Question

the difference between the upper and lower control limits of an x - bar chart for a process is 16 units. samples of size 7 have been collected every hour for the past several months. assuming that $sigma$ is not known, what is the implied average range from the above results (assume $z = 3$)? (round it to two decimal points)

Explanation:

Step1: Recall the formula for control - limits of X - bar chart

The formula for the difference between the upper and lower control limits (UCL - LCL) of an X - bar chart when $\sigma$ is unknown is $UCL - LCL = A_2\bar{R}\times2$, where $A_2$ is a constant related to the sample size $n$, and $\bar{R}$ is the average range. For a sample size $n = 7$, from the control - chart constants table, $A_2=\frac{3}{\sqrt{n}}=\frac{3}{\sqrt{7}}\approx1.134$.
We know that $UCL - LCL = 16$.

Step2: Solve for $\bar{R}$

Since $UCL - LCL=A_2\bar{R}\times2$, we can re - arrange the formula to solve for $\bar{R}$.
$\bar{R}=\frac{UCL - LCL}{2A_2}$.
Substitute $UCL - LCL = 16$ and $A_2\approx1.134$ into the formula:
$\bar{R}=\frac{16}{2\times1.134}=\frac{16}{2.268}\approx7.05$.

Answer:

$7.05$