The following table gives the distribution of the life time of 400 neon lamps : Life time (in hours) Number of lamps 1500 - 2000 14 2000 - 2500 56 2500 - 3000 60 3000 - 3500 86 3500 - 4000 74 4000 - 4500 62 4500 - 5000 48 Find the median life time of a lamp.
Step-by-Step Solution
Key Concept: For grouped data, the median is found using the formula \(\text{Median}=L+\left(\frac{\frac{N}{2}-c_f}{f}\right)h\), where \(L\) is the lower class boundary of the median class, \(c_f\) is the cumulative frequency before the median class, \(f\) is the frequency of the median class, \(h\) is the class width, and \(N\) is the total number of observations.
1. Total number of lamps: \(N = 400\).
2. Find \(\frac{N}{2}\): \(\frac{N}{2}=200\).
3. Prepare cumulative frequencies:
- 1500‑2000: \(f=14\), \(c_f=0\), cumulative = 14
- 2000‑2500: \(f=56\), cumulative = 70
- 2500‑3000: \(f=60\), cumulative = 130
- 3000‑3500: \(f=86\), cumulative = 216
- 3500‑4000: \(f=74\), cumulative = 290
- 4000‑4500: \(f=62\), cumulative = 352
- 4500‑5000: \(f=48\), cumulative = 400
4. Locate the median class: The cumulative frequency just exceeds \(200\) in the class 3000‑3500 (cumulative = 216). Hence, median class = 3000‑3500.
5. Identify required quantities:
- Lower class boundary \(L = 3000\) (assuming continuous class limits).
- Class width \(h = 3500-3000 = 500\).
- Cumulative frequency before median class \(c_f = 130\).
- Frequency of median class \(f = 86\).
6. Apply the median formula:
\[
\text{Median}=L+\left(\frac{\frac{N}{2}-c_f}{f}\right)h
=3000+\left(\frac{200-130}{86}\right)\times 500
\]
\[
=3000+\left(\frac{70}{86}\right)\times 500
=3000+0.813953\times 500
\]
\[
=3000+406.98\approx 3407\text{ hours}
\]
7. Result: The median life time of a lamp is approximately 3407 hours (rounded to the nearest hour).
Correct Answer: ≈ 3407 hours