A manufacturing process is in statistical control with an average mean of $\frac{?}{\text{b}\text{a}\text{r}}(\stackrel{}{x})=197\mathrm{.}5$ and average range of $\frac{?}{\text{b}\text{a}\text{r}}(R)=4$ on the quality characteristic of a manufactured part. The control chart uses a sample size of $n=4.$ Specifications of the quality characteristic are at $200+-7\mathrm{.}5$ and the values of the quality characteristic are normally distributed.

a$)$ Estimate the potential capability of the process.

b$)$ Based on part a$),$ what percentage of the total specification band is being used by the process?

c$)$ Estimate the actual process capability.

d$)$ What percentage of the manufactured parts are nonconforming at the current state?

e$)$ How much improvement could be made in the nonconforming percentage if the mean could be centered at the nominal value?