Refer to

Table S$6\mathrm{.}1-$ Factors for Computing$\_$Control Chart Limits $(3$ sigma$)$ for this problem.

Eagletrons are all$-$electric automobiles produced by Mogul Motors, Inc. One of the concerns of Mogul Motors is that the Eagletrons be capable of achieving appropriate maximum speeds. To monitor this, Mogul executives take samples of tenEagletrons at a time.

For each sample, they determine the average maximum speed and the range of the maximum speeds within the sample.

They repeat this with $35$ samples to obtain $35$ sample means and $35$ ranges. They find that the average sample mean is $98\mathrm{.}50$ miles per hour, and the average range is $4\mathrm{.}50$ miles per hour. Using these results, the executives decide to establish an R$-$chart. They would like this chart to be established so that when it shows that the range of a sample is not within the control limits$,$ there is only approximately a $0\mathrm{.}0027$ probability that this is due to natural variation.

The control limits for the chart based on the above requirement for the given information are:

$UC{L}_{\frac{?}{\text{b}\text{a}\text{r}}(R)}=$ miles per hour $($round the reponse to three decimal places$).$