Question 5

We need to find the mode of the given grouped data.

Step 1 — Identify modal class

Let's find the class with the highest frequency. The maximum frequency is 18. This frequency belongs to the class interval 4000-5000. So, the modal class is 4000-5000.

The lower limit ($l$) of the modal class is 4000. The frequency ($f_1$) of the modal class is 18. The frequency ($f_0$) of the class before the modal class is 4. The frequency ($f_2$) of the class after the modal class is 9. The class size ($h$) is the difference between class limits.

$h = 5000 - 4000$

$\boxed{h = 1000}$

Step 2 — Calculate the mode

Let's use the formula for the mode of grouped data.

$Mode = l + \left[ \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right] \times h$

Now, we substitute the values we found.

$Mode = 4000 + \left[ \frac{18 - 4}{2 \times 18 - 4 - 9} \right] \times 1000$

$Mode = 4000 + \left[ \frac{14}{36 - 4 - 9} \right] \times 1000$

$Mode = 4000 + \left[ \frac{14}{23} \right] \times 1000$

$Mode = 4000 + \frac{14000}{23}$

$Mode = 4000 + 608.69565...$

$Mode \approx 4608.7$

The mode of the data is approximately 4608.7 runs.

Answer

(i) The mode of the given data is 4608.7 runs.