$1.$ Given the following process control data for a quality attribute $($three samples of size $400$ each$)$:

Sample Defectives

#$136$

#$232$

#$352$

a$)$ If the process or population proportion of defectives is unknown, what is its estimated value $?($Note: The population or process proportion is NOT given, but we estimate it by using the expected value of the sample proportion$-$simply average the given sample proportions and this is your process or population proportion

b$)$ Hypothetically speaking, if this process is known to produce $11$ percent defectives on average, using $3\backslash$sigma control limits$,$ do any of the sample proportions indicate that the process is out of control, viz., that there is the existence of assignable$-$cause variation $?($Use the binomial distribution$$s mean and S$.$D$.$ for calculating the upper and lower control limits$)$

$2.$ What is the operational implication of the Type $1$ error or so$-$called $$Alpha Risk$?$ Depict what we mean by Type $1$ error on a diagram of the standard normal distribution. Please ensure that you precisely label the Critical Region $($set of values of the sample statistic pertinent to rejection of H$0,$ the null hypothesis$).$ Explain the primary difference between Type $1$ and Type $2$ errors and which of them is likely to cause a greater problem for an Operations Manager $/$Industrial Engineer $?$