Question

the defect rate for a product (or p bar) has historically been about .016. what is the upper control limit for a p-chart, if you wish to use a sample size of 100 and 3-sigma limits? (answer to four decimal places)

Answer 1

The upper control limit (UCL) for a p-chart with a historical defect rate of 0.016 and a sample size of 100 using 3-sigma limits is calculated as 0.016 + 3*(sqrt(0.016*0.984/100)), resulting in a UCL of 0.0536.

Step-by-step explanation:

To calculate the upper control limit (UCL) for a p-chart with 3-sigma limits, we use the formula UCL = p-bar + z*(sqrt(p-bar*(1-p-bar)/n)), where p-bar is the historical defect rate, z is the z-value corresponding to the desired sigma level (3-sigma in this case), and n is the sample size.

Given that the historical defect rate p-bar is 0.016 and the sample size n is 100, we first need to find the z-value for 3-sigma limits in a standard normal distribution, which is z=3. Then, we substitute these values into the formula:

UCL = 0.016 + 3*(sqrt(0.016*(1-0.016)/100))
UCL = 0.016 + 3*(sqrt(0.016*0.984/100))
UCL = 0.016 + 3*(sqrt(0.015744/100))
UCL = 0.016 + 3*(sqrt(0.00015744))
UCL = 0.016 + 3*(0.012548)
UCL = 0.016 + 0.037644
UCL = 0.053644

Therefore, the upper control limit for the p-chart, to four decimal places, is 0.0536.

Answer 2

The upper control limit for a p-chart with a p-bar of 0.016, a sample size of 100, and 3-sigma limits is 0.0536.

The formula for the upper control limit (UCL) of a p-chart is:

UCL = p + (z * σ)

where:

p is the average defect rate (p-bar)

z is the number of standard deviations from the mean (usually 3 for a 3-sigma control chart)

σ is the standard deviation of the proportion, which is calculated as:

σ = √(p * (1 - p) / n)

where:

n is the sample size

Plugging in the given values, we get:

σ = √(0.016 * (1 - 0.016) / 100) = 0.013

UCL = 0.016 + (3 * 0.013) = 0.0536

Therefore, the upper control limit for a p-chart with a p-bar of 0.016, a sample size of 100, and 3-sigma limits is 0.0536.