$6\mathrm{.}8.$ The Long Life Insurance Company receives applications to buy insurance from its salespeople, who are specially trained in selling insurance to new customers. After the applications are received, they are processed through a computer. The computer is programmed so that it prints messages whenever it runs across an item that is not consistent with company policies. The company is concerned with the accuracy of the training that its salespeople receive, and it contemplates recalling them for more training if the quality of their performance is below certain limits$.$ Five samples of $20$ applications received from specific market areas were collected and inspected with the following results:

$\backslash$table$[[$Sample$,$No$.$ of Applications with Errors$],[1,2],[2,2],[3,1],[4,3],[5,2]]$

a$.$ Determine the upper and lower control limits for a $p-$chart using a sample size of $20.$

b$.$ After the control limits were established, a sample was taken and four applications were found to have mistakes. What can we conclude from this?