Question 7

We need to find the median weight using the given frequency distribution.

Step 1 — Calculate Cumulative Frequency

Let's first find the total number of students. We will also create a cumulative frequency table.

| Weight (in kg) | Number of students (f) | Cumulative Frequency (cf) | | :------------- | :--------------------- | :------------------------ | | 40 - 45 | 2 | 2 | | 45 - 50 | 3 | 5 | | 50 - 55 | 8 | 13 | | 55 - 60 | 6 | 19 | | 60 - 65 | 6 | 25 | | 65 - 70 | 3 | 28 | | 70 - 75 | 2 | 30 |

The total number of students is $N$.

$N = 2 + 3 + 8 + 6 + 6 + 3 + 2$

$\boxed{N = 30}$

Step 2 — Identify Median Class

We need to find the value of $N/2$.

$\frac{N}{2} = \frac{30}{2}$

$= 15$

Now, we find the class interval whose cumulative frequency is just greater than 15. From the table, the cumulative frequency just greater than 15 is 19. This cumulative frequency corresponds to the class interval 55 - 60. So, the median class is 55 - 60.

Step 3 — Apply Median Formula

Let's list the values for the median formula. The lower limit of the median class ($L$) is 55. The cumulative frequency of the class preceding the median class ($cf$) is 13. The frequency of the median class ($f$) is 6. The class size ($h$) is $45 - 40$.

$h = 45 - 40$

$= 5$

The formula for the median is:

$\text{Median} = L + \left(\frac{\frac{N}{2} - cf}{f}\right) \times h$

Substitute the values into the formula.

$\text{Median} = 55 + \left(\frac{15 - 13}{6}\right) \times 5$

$\text{Median} = 55 + \left(\frac{2}{6}\right) \times 5$

$\text{Median} = 55 + \left(\frac{1}{3}\right) \times 5$

$\text{Median} = 55 + \frac{5}{3}$

$\text{Median} = 55 + 1.666...$

$\text{Median} = 56.666...$

Rounding to two decimal places, we get:

$\boxed{56.67 \text{ kg}}$

Answer

The median weight of the students is 56.67 kg.