Explanation:
Specifications for dimension #11 are [tex]550 \pm 10[/tex]
This means, mean is 550, and margin of error on each side is 10
Sample size is n= 5
Standard Deviation [tex](\sigma_{1})=\frac{\bar{R}}{d_{2}}[/tex]
Where [tex]\bar{R}=10.23: d_{2}=2.326[/tex] (d2 is a constant dependant on sample size)
[tex]\begin{aligned}&_{ SO ,} \sigma=\frac{10.23}{2.326}=4.398\\&\text { Z-value }=\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{N}}}\end{aligned}[/tex]
Here, 547.9 is the process mean and 550 is population mean, subgroup size is 5 and standard deviation in each subgroup is 4.398
[tex]\text { Z-value }=\frac{547.9-550}{\frac{4.398}{\sqrt{5}}}=\frac{-2.1}{1.966}=-1.068[/tex]
p-value corresponding to z- value of -1.068 is 0.1446
So, 14.46% area on each side would be outside the range -1.068, +1.068
Total % of shaft outside the specifications are: 14.46*2 =28.92%
