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Problem #1: An automotive parts manufacturer employs a CNC lathe in the machining of a vehicle part. Each part is inspected for its length and the specifications provided by the customer are 80.00±0.50 (cm). Based on 30 subgroups of sample size n=3, the following statistics are collected: $\sum_{i=1}^{30} \bar{x}_i = 2403.00$ and $\sum_{i=1}^{30} R_i = 6.978$ (cm). (a) Find the control limits and the center line for both $\bar{X}$ and $R$ chart. (b) Estimate the process standard deviation. (Hint: $d_2 = \bar{R}/\sigma$ from Table) (c) Suppose the process mean has suddenly shifted to 80.3379 (cm) but the process variation remains the same, find the $\beta$ error using the $\bar{X}$ control limits calculated in (a). In addition, compute the ARL after this shift in the process mean.

          Problem #1: An automotive parts manufacturer employs a CNC lathe in the machining of a vehicle part. Each
part is inspected for its length and the specifications provided by the customer are 80.00±0.50 (cm). Based on
30 subgroups of sample size n=3, the following statistics are collected:
$\sum_{i=1}^{30} \bar{x}_i = 2403.00$
and $\sum_{i=1}^{30} R_i = 6.978$ (cm).
(a) Find the control limits and the center line for both $\bar{X}$ and $R$ chart.
(b) Estimate the process standard deviation. (Hint: $d_2 = \bar{R}/\sigma$ from Table)
(c) Suppose the process mean has suddenly shifted to 80.3379 (cm) but the process variation remains the same,
find the $\beta$ error using the $\bar{X}$ control limits calculated in (a). In addition, compute the ARL after this shift
in the process mean.

Problem #1: An automotive parts manufacturer employs a CNC lathe in the machining of a vehicle part. Each
part is inspected for its length and the specifications provided by the customer are 80.00±0.50 (cm). Based on
30 subgroups of sample size n=3, the following statistics are collected:
∑i=1^30x̅i = 2403.00
and ∑i=1^30 Ri = 6.978 (cm).
(a) Find the control limits and the center line for both X̅ and R chart.
(b) Estimate the process standard deviation. (Hint: d2 = R̅/σ from Table)
(c) Suppose the process mean has suddenly shifted to 80.3379 (cm) but the process variation remains the same,
find the β error using the X̅ control limits calculated in (a). In addition, compute the ARL after this shift
in the process mean.

Added by Walter C.

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