The defect rate for a product has historically been about 2.0%. What are the upper and lower control chart limits if you wish to use a sample size of 100 and 3-sigma limits? All answers are in (LCL,UCL) format.

Question

BlackTom AI · Accepted Answer

We’re dealing with a p-chart style calculation for a defect rate, using the historical proportion p0 = 0.02 and a sample size n = 100 with 3-sigma limits.
First, compute the standard error term: sqrt[p0(1 - p0) / n] = sqrt[0.02 * 0.98 / 100] = sqrt[0.0196 / 100] = sqrt[0.000196] ≈ 0.0140.
Then, multiply by 3 to obtain the margin of control: 3 * 0.0140 ≈ 0.042.
Now, find the upper and lower control limits using p0 ± 3*standard_error:
- Lower Control Limit (LCL) = p0 − 3*SE ≈ 0.02 − 0.042 ≈ −0.022. Since a defect rate cannot be negative, the LCL is set at 0.00.
- Upper Control Limit (UCL) = p0 + 3*SE ≈ 0.02 + 0.042 ≈ 0.062.
Thus the practical limits are (0.00, 0.062).
Option-by-option analysis:
- Option (0.00, 0.062): This matches the calculated LCL and UCL exactly after applying the non-negativity floor on the LCL. Correct considering the standard error and 3-sigma spread.
- Option (0.00, 0.059): The UCL would require a margin of 0.039, which implies a smaller SE than computed; this would occur if the SE were 0.013, not 0.014, which is inconsistent with the given p0 and n.
- Option (0.00, 0.048): An even smaller UCL of 0.048 would require a margin of 0.028, corresponding to an SE around 0.0093, far from the calculated 0.014.
- Option (0.00, 0.067): A UCL of 0.067 implies a margin of 0.047, implying a larger SE than the actual 0.014; this overstates the variability.
In summary, only (0.00, 0.062) aligns with the standard 3-sigma calculation using p0 = 0.02 and n = 100, with the LCL adjusted to zero whenever the raw calculation would yield a negative value.

The defect rate for a product has historically been about 2.0%. What are the upper and lower control chart limits if you wish to use a sample size of 100 and 3-sigma limits? All answers are in (LCL,UCL) format. 单项选择题

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