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

question 121

Essay

Construct an R chart and determine whether the process variation is within statistical control. $\text { Control Chart Constants }$
$\quad$$\quad$$\quad$$\quad$$\quad$$\bar{x}$ $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$ $\boldsymbol{s}$ $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$ $\boldsymbol{R}$
$\begin{array}{ccccccc}\hline n & A_{2} & A_{3} & B_{3} & B_{4} & D_{3} & D_{4} \\\hline 2 & 1.880 & 2.659 & 0.000 & 3.267 & 0.000 & 3.267 \\3 & 1.023 & 1.954 & 0.000 & 2.568 & 0.000 & 2.574 \\4 & 0.729 & 1.628 & 0.000 & 2.266 & 0.000 & 2.282 \\5 & 0.577 & 1.427 & 0.000 & 2.089 & 0.000 & 2.114 \\6 & 0.483 & 1.287 & 0.030 & 1.970 & 0.000 & 2.004 \\7 & 0.419 & 1.182 & 0.118 & 1.882 & 0.076 & 1.924 \\8 & 0.373 & 1.099 & 0.185 & 1.815 & 0.136 & 1.864 \\9 & 0.337 & 1.032 & 0.239 & 1.761 & 0.184 & 1.816 \\10 & 0.308 & 0.975 & 0.284 & 1.716 & 0.223 & 1.777 \\\hline\end{array}$
A machine is supposed to fill cans that contain 12 ounces. Each hour, a sample of four cans is tested; the results of 15 consecutive hours are given below.
$\text { Sample }\mid$ $\quad \text { ~~~Can Volume (oz) } \quad\mid$$\quad\overline{\boldsymbol{~~x}} \quad\mid$$\text { Range }$
$\begin{array}{c|cccc|c|c}\hline 1& 11.2 & 11.7 & 11.8 & 11.7 & 11.600 & 0.6 \\ 2 & 11.8 & 11.3 & 11.6 & 11.6 & 11.575 & 0.5 \\3 & 11.3 & 12.0 & 11.8 & 11.7 & 11.700 & 0.7 \\4 & 12.1 & 11.6 & 12.1 & 12.0 & 11.950 & 0.5 \\5 & 11.8 & 11.8 & 11.9 & 11.9 & 11.850 & 0.1 \\6 & 12.0 & 11.9 & 11.8 & 11.7 & 11.850 & 0.3 \\7 & 11.6 & 12.0 & 11.9 & 11.8 & 11.825 & 0.4 \\8 & 11.5 & 12.1 & 11.9 & 12.0 & 11.875 & 0.6 \\9 & 12.1 & 12.1 & 11.7 & 11.9 & 11.950 & 0.4 \\10 & 11.7 & 12.0 & 11.6 & 11.9 & 11.800 & 0.4 \\11 & 12.1 & 12.4 & 11.9 & 12.2 & 12.150 & 0.5 \\12 & 12.5 & 12.0 & 12.4 & 12.3 & 12.300 & 0.5 \\13 & 12.5 & 12.0 & 12.1 & 12.1 & 12.175 & 0.5 \\14 & 12.4 & 12.0 & 12.0 & 12.4 & 12.200 & 0.4 \\ \quad15 \quad& 12.4 & 12.4 & 12.6 & 12.1 & 12.375 & 0.5\end{array}$

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Definitions:

Pitch Discrimination

The ability to distinguish between different tones or musical notes based on their frequency, an important aspect of auditory perception.

Place Theory

A theory in psychology and physiology that suggests the perception of sound depends on where each component frequency produces vibrations along the basilar membrane of the inner ear.

Frequency Theory

The hypothesis that the frequency of a sound wave is directly related to the perception of the sound's pitch, as the auditory nerve fires at the same rate as the sound frequency.

Timbre

The quality or color of sound that distinguishes different types of sound production, such as voices or musical instruments.