- 2075 Chaitra · 6 marks
Significance of statistics; missing frequencies
Short answer
Statistics in engineering aids quality control, reliability testing, design of experiments, decision making under uncertainty, error analysis, and modeling. For the plant heights: total 51 and mean 40 give equations a+b=20, 30a+50b=820, so missing frequencies: f30=9, f50=11.
Write down the significance of statistics in engineering. An experiment shows the height of 51 plants given below. If average heights of all the 51 plants are 40 cm find the missing frequencies corresponding to the height 30 and 50 cm.
| Height (cm) | 10 | 20 | 30 | 40 | 50 | 60 |
|---|---|---|---|---|---|---|
| No. of plant | 2 | 3 | - | 21 | - | 5 |
Significance of statistics in engineering
- Quality control: control charts and sampling inspection keep product quality within limits.
- Reliability and testing: life of components, failure rates and warranty periods are estimated from sample data.
- Design of experiments: helps to choose the best material, process or settings with fewer trials.
- Decision making under uncertainty: forecasting load, traffic or demand, and risk analysis.
- Measurement and error analysis: handling noise, variation and measurement errors in data.
- Research and modelling: regression and hypothesis tests validate models and compare designs.
Missing frequencies
Let the missing frequencies be f₃₀ = a and f₅₀ = b.
Total plants: 2 + 3 + a + 21 + b + 5 = 51, so a + b = 51 − 31 = 20 ... (1)
Mean = Σfx / Σf = 40, so Σfx = 40 × 51 = 2040.
| Height x | f | fx |
|---|---|---|
| 10 | 2 | 20 |
| 20 | 3 | 60 |
| 30 | a | 30a |
| 40 | 21 | 840 |
| 50 | b | 50b |
| 60 | 5 | 300 |
| Total | 51 | 1220 + 30a + 50b |
1220 + 30a + 50b = 2040
30a + 50b = 820 → 3a + 5b = 82 ... (2)
Substitute a = 20 − b in (2):
3(20 − b) + 5b = 82
60 + 2b = 82
b = 11, a = 9
Check: 20 + 60 + 270 + 840 + 550 + 300 = 2040, and 2040/51 = 40. ✓
Answer: the frequency for 30 cm is 9 and for 50 cm is 11.
✓ Answer checked by Claude Sonnet 5.5
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