Question

Finished crankshafts take a lot of time and money to make, and are critical to the proper function of an automobile engine. Because of the critical nature of this part, one company measures several critical dimensions on each and every finished crankshaft. If any of the dimensions measured do not meet the engineering specifications, the crankshaft is scrapped Since measurements are being taken for inspection purposes, data is readily available for control charts as well. Dimension #11 is one of the many critical dimensions that is measured, and has been the subject of much debate. Lately, all of the crankshafts that are scrapped were scrapped because of dimension #11 being out of spec. The control charts for this dimension have been in statistical control, with a centerline on the X-bar chart of 547.9, and a centerline on the R-chart of 10.23. (Subgroups were formed by taking the first 5 crankshafts inspected each hour to form subgroups of n-6). Specifications for dimension #11 are 550.0 ± 10.0 (Assume a normal distribution of all characteristics measured when working this problem.) a) Using the above information, estimate the percentage of crankshafts that would not meet the specs for dimension #11 . Calculate Z-scores, and use the "Z-table" to find your answer.

Answer 1

Step-by-step explanation:

Specifications for dimension #11 are
`550 \pm 10`

This means, mean is 550, and margin of error on each side is 10

Sample size is n= 5

Standard Deviation
`(\sigma_(1))=\frac{\bar{R}}{d_(2)}`

Where
`\bar{R}=10.23: d_(2)=2.326` (d2 is a constant dependant on sample size)

`\begin{aligned}&_( SO ,) \sigma=(10.23)/(2.326)=4.398\\&\text { Z-value }=\frac{\bar{X}-\mu}{(\sigma)/(√(N))}\end{aligned}`

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

`\text { Z-value }=(547.9-550)/((4.398)/(√(5)))=(-2.1)/(1.966)=-1.068`

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%