The question asks us to calculate the modal lifetime from the given frequency distribution table. The modal lifetime represents the most frequent lifetime range in the data.

Identifying the Modal Class

First, we need to identify the modal class, which is the class interval with the highest frequency.

From the table, the highest frequency is 61, which corresponds to the lifetime interval 30-40 days. Therefore, the modal class is 30-40.

Modal Lifetime Calculation Formula

We use the formula for calculating the mode of grouped data:

Mode = \(l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h\)

Where:

• \(l\) = Lower limit of the modal class
• \(f_1\) = Frequency of the modal class
• \(f_0\) = Frequency of the class preceding the modal class
• \(f_2\) = Frequency of the class succeeding the modal class
• \(h\) = Class width

Applying the Formula

From the identified modal class (30-40):

• \(l = 30\)
• \(f_1 = 61\)
• \(f_0 = 52\) (Frequency of the 20-30 class)
• \(f_2 = 38\) (Frequency of the 40-50 class)
• \(h = 10\) (Class width, e.g., 40 - 30 = 10)

Substitute these values into the formula:

Mode = \(30 + \frac{61 - 52}{2 \times 61 - 52 - 38} \times 10\)

Mode = \(30 + \frac{9}{122 - 52 - 38} \times 10\)

Mode = \(30 + \frac{9}{122 - 90} \times 10\)

Mode = \(30 + \frac{9}{32} \times 10\)

Mode = \(30 + \frac{90}{32}\)

Mode = \(30 + 2.8125\)

Mode = \(32.8125\)

Rounding the Result

The question requires the answer to be rounded up to one decimal place. Rounding 32.8125 gives 32.8.

Therefore, the modal lifetime is approximately 32.8 days.