For a production process using statistical process control, where each sample has 4 observations in it, let
𝑥
¯
¯
= 9.35 inches and
𝑅
¯
= 3 inches. What is the lower control limit for the
𝑥
¯
-chart?

Question

For a production process using statistical process control, where each sample has 4 observations in it, let 
𝑥
¯
¯
 = 9.35 inches and 
𝑅
¯
 = 3 inches. What is the lower control limit for the 
𝑥
¯
-chart?

BlackTom AI · Accepted Answer

We need to determine the lower control limit (LCL) for the x-bar chart in a statistical process control context with n = 4 observations per sample.

First, recall the standard relationship for an x-bar chart:
LCL = Xbar-bar − A2 × R-bar
UCL = Xbar-bar + A2 × R-bar
where Xbar-bar is the process mean of the sample means, R-bar is the average range, and A2 is a control-chart constant that depends on sample size n.

From the given data: Xbar-bar = 9.35 inches and R-bar = 3 inches.

We need the appropriate A2 value for n = 4. For n = 4, the conventional SPC constants give A2 ≈ 0.729. (These constants come from standard reference tables and are derived from the distribution of ranges for samples of size n.)

Compute A2 × R-bar:
A2 × R-bar = 0.729 × 3 = 2.187 inches (approximately).

Now subtract to obtain the LCL:
LCL = Xbar-bar − A2 × R-bar = 9.35 − 2.187 ≈ 7.163 inches.

Rounding to a reasonable presentation, the lower control limit is about 7.16 inches.

Note on options: the provided options list contains a single value, 7.16 inches, which matches this calculation. In cases with more options, you would compare each to the computed LCL and choose the one that aligns with the formula and rounding convention used.

If any nuance arises (e.g., different rounding rules or alternate A2 tables), the essential steps remain: identify n, retrieve the correct A2, multiply by R-bar, and subtract from Xbar-bar to obtain LCL.

Section Merge: SCM 405: Mfg & Svcs Strategy Midterm Exam 2

For a production process using statistical process control, where each sample has 4 observations in it, let 𝑥 ¯ ¯ = 9.35 inches and 𝑅 ¯ = 3 inches. What is the lower control limit for the 𝑥 ¯ -chart?

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