A quality analyst wants to construct a control chart for determining whether four machines, all producing the same product, are in control with regard to a particular quality attribute. Accordingly, she inspected 1,000 units of output from each machine in random samples, with the following results: Machine Total Defectives #1 23 #2 15 #3 29 #4 13 What is the estimate of the standard deviation of the sampling distribution of sample proportions for this process?

2) Estimate of the process proportion of defectives is the average of the proportion of defectives from all samples. In this case, it is : (23+15+29+13)/{4*(1000)}=80/4000=0.02.

3) Estimate of the Standard Deviation: Let us denote the mean (average) of the proportion of defectives by p. Then, the estimate for the standard deviation is : sqrt{p*(1 - p)/n}. Where n is the sample size. Putting p = 0.02, and n = 1000, we get: σ=0.0044.

4) The control Limits for this case, at Alpha risk of 0.05 (i.e. equivalent to 95% confidence interval), can be found out using the formulas given below:

Lower Control Limit : p - (1.96)*σ = 0.02 - (1.96)*0.0044=0.0113.

& Upper Control Limit: p + (1.96)*σ = 0.02 + (1.96)*0.0044 = 0.0287.

5) The proportion defective in each case is : Machine #1: 0.023; Machine #2: 0.015; Machine# 3: 0.029; Machine# 4: 0.013. For the Lower & Upper control limits of 0.014 & 0.026; It is easy to see that Machines #3 & #4 appear to be out of control.