Question

Random samples of size n = 420 are taken from a population with p = 0.10.

Calculate the centerline, the upper control limit (UCL), and the lower control limit (LCL) for the p¯p¯ chart. (Round the value for the centerline to 2 decimal places and the values for the UCL and LCL to 3 decimal places.)

Centerline

Upper Control Limit - ?

Lower Control Limit - ?

Answer 1

The centerline of the ¯pp chart is the population proportion p, with the upper and lower control limits calculated using the z-score and the standard deviation of the sample proportion. A 95% confidence level corresponds to a z-score of 1.96, which is used unless a different confidence level is specified.

Step-by-step explanation:

To calculate the centerline (CL), upper control limit (UCL), and lower control limit (LCL) of the ¯pp chart, we use the formulas CL = p, UCL = p + z*sqrt(pq/n), and LCL = p - z*sqrt(pq/n), where p is the population proportion, q = 1 - p, and z is the z-score corresponding to the desired confidence level. Since we don't have a specific confidence level provided in the question, we can assume a standard 95% confidence level, which corresponds to a z-score of 1.96. If we were given a different confidence level, the z-score would change accordingly.

Using the given sample size (n = 420) and population proportion (p = 0.10), we can find q as 1 - p = 0.90. Now, we calculate the standard deviation of the sample proportion, σ = sqrt(npq), to use in the UCL and LCL formulas. The centerline (CL) will be equal to the population proportion (p).

Please note that the confidence interval around p can only be used if both np and nq are greater than five, ensuring the normal approximation to the binomial distribution is valid.