The Personnel Department of the ACME Textile Co. administers an aptitude test to all prospective assembly line employees. The average number of garments per hour that an employee can produce is approximately proportional to the score received on the aptitude test. In particular, the relationship is represented by

\(Y_{i}=\mathbf{x}_{i} \beta+\varepsilon_{i}\)

where

\(y_{i}=\) average number of garments/hour produced by employee \(i\),

\(x_{i}=\) score of employee \(i\) on aptitude test,

\(\beta=\) proportionality factor,

\(\varepsilon_{i}=\) error term, representing the deviation between actual average number of garments per hour produced by employee \(i\) and the production level implied by \(x_{i} \beta\).

You may assume that the \(\varepsilon_{i}^{\prime}\) s are independent, and you may also assume that \(\mathrm{E}\left(\varepsilon_{i}ight)=0\) and \(\mathrm{E}\left(\varepsilon_{i}{ }^{2}ight)=\sigma^{2}, \forall i\). Suppose you had the outcome, \(\left\{y_{1}, y_{2}, \ldots, y_{n}ight\}\), of a random sample of average production rates for \(n\) employees, together with their associated scores, \(\left\{x_{1}, x_{2}, \ldots, x_{n}ight\}\), on the aptitude test.

(a) Should the random sample \(\left\{Y_{1}, \ldots, Y_{n}ight\}\) be interpreted as a random sample from some population distribution, or should it be interpreted as a random sample generated from a general experiment? (Note: It cannot be expected that the aptitude scores will all be the same.)

(b) Derive the functional form of the least-squares estimator of the proportionality factor, \(\beta\). Is the least squares estimator BLUE in this case? Is it the MVUE of \(\beta\) ?

(c) Presuming that you could increase the sample size without bound, is the estimator you derived in

(b) a consistent estimator of \(\beta\) ? Is it asymptotically normally distributed? Justify your answer, being explicit about any assumptions you have made about the behavior of the \(x_{i}\) values and/or the \(\varepsilon_{i}\) values.

(d) From a sample of size \(n=100\), the sample outcome resulted in

\(\sum_{i=1}^{100} x_{i} y_{i}=92,017\) and \(\sum_{i=1}^{n} x_{i}^{2}=897,235\)

Use the estimator you derived in

(b) to generate an estimate of \(\beta\).