In Any Production Process in Which One or More Workers $( y )$

question 28

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 $\left( x _ { 1 } \right)$ and the number of checks cashed per day $\left( x _ { 2 } \right)$ . 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:

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array} { l r r r r r } \text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \text { F VALUE } & \text { PROB > F } \\\\\text { MODEL } & 2 & 7089.06512 & 3544.53256 & 13.267 & 0.0003 \\\text { ERROR } & 17 & 4541.72142 & 267.16008 & & \\\text { C TOTAL } & 19 & 11630.78654 & & &\end{array}\end{array}$

$\begin{array}{llll}\text { ROOT MSE } & 16.34503 & \text { R-SQUARE } & 0.6095 \\\text { DEP MEAN } & 93.92682 & \text { ADJR-SQ } & 0.5636 \\\text { C.V. } & 17.40188 & &\end{array}$

Parameter Estimates
PARAMETER STANDARD T FOR 0:
VARIABLE DF ESTIMATE ERROR PARAMETER $=0 \quad$ PROB $>|\mathrm{T}|$

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




$\begin{array}{rrrrrrrr} & & & \text { Actual } & \text { Predict } & & \text { Lower 95\% CL } & \text { Upper 95\% CL } \\\text { OBS } & \mathrm{X} 1 & \mathrm{X} 2 & \text { Value } & \text { Value } & \text { Residual } & \text { Predict } & \text { Predict } \\1 & 7781 & 644 & 74.707 & 83.175 & -8.468 & 47.224 & 119.126 \\\hline\end{array}$

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 $\alpha = .05$ .

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Definitions:

Value of K

A specific value or constant used in calculations, varying in meaning across different mathematical or scientific contexts.

Cyclical Effect

A pattern of fluctuations observed in economic or business activities corresponding to the phases of the business cycle, such as expansion, peak, contraction, and trough.

Seasonal Effect

A periodic effect that recurs or fluctuates in a predictable pattern over a year due to the change in seasons.

Time Series

A sequence of data points collected or recorded at successive time intervals, often analyzed to identify trends, cycles, and forecasts.