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.
3406.98 hours
Explanation
To find the median life time, we first calculate the cumulative frequency for the given data. The total number of lamps (N) is 400, so N/2 is 200. We identify the median class by finding the class interval where the cumulative frequency is just greater than 200. Using the median formula for grouped data, we substitute the values for lower limit (l), class size (h), frequency (f), and cumulative frequency of the preceding class (cf) to find the exact median value.
Solution Steps
Step 1: Calculate the cumulative frequencies. The cumulative frequencies are 14, 70 (14+56), 130 (70+60), 216 (130+86), 290 (216+74), 352 (290+62), and 400 (352+48).
Step 2: Determine N/2. Here, N = 400, so N/2 = 200.
Step 3: Identify the median class. The cumulative frequency just greater than 200 is 216, which corresponds to the class interval 3000 - 3500.
Step 4: Identify the values for the median formula: Lower limit (l) = 3000, Frequency (f) = 86, Cumulative frequency of preceding class (cf) = 130, and Class size (h) = 500.
Step 5: Apply the formula: Median = l + [(N/2 - cf) / f] * h Median = 3000 + [(200 - 130) / 86] * 500 Median = 3000 + (70/86) * 500 Median = 3000 + 406.98 Median = 3406.98 hours.