Ten Samples of N = 5 Were Collected to Construct $\overline { x ^ { * } }$

question 96

Essay

Ten samples of n = 5 were collected to construct an $\overline { x ^ { * } }$ chart. The sample means and ranges are in the table below. Calculate the control limits for the $\overline { x ^ { * } }$ chart and R-chart. After plotting the 10 sample means and ranges on the $\overline { x ^ { * } }$ and R charts, what is your conclusion? Note: You will need a table of control chart factors. $$\begin{array} { | c c c | c c c | } \hline \text { Sample } & \text { Mean } & \text { Range } & \text { Sample } & \text { Mean } & \text { Range } \\\hline 1 & 215 & 15 & 6 & 224 & 21 \\2 & 204 & 20 & 7 & 205 & 27 \\3 & 160 & 10 & 8 & 184 & 12 \\4 & 226 & 19 & 9 & 207 & 16 \\5 & 232 & 24 & 10 & 212 & 17 \\\hline\end{array}$$ $$\begin{array} { c c l c } \hline \text { Subgroup Size } & d _ { 2 } & D _ { 3 } & D _ { 4 } \\\hline 2 & 1.128 & 0 & 3.267 \\3 & 1.693 & 0 & 2.574 \\4 & 2.059 & 0 & 2.282 \\5 & 2.326 & 0 & 2.114 \\6 & 2.534 & 0 & 2.004 \\7 & 2.704 & 0.076 & 1.924 \\8 & 2.847 & 0.136 & 1.864 \\9 & 2.970 & 0.184 & 1.816 \\\hline\end{array}$$

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Definitions:

Causal Variable

A causal variable is a factor that brings about change in another variable or set of variables, establishing a cause-and-effect relationship.

Correlational Method

A research technique that establishes the relationship (not necessarily causal) between two variables, traditionally denoted x and y, by measuring both variables in a sample of participants.

Random Assignment

Random assignment is a method used in experiments to randomly allocate participants to different groups, ensuring that each group is comparable at the start of the experiment.

Measurement Biases

Involves errors or inaccuracies in data collection, interpretation, or analysis due to systemic factors.