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 of an X - bar chart when $\sigma$ is unknown is $UCL - LCL=2A_2\bar{R}$, where $UCL$ is the upper control limit, $LCL$ is the lower control limit, $A_2$ is a constant related to sample size, and $\bar{R}$ is the average range. For a sample size $n = 7$, from control - chart constants table, $A_2=\frac{3}{\sqrt{n}}$. Here $n = 7$, so $A_2=\frac{3}{\sqrt{7}}\approx1.134$. We know that $UCL - LCL = 16$.

Step2: Solve for $\bar{R}$

Since $UCL - LCL=2A_2\bar{R}$, we can re - arrange the formula to find $\bar{R}=\frac{UCL - LCL}{2A_2}$. Substitute $UCL - LCL = 16$ and $A_2=\frac{3}{\sqrt{7}}$ into the formula. $\bar{R}=\frac{16}{2\times\frac{3}{\sqrt{7}}}=\frac{8\sqrt{7}}{3}$.
Calculate $\frac{8\sqrt{7}}{3}=\frac{8\times2.64575}{3}\approx7.06$.

Answer:

$7.06$