Question 2

We will use the given median and total frequency to find the unknown values.

Step 1 — Find the total frequency equation

Let's sum all the frequencies. The sum of frequencies is equal to the total frequency. We are given the total frequency is 60.

$5 + x + 20 + 15 + y + 5 = 60$

$45 + x + y = 60$

$x + y = 60 - 45$

$\boxed{x + y = 15}$

Step 2 — Identify the median class

The median of the distribution is 28.5. We need to find the class interval that contains 28.5. The class interval 20 - 30 contains 28.5. So, the median class is 20 - 30.

Step 3 — Extract values for the median formula

For the median class 20 - 30: The lower limit ($L$) is 20. The frequency of the median class ($f$) is 20. The class size ($h$) is $30 - 20 = \textbf{10}$. The total frequency ($N$) is 60, so $N/2 = 60/2 = \textbf{30}$. The cumulative frequency ($cf$) of the class preceding the median class is $5 + x$.

Step 4 — Apply the median formula

Let's use the formula for the median of grouped data. Median $= L + \left(\frac{\frac{N}{2} - cf}{f}\right) \times h$

$28.5 = 20 + \left(\frac{30 - (5 + x)}{20}\right) \times 10$

$28.5 - 20 = \left(\frac{30 - 5 - x}{20}\right) \times 10$

$8.5 = \left(\frac{25 - x}{20}\right) \times 10$

$8.5 = \frac{25 - x}{2}$

$8.5 \times 2 = 25 - x$

$17 = 25 - x$

$x = 25 - 17$

$\boxed{x = 8}$

Step 5 — Find the value of $y$

We use the equation from Step 1: $x + y = 15$. Substitute the value of $x = \textbf{8}$.

$8 + y = 15$

$y = 15 - 8$

$\boxed{y = 7}$

Answer

The value of $x$ is 8. The value of $y$ is 7.