Question 2 (c)
Mean and standard deviation of undamaged shipments (binomial)
4 marks
- Time to write
- About 8 minutes
Question
Question 2 (c) · 4 marksA company gets 75% of its shipments from its supplier undamaged. The quality control inspector checks 12 shipments from this supplier. Find the average number (mean) and the standard deviation of the shipments arriving undamaged.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- FindFormula first, then the working, then the answer with its unit (₹, %, units).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Identifying the binomial distribution and its parameters 1 mark 2 min
- 2 Mean 1½ marks 3 min
- 3 Standard deviation 1½ marks 2 min
Try it first. About 8 minutes for 4 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
The distribution. Each of the 12 shipments either arrives undamaged (success) or does not; the chance of success, 0.75, is the same for each, and shipments are independent. The number X of undamaged shipments is therefore binomial, X ~ B(n, p) with n = 12, p = 0.75 and q = 1 − p = 0.25. For a binomial variable the mean is np and the variance npq.
Mean. μ = np = 12 × 0.75 = 9 shipments.
Standard deviation. σ² = npq = 12 × 0.75 × 0.25 = 2.25, so σ = √2.25 = 1.5 shipments.
Interpretation. Out of every 12 shipments the inspector checks, 9 are expected to arrive undamaged on average, and the number undamaged typically varies by about 1.5 shipments from one batch of 12 to another. As for every binomial distribution, the variance (2.25) is less than the mean (9).
Where the 4 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Identifying the binomial distribution and its parameters | 1 |
| 2 | Mean | 1.5 |
| 3 | Standard deviation | 1.5 |
| Total | 4 |
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