Construct an R Chart and Determine Whether the Process Variation $\text { Control Chart Constants }$

question 90

Essay

Construct an R chart and determine whether the process variation is within statistical control.
Control Chart Constants $\text { Control Chart Constants }$
$\begin{array}{ccccccc}&\bar{x}&&s&&R\\\hline n & A_{2} & A_{3} & B_{3} & B_{4} & D_{3} & D_{4} \\\hline 2 & 1.880 & 2.659 & 0.000 & 3.267 & 0.000 & 3.267 \\3 & 1.023 & 1.954 & 0.000 & 2.568 & 0.000 & 2.574 \\4 & 0.729 & 1.628 & 0.000 & 2.266 & 0.000 & 2.282 \\5 & 0.577 & 1.427 & 0.000 & 2.089 & 0.000 & 2.114 \\6 & 0.483 & 1.287 & 0.030 & 1.970 & 0.000 & 2.004 \\7 & 0.419 & 1.182 & 0.118 & 1.882 & 0.076 & 1.924 \\8 & 0.373 & 1.099 & 0.185 & 1.815 & 0.136 & 1.864 \\9 & 0.337 & 1.032 & 0.239 & 1.761 & 0.184 & 1.816 \\10 & 0.308 & 0.975 & 0.284 & 1.716 & 0.223 & 1.777 \\\hline\end{array}$


A machine is supposed to fill cans that contain 12 ounces. Each hour, a sample of four cans is tested; the results of 15 consecutive hours are given below.

$\begin{array}{c|cccc|c|c}\text { Sample } &&&{\text { Can Volume (oz) }} && \overline{\boldsymbol{x}} & \text { Range } \\\hline 1 & 11.2 & 11.7 & 11.8 & 11.7 & 11.600 & 0.6 \\2 & 11.8 & 11.3 & 11.6 & 11.6 & 11.575 & 0.5 \\3 & 11.3 & 12.0 & 11.8 & 11.7 & 11.700 & 0.7 \\4 & 12.1 & 11.6 & 12.1 & 12.0 & 11.950 & 0.5 \\5 & 11.8 & 11.8 & 11.9 & 11.9 & 11.850 & 0.1 \\6 & 12.0 & 11.9 & 11.8 & 11.7 & 11.850 & 0.3 \\7 & 11.6 & 12.0 & 11.9 & 11.8 & 11.825 & 0.4 \\8 & 11.5 & 12.1 & 11.9 & 12.0 & 11.875 & 0.6 \\9 & 12.1 & 12.1 & 11.7 & 11.9 & 11.950 & 0.4 \\10 & 11.7 & 12.0 & 11.6 & 11.9 & 11.800 & 0.4 \\11 & 12.1 & 12.4 & 11.9 & 12.2 & 12.150 & 0.5 \\12 & 12.5 & 12.0 & 12.4 & 12.3 & 12.300 & 0.5 \\13 & 12.5 & 12.0 & 12.1 & 12.1 & 12.175 & 0.5 \\14 & 12.4 & 12.0 & 12.0 & 12.4 & 12.200 & 0.4 \\15 & 12.4 & 12.4 & 12.6 & 12.1 & 12.375 & 0.5\end{array}$

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Definitions:

How Much Decision

A consumer's determination of the quantity of a good or service to purchase, balancing the marginal utility and the price.

Total Revenues

The total amount of money generated by a firm from its sales activities before any costs or expenses are subtracted.

Total Costs

An alternate formulation for the sum of all expenses incurred in the production of goods or services, reflecting both fixed and variable components.

Profit

The financial gain obtained when the revenue from business activities exceeds the costs and expenses associated with operating the business.