5. A control chart for a proportion

5. A control chart for a proportion is set up such that:The center line is the assumed true proportion of successes.The control limits are set at two standard deviations above and below the mean.As sample proportions are collected, they are graphed on the chart in the order that they are collected.a. A distribution that is labeled “out of control” occurs when:i. A sample proportion greater than 3 standard deviations from the mean is found.ii. Nine or more consecutive sample proportions are either all greater or all less than the assumed true proportion of successes.iii. At least two of three consecutive sample proportions greater than 2 standard deviations from the mean are found.Find the probability of each of the three events above occurring, assuming the samplingdistribution is actually centered at the true proportion of successes. (10)b. If the true proportion is supposed to be 60%, find the sample size needed to ensure thattwo proportions greater than 63% are enough to ensure an out-of-control distribution.