You are the newly appointed assistant administrator at a local hospital, and your first project is to investigate the quality of the patient meals put out by the food$-$service department. You conducted a $10-$day survey by submitting a simple questionnaire to the $400$ patients with each meal, asking that they simply check off that the meal was either satisfactory or unsatisfactory. For simplicity in this problem, assume that the response was $1,000$ returned questionnaires from the $1,200$ meals each day. The results are as follows.

$\backslash$table$[[,$NUMBER OF$,$SAMPLE$],[$UNSATISFACTORY$,$MEALS,SIZE$],[$December $1,42,1,000],[$December $2,47,1,000],[$December $3,63,1,000],[$December $4,48,1,000],[$December $5,60,1,000],[$December $6,62,1,000],[$December $7,43,1,000],[$December $8,62,1,000],[$December $9,66,1,000],[$December $10,53,1,000],[,546,10,000]]$

a$.$ Determine the $\frac{?}{\text{b}\text{a}\text{r}}(p),S_p,$ UCL and LCL based on the questionnaire results, using a confidence interval of $95\mathrm{.}5$ percent, which is two standard deviations. $($Round your answers to $4$ decimal places.$)$

$\backslash$table$[[$P$,0\mathrm{.}054d$