The diameters of blank compact disks manufactured by the Dandy Disk Co. can be represented as outcomes of a random variable

\(Z \sim f(z ; \Theta)=\frac{1}{\Theta} I_{(4,4+\Theta)}(z)\), for some \(\Theta>0\), where \(z\) is measured in inches. You will be using a random sample \(\left(X_{1}, X_{2}, \ldots X_{n}ight)\) from the population distribution \(f(z ; \Theta)\) to answer the questions below.

a. Based on the random sample, define an unbiased estimator of the parameter \(\Theta\).

b. Is the estimator you defined in

(a) a BLUE for \(\Theta\) ? If not, find a BLUE for \(\Theta\), if it exists.

c. Is your estimator a consistent estimator for \(\Theta\) ? Why or why not?

d. Define an asymptotic distribution for your estimator.

e. A random sample of size \(n=1,000\) from \(f(z ; \Theta)\) results in \(\sum_{i=1}^{1000} x_{i}=4,100\). Use your estimator to estimate the value of \(\Theta\). Using your estimate of \(\Theta\), what is the estimated probability that \(z \in(4.05,4.15)\) ?