In Any Production Process in Which One or More Workers $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$$

Essay

In any production process in which one or more workers are engaged in a variety of tasks, the total time spent in production varies as a function of the size of the workpool and the level of output of the various activities. In a large metropolitan department store, it is believed that the number of man-hours worked (y) per day by the clerical staff depends on the number of pieces of mail processed per day (x1) and the number of checks cashed per day (x2). Data collected for n = 20 working days were used to fit the model: $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$$
A printout for the analysis follows:

$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Analysis of Variance }$
$\begin{array}{llllll}\text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \text { FVALUE } & \text { PROB }>F\end{array}$

$\begin{array}{lrrrrr}\text { MODEL } & 2 & 7089.06512 & 3544.53256 & 13.267 & 0.0003 \\\text { ERROR } & 17 & 4541.72142 & 267.16008 & & \\\text { C TOTAL } & 19 & 11630.78654 & & &\end{array}$

$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Parameter Estimates }$

$\begin{array}{rrrrl} & \text { PARAMETER } & \text { STANDARD } & \text { TFOR 0: } \\\text { VARIABLE } \text { DF } & \text { ESTIMATE } & \text { ERROR } & \text { PARAMETER }=0 \quad \text { PROB }>|T|\end{array}$

$\begin{array}{lrrrrr}\text { INTERCEPT } & 1 & 114.420972 & 18.68485744 & 6.124 & 0.0001 \\\text { X1 } & 1 & -0.007102 & 0.00171375 & -4.144 & 0.0007 \\\text { X2 } & 1 & 0.037290 & 0.02043937 & 1.824 & 0.0857\end{array}$

$\begin{array}{lrrrrrrr}&&&\text { Actual}& \text {Predict }&&\text {Lower 95 \% CL}& \text { Upper 95 \% CL}\\\text { OBS } & \text { X1 } & \text { X2 } & \text { Value } & \text { Value } & \text { Residual } & \text { Predict } & \text { Predict } \\1 & 7781 & 644 & 74.707 & 83.175 & -8.468 & 47.224 & 119.126 \\\hline\end{array}$

Test to determine if there is a positive linear relationship between the number of man-hours worked, $y$ , and the number of checks cashed per day, $x _ { 2 }$ . Use $\alpha = .05$ .