1 Using JMP® to Develop a Sample Design to Measure Supplier Standards Compliance Ned Jones, Statistician, 1-alpha Solutions Wake Forest, NC Abstract Assumptions & Factors Results Assume the probability of an unfavorable response is equal across all 6 questions, Assume binary responses (respondents will give clear yes/no answers), Assume 100% timely response to survey. Factor consider the effect of responses accuracy. 0.114 Probability of non-compliance based on binomial distribution & 100% response accuracy Sample sizes with 100% response accuracy Response accuracy has a direct effect on compliance measures and survey margin of error A large national business needed to sample 1300 service suppliers to evaluate their compliance with standards. The suppliers all provide similar services. The sampled suppliers will be asked to respond to 6 questions. The responses will be yes/no. The business believes ten percent of the questions will have an unfavorable response. Suppliers in compliance will have 5 or more favorable responses. What is the probability of a non-compliant supplier? What assumptions were made? What other factors need to be considered in this evaluation? What sample size is needed to measure the supplier compliance rate with in a chosen margin of error? The JMP® profiler was used to investigate these questions and more. Sample sizes were developed based on the normal distribution and the hypergeometric distribution. Margin of error Sample size .95 confidence based on normal distribution Sample size .95 confidence based on hypergeometric distribution 2% 585 559 3% 351 305 4% 227 188 Objectives Estimate the probability of a noncompliant supplier. Estimate the sample size needed to be within a 2, 3 & 4% margin of error. Demonstrate the effects inaccurate responses. Conclusions As expected the hypergeometric distribution provided smaller sample. Verifying response accuracy should be included as a part of survey follow-up. The JMP® Profiler provided a valuable tool to develop sample size. Method Supplier compliance estimated using binomial distribution. Sample size estimates using both normal & hypergeometric distribution Use the JMP® Profiler develop sample size. References Cochran, W. G. (1977). Sampling Techniques, WILEY. SAS_Institute_Inc. (2011). JMP.
2 Using JMP® to Develop a Sample Design to Measure Supplier Standards Compliance Ned Jones, Statistician, 1-alpha Solutions Wake Forest, NC Probability of Non-compliance The probability of non-compliance was easily calculated using the JMP® binomial function. In the survey each supplier will asked to respond to 6 questions. It is expected that there will be a 10% chance of an unfavorable response, pr(neg) for each of the questions. We assume the probability of an unfavorable response is equal across all 6 questions. Suppliers in compliance will have 5 or more favorable survey responses. In the binomial function the answer accuracy (z) was applied (multiplied) directly to p(neg). Applying these facts in the binomial function we get the probability of a compliant supplier pr(pass) as follow: Pr(pass)=Binomial Distribution(p(neg)(accuracy (z)), 6,1), Where according JMP® : the Binomial Distribution(p, n, k), Returns the cdf for the binomial distribution with n trials, probability p of success for each trial, and k successes. If we subtract this result from 1 we get the probability of a non-compliant supplier survey response. The image to the right is of the profiler used to present the probability of a non-compliant supplier. We can see that as the probability of negative response to survey questions, pr(neg), increases the probability of a non-compliant supplier, pr(fail) increases. In the profiler image we can see the effect of less than 100% accuracy (z). As accuracy decreases the probability of a non-compliant, pr(fail), declines and the probability of compliance, pr(pass) increases.
3 Using JMP® to Develop a Sample Design to Measure Supplier Standards Compliance Ned Jones, Statistician, 1-alpha Solutions Wake Forest, NC Sample Development – Normal Distribution The sample development using normal distribution takes advantage of Cochran’s normal approximation for the binomial distribution. The approximation is as follows: Where p=pr(fail), the probability of a non-compliant supplier survey respondent, q= pr(pass), the probability of a compliant supplier survey respondent, n= the sample size, N= the population of suppliers and t(0.95) = the p-th quantile from the Student’s t distribution with degrees of freedom df. NonCentrality defaults to 0. . The portion of Cochran’s approximation, the expression after the +/- is set equal to the margin of error. We know N and p & q from p(neg) so the equation can be solved for, n, the sample size by using an iterative approach. If the input variables, p, q & N and Cochran approximation expression for margin of error are entered into a JMP® file we can view the expression relationship in the profiler. After setting the known inputs we can adjust n to find the sample size for the desired margin of error. An image of the profiler is as follows: Response accuracy shows a positive relationship with margin of error as accuracy increases the margin of error increases; however, as p(neg) increases the accuracy relationship changes eventually taking on the form of the p(neg) relationship. Also as accuracy decreases the p(neg) relationship changes eventually taking the form of the accuracy relationship.
4 Using JMP® to Develop a Sample Design to Measure Supplier Standards Compliance Ned Jones, Statistician, 1-alpha Solutions Wake Forest, NC Sample Development – Hypergeometric Distribution The sample development using hypergeometric distribution takes advantage of the features of the JMP® hypergeometric distribution function. The JMP® hypergeometric distribution function is as follows: Hypergeometric Distribution(N, K, n, x,