A canning company defines a “nonconformance” as an orange juice can whose diameter is beyond the upper specification limit. A manager at the canning company wants to monitor the number of such orange juice cans produced per day. Each day, an employee will randomly select 120 cans over a period of four weeks. The employee then determines and plots the number of nonconformance cans in each sample. The appropriate control chart is a(n)

Question

BlackTom AI · Accepted Answer

Because we're monitoring the count of defective cans in repeated samples of the same size, an np-chart is appropriate.

1. Relevant knowledge points and concepts
- Attribute data vs. variable data: "nonconformance" (beyond upper spec) is attribute (defective or not), so an attributes control chart is required.
- p-chart vs. np-chart: p-chart monitors the proportion defective (p̂) and is used when sample sizes may vary; np-chart monitors the number defective (np̂) when sample size is constant across samples.
- u-chart: monitors defects per unit (for count of defects where a unit can have multiple defects) and is used when the area/unit size may vary.
- R-chart: a variable-chart for subgroup ranges, not applicable to attribute counts.

2. Complete reasoning process step by step
- Each day an employee samples n = 120 cans and counts how many are nonconforming (defective). That gives a count of defectives per fixed-size sample.
- Because n is constant and the measurement is the count of defectives, the np-chart (which plots the number defective in each sample) is the standard choice.

3. Calculations and intermediate steps (formulas)
- Let p̄ be the overall proportion defective estimated from multiple samples.
- Center line (CL) for an np-chart: CL = n * p̄.
- Control limits: UCL = n p̄ + 3 * sqrt(n p̄ (1 − p̄)); LCL = n p̄ − 3 * sqrt(n p̄ (1 − p̄)). If LCL < 0, set LCL = 0.
- Here n = 120, so CL = 120 p̄ and limits use sqrt(120 p̄ (1 − p̄)).

4. Why the correct answer is correct
- The np-chart directly plots the number defective in each fixed-size sample, matching the situation exactly (daily counts of nonconforming cans with n fixed at 120).

5. Why the other options are incorrect
- u-chart: for defects per unit (often when units differ in size or multiple defects per unit), not for counting defective units in a fixed sample.
- R-chart: used for ranges of variable (continuous) measurements within subgroups, not for attribute counts.
- p-chart: monitors proportion defective and is used when sample sizes vary; with constant n, np-chart is preferred because plotting counts is simpler and equivalent but conventional.

6. Conclusion
The correct control chart is the np-chart, which matches the provided answer: np-chart.

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