Question

Five samples of size 80 are inspected. A p-chart with ±3σ limits is used. The defective counts are 4, 10, 5, 6, and 5. The upper control limit (fraction defective) is \(\underline{\hspace{2cm}}\) (round off to two decimal places).

Hint:

For p-charts, use \( \bar{p} \pm 3\sigma_p \), where \( \sigma_p = \sqrt{\bar{p}(1-\bar{p})/n} \).

Answer 1

Correct Answer: 0.15

Total sample size: \[ 5 \times 80 = 400. \] Total number of defectives: \[ 4 + 10 + 5 + 6 + 5 = 30. \] Fraction defective: \[ \bar{p} = \frac{30}{400} = 0.075. \] Standard deviation: \[ \sigma_p = \sqrt{\frac{\bar{p}(1-\bar{p})}{n}} = \sqrt{\frac{0.075 \times 0.925}{80}} = \sqrt{0.000867} = 0.02947. \] Upper control limit: \[ UCL = \bar{p} + 3\sigma_p = 0.075 + 3(0.02947) = 0.075 + 0.08841 = 0.16341. \] Rounded to two decimals: \[ UCL = 0.16. \] Final Answer: \(0.16\)