The following data gives the information on the observed lifetimes (in hours) of 225 electrical components:
| Lifetimes (in hours) | 0 - 20 | 20 - 40 | 40 - 60 | 60 - 80 | 80 - 100 | 100 - 120 |
|---|---|---|---|---|---|---|
| Frequency | 10 | 35 | 52 | 61 | 38 | 29 |
Determine the modal lifetimes of the components.
The modal lifetime of the components is 65.63 hours.
Explanation
This question requires finding the mode for grouped data. The modal class is identified as the class with the highest frequency (61 in the 60-80 class). The mode formula for grouped data is then applied with the appropriate values from the frequency distribution.
Solution Steps
Step 1: Identify the modal class. The class with the highest frequency is 60-80 with frequency 61. So, modal class = 60-80.
Step 2: Write the formula for mode: Mode = l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h
Step 3: Identify the values: l (lower limit of modal class) = 60, f₁ (frequency of modal class) = 61, f₀ (frequency of preceding class) = 52, f₂ (frequency of succeeding class) = 38, h (class size) = 20
Step 4: Substitute values in the formula: Mode = 60 + [(61 - 52) / ( - 52 - 38)] × 20
Step 5: Simplify: Mode = 60 + [9 / (122 - 52 - 38)] × 20 = 60 + [9 / 32] × 20
Step 6: Calculate: Mode = 60 + (180 / 32) = 60 + 5.625 = 65.625 hours