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 following data gives the information on the observed lifetimes…

Question 11 of 21intermediate🔧 ApplyNumerical3 marks

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.

Correct Answer

The modal lifetime of the components is 65.625 hours.

Exercise: EXERCISE 13.2 | Q: 2 | (Page 16)

For More Understanding

Explanation

This question is from Statistics chapter dealing with calculation of mode for grouped data. The mode is the value that occurs most frequently. For grouped data, we first identify the modal class (class with highest frequency) and then apply the mode formula. The modal class here is 60-80 with frequency 61.

Solution Steps

Step 1: Identify the modal class. The highest frequency is 61, corresponding to class 60-80. So, modal class = 60-80.
Step 2: Identify the values for the formula: l (lower limit) = 60, f₁ (frequency of modal class) = 61, f₀ (frequency of preceding class) = 52, f₂ (frequency of succeeding class) = 38, h (class width) = 20.
Step 3: Apply the mode formula: Mode = l + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h
Step 4: Substitute values: Mode = 60 + [(61 - 52)/( $2 \times 61$ - 52 - 38)] × 20
Step 5: Simplify: Mode = 60 + [9/(122 - 90)] × 20 = 60 + (9/32) × 20 = 60 + 180/32 = 60 + 5.625 = 65.625 hours.

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