Solve the Problem $\bar { x }$ Chart Based on Standard Deviations (Instead of Ranges) Is Made

question 7

Essay

Solve the Problem.
-An $\bar { x }$ chart based on standard deviations (instead of ranges) is made by plotting sample means with a centerline at $\bar {\bar { x} }$ and control limits at $\bar {\bar { x} } + A _ { 3 } \bar { s }$ and $\bar {\bar { x} } - A _ { 3 } \bar { s }$ where $A _ { 3 }$ is found in the Control Chart Constants table and $\bar { s }$ is the mean of the sample standard deviations. Construct an $\bar { x }$ chart based on standard deviations for the data belo
A manufacturer of high intensity video projector lightbulbs picks 6 lightbulbs at random each day for quality cor The mean, standard deviation, and range (in hours) of the lifetimes of the 6 lightbulbs are calculated. The results consecutive days are shown in the table below.
$\begin{array}{lllll}\text { Day } & \bar{x} & \text { Range } & \text { s } \\1 & 21.13 .4 & & 1.2 & \\2 & 23.44 .7 & & 1.7 & \\3 & 20.93 .1 & & 0.9 & \\4 & 25.66 .2 & & 2.3 & \\5 & 21.3 & 4.7 &{1.3} \\6 & 20.5 & 3.7 &{1.7} \\7 & 25.85 .8 & &2.1 & \\8 & 23.76 .2 & & 2.9 & \\9 & 29.82 .1 & & 0.8 & \\10 & 22.7 & 4.4 &{1.6} \\11 & 20.02 .7 & & 0.7 & \\12 & 24.67 .8 & & 3.0\end{array}$ $$\text { Answer: }$$

---

Definitions:

Independent Samples T-Test

A statistical test used to determine if there is a significant difference between the means of two independent samples.

Sum of Squares for Error

The sum of the squared differences between the observed values and the predicted values in a dataset, used in statistical analysis.

Randomized Block Design

An experimental design in which subjects are first divided into blocks based on a certain characteristic, and then randomly assigned to different treatment groups within each block.

Randomized Block Design

A statistical design for experiments that aims to reduce the effects of confounding variables by organizing subjects into blocks based on these variables before randomly assigning treatments.