Question 8

We need to find the mean number of absent days.

Step 1 — Find class marks

First, we find the class mark ($x_i$) for each interval. The class mark is the midpoint of the interval. We use the formula: (Upper limit + Lower limit) / 2.

For 0 - 6 days, $x_i = (0 + 6) / 2 = \mathbf{3}$.

For 6 - 10 days, $x_i = (6 + 10) / 2 = \mathbf{8}$.

For 10 - 14 days, $x_i = (10 + 14) / 2 = \mathbf{12}$.

For 14 - 20 days, $x_i = (14 + 20) / 2 = \mathbf{17}$.

For 20 - 28 days, $x_i = (20 + 28) / 2 = \mathbf{24}$.

For 28 - 38 days, $x_i = (28 + 38) / 2 = \mathbf{33}$.

For 38 - 40 days, $x_i = (38 + 40) / 2 = \mathbf{39}$.

Step 2 — Calculate mean

We will use the assumed mean method. Let's choose the assumed mean ($a$) as 17. Now, we calculate $d_i = x_i - a$.

For $x_i = 3$, $d_i = 3 - 17 = \mathbf{-14}$.

For $x_i = 8$, $d_i = 8 - 17 = \mathbf{-9}$.

For $x_i = 12$, $d_i = 12 - 17 = \mathbf{-5}$.

For $x_i = 17$, $d_i = 17 - 17 = \mathbf{0}$.

For $x_i = 24$, $d_i = 24 - 17 = \mathbf{7}$.

For $x_i = 33$, $d_i = 33 - 17 = \mathbf{16}$.

For $x_i = 39$, $d_i = 39 - 17 = \mathbf{22}$.

Next, we calculate $f_i d_i$.

For $f_1 = 11, d_1 = -14$, $f_1 d_1 = 11 \times (-14) = \mathbf{-154}$.

For $f_2 = 10, d_2 = -9$, $f_2 d_2 = 10 \times (-9) = \mathbf{-90}$.

For $f_3 = 7, d_3 = -5$, $f_3 d_3 = 7 \times (-5) = \mathbf{-35}$.

For $f_4 = 4, d_4 = 0$, $f_4 d_4 = 4 \times 0 = \mathbf{0}$.

For $f_5 = 4, d_5 = 7$, $f_5 d_5 = 4 \times 7 = \mathbf{28}$.

For $f_6 = 3, d_6 = 16$, $f_6 d_6 = 3 \times 16 = \mathbf{48}$.

For $f_7 = 1, d_7 = 22$, $f_7 d_7 = 1 \times 22 = \mathbf{22}$.

Now, we find the sum of frequencies ($\sum f_i$). $\sum f_i = 11 + 10 + 7 + 4 + 4 + 3 + 1$ $= \mathbf{40}$

Then, we find the sum of $f_i d_i$ ($\sum f_i d_i$). $\sum f_i d_i = -154 - 90 - 35 + 0 + 28 + 48 + 22$ $= -279 + 98$ $= \mathbf{-181}$

We use the mean formula. It is $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$. $\bar{x} = 17 + \frac{-181}{40}$ $= 17 - 4.525$

$\boxed{12.475 \text{ days}}$

Rounding to two decimal places, the mean is 12.48 days.

Answer

The mean number of days a student was absent is 12.48 days.