in a sample of size n. 

By solving the equation

for p_U, we obtain an upper 100(1-a/2)% limit for p. 

Next we solve

for p_L, and that gives us a lower 100(1-a/2)% limit for p. The interval (p_L, p_U) is an exact 100(1-a)% confidence interval for p.

Note: The exact confidence intervals constructed by the above method will not be symmetric around the point estimate (x/n)..

Calculation of exact binomial confidence bounds using EXCEL

Exact Binomial Confidence Bounds Using EXCEL

The equations above that determine p_L and p_U can easily be solved using functions built into EXCEL. The steps are as follows:

To solve for p_U 

1. Open an EXCEL spreadsheet and put the starting value of .5 in the A1 cell and put =BINOMDIST(X, N, A1, TRUE) in B1, where X and N are the number of defectives and the sample size, respectively.Your spreadsheet will look like the picture below.



2. Open the Tools menu and click on GOAL SEEK.

3. The GOAL SEEK box requires 3 entries: 

• Put B1 in the first "Set Cell" box
• Put a/2 in the second "To Value" box
• Put A1 in the third "By Changing Cell" box
Your spreadsheet will look like the picture below.



4. Click OK in the GOAL SEEK box and the number in A1 changes from .5 to  p_U. 
 

To Solve for p_L

1. Open an EXCEL spreadsheet and put the starting value of .5 in the A1 cell and put =BINOMDIST(X-1, N, A1, TRUE) in B1.

2. Open the Tools menu and click on GOAL SEEK.

3. The GOAL SEEK box requires 3 entries: 

• Put B1 in the first "Set Cell" box
• Put 1-a/2 in the second "To Value" box
• Put A1 in the third "By Changing Cell" box.

4. Click OK on the GOAL SEEK box and the number in A1 changes from .5 to p_L . 

Example of Computuing Exact Binomial Confidence Bounds

Assume we take a sample of 20 units off a continuous production line and find 4 defectives. is 4/20 = 0.20. A 90% confidence interval for p is calculated as shown in the series of spreadsheet shots below.

Upper Bound Calculation:

 

 

 

Lower Bound Calculation:

 

90% Confidence Interval:

An exact 90% confidence interval for p is (0.071, 0.400) 

Note: The downloadable software package SEMSTAT contains a menu item "Hypothesis Testing and Confidence Intervals" . Selecting this brings up a menu that contains "Confidence Limits on Binomial Parameter". That option can be used to quickly calculate exact binomial confidence limits as shown in the screen shot below:

 

Correction factor for sampling from a finite population

Finite Population Correction (fpc) Factor

Thus far, we assumed sampling from a continuous production line or a very large or infinite population. Occasionally, we sample from a relatively small population, and the sampling is done without replacement.  In this case, a correction factor, called the fpc factor, is applied.  The 100(1-a) % confidence interval estimate of the proportion becomes 

The last term, , is the fpc factor, with N = the finite population size and n = the sample size.

For example, if a department manager selects 200 items from a lot of 100,000 (e.g. the finite population), and 35 defectives are observed, the 90% confidence interval estimate is:

This, of course, is not too much of a correction. But if the lot size had been 1000, the correction factor would be fpc = 0.775, and the corresponding 90% confidence interval is from 0.142 to 0.208.