- 20 Marks
Question
Nawaoo is a registered company in the Central Region specialized in electronic products. Nawaoo is involved in manufacturing high-quality electrical equipment. Each item produced costs GHS 6,000.00 and the total output is 500 items. At the end of the production process, each item is individually tested for quality and safety. If the item is defective in any way it is scrapped at a complete loss to the firm, since it has been found not to be cost-effective to repair such items. Historically, one item in 1000 is found to be faulty in some way.
(a) What is the relationship between the mean, mode and median in a non-symmetric data? [7 Marks]
(b) Determine the probability that the firm will produce zero faulty items in a year. [6 Marks]
(c) Determine the probability that the firm will produce no more than three faulty items in a year. [6 Marks]
(d) The firm is considering employing a quality inspector at an additional cost of GHS1000 per year. The Inspector, however, will be able to prevent any item from being defective. Suggest how the firm might evaluate whether employing the inspector would be cost-effective. [6 Marks]
Answer
(a) In non-symmetric (skewed) data, the mean, median, and mode differ in position. For positive skew (tail right), mean > median > mode, as outliers pull mean up. For negative skew (tail left), mean < median < mode. This relates to faulty items data, potentially positively skewed if rare high defects occur, affecting risk assessments in banking operations like loan portfolio quality.
(b) Faulties follow Poisson (rare events, lambda=500*(1/1000)=0.5). P(0) = e^{-0.5} ≈ 0.6065. To arrive: Poisson pmf P(k)=e^{-λ} λ^k / k!, for k=0: e^{-0.5} ≈ 0.6065 (using exp(-0.5)=1/sqrt(e)≈0.6065).
(c) P(≤3) = Σ P(k=0 to 3) ≈ 0.9982. Using cdf of Poisson, sum e^{-0.5} (1 + 0.5 + 0.5²/2 + 0.5³/6) ≈ 0.6065 + 0.3033 + 0.0758 + 0.0126 ≈ 0.9982.
(d) Evaluate by comparing expected cost without inspector: Expected faulties=0.5, cost=0.5*6000=GHS3000. With inspector: Cost=1000, faulties=0, savings=3000-1000=2000. Hire if savings positive. In practice, factor probability of perfect prevention and indirect benefits like reputation, aligning with Basel operational risk standards in Ghanaian banks.
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