7.2.4.2. Confidence intervals for small sample sizes

7. Product and Process Comparisons
7.2. Comparisons based on data from one process
7.2.4. Does the proportion of defectives meet requirements?

7.2.4.2. Confidence intervals for small sample sizes

Exact binomial confidence limits

If the sample size, N, is small, confidence limits that are approximated using the normal distribution may not be accurate enough for some applications. An "exact" method based on the binomial distribution is discussed on this page.

Constrution of exact two-sided confidence intervals based on the binomial distribution

To construct a two-sided confidence interval at the 100(1 -

)% confidence level for the true proportion defective p where N_d defects are found in a sample of size N follow the steps below.

Solve the equation

$SUM[k=0 to Nd][(N k)*p(U)**k*(1-p(U))**(N-k)] \= alpha/2$
for p_U to obtain the upper 100(1-)% limit for p.
Next solve the equation
$SUM[k=0 to Nd-1][(N k)*p(L)**k*(1-p(L))**(N-k)] \= 1 - alpha/2$

for p_L to obtain the lower 100(1 - )% limit for p.

Note

The interval {p_L, p_U} is an exact 100(1 - alpha

)% confidence interval for p. However, it is not symmetric about the observed proportion defective, phat = Nd/N

Example of calculation of upper limit for binomial confidence intervals using EXCEL

The equations above that determine p_L and p_U can easily be solved using functions built into EXCEL. Take as an example the situation where twenty units are sampled from a continuous production line and four items are found to be defective. The proportion defective is estimated to be phat

= 4/20 = 0.20. The calculation of a 90% confidence interval for the true proportion defective, p, is demonstrated using EXCEL spreadsheets.

Upper confidence limit from EXCEL

To solve for p_U:

Open an EXCEL spreadsheet and put the starting value of 0.5 in the A1 cell.
Put =BINOMDIST(Nd, N, A1, TRUE) in B1, where Nd = 4 and N = 20.
Open the Tools menu and click on GOAL SEEK. The GOAL SEEK box requires 3 entries.
- B1 in the "Set Cell" box
- /2 = 0.05 in the "To Value" box
- A1 in the "By Changing Cell" box.
The picture below shows the steps in the procedure.

Excel spread sheet showing initial values

Final step

Click OK in the GOAL SEEK box. The number in A1 will change from 0.5 to P_U. The picture below shows the final result.

Excel spread sheet showing the computation of P(U)

Example of calculation of lower limit for binomial confidence limits using EXCEL

The calculation of the lower limit is similar. To solve for p_L:

Open an EXCEL spreadsheet and put the starting value of 0.5 in the A1 cell.
Put =BINOMDIST(Nd -1, N, A1, TRUE) in B1, where Nd -1 = 3 and N = 20.
Open the Tools menu and click on GOAL SEEK. The GOAL SEEK box requires 3 entries.
- B1 in the "Set Cell" box
- 1 - /2 = 1 - 0.05 = 0.95 in the "To Value" box
- A1 in the "By Changing Cell" box.
The picture below shows the steps in the procedure.

Excel spread sheet showing the initial values

Final step

Click OK in the GOAL SEEK box. The number in A1 will change from 0.5 to p_L. The picture below shows the final result.

Excel spread sheet showing the comptation of p(L)

Interpretation of result

A 90% confidence interval for the proportion defective, p, is {0.071, 0.400}. Whether or not the interval is truly "exact" depends on the software. Notice in the screens above that GOAL SEEK is not able to find upper and lower limits that correspond to exact 0.05 and 0.95 confidence levels; the calculations are correct to two significant digits which is probably sufficient for confidence intervals. The calculations using a package called SEMSTAT agree with the EXCEL results to two significant digits.

Calculations using SEMSTAT

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

SEMSTAT output for exact confidence interval

Calculations using Dataplot

This computation can also be performed using the following Dataplot program.

. Initalize
let p = 0.5
let nd = 4
let n = 20
. Define the functions
let function fu = bincdf(4,p,20) - 0.05
let function fl = bincdf(3,p,20) - 0.95
. Calculate the roots
let pu = roots fu wrt p  for p = .01 .99
let pl = roots fl wrt p  for p = .01 .99
. print the results
let pu1 = pu(1)
let pl1 = pl(1)
print "PU = ^pu1"
print "PL = ^pl1"

Dataplot generated the following results.

 PU = 0.401029
 PL = 0.071354