(a) The mean of the following frequency distribution is 28. If sum of all frequencies is 100, then find the values of p and q:

Class Interval $\displaystyle 0$-$\displaystyle 10$ $\displaystyle 10$-$\displaystyle 20$ $\displaystyle 15$ $\displaystyle 27$ $\displaystyle 675$ $\displaystyle 700$ $\displaystyle 40$-$\displaystyle 50$ $\displaystyle 50$-$\displaystyle 60$ $\displaystyle 55$ \(\displaystyle 65+\mathrm{p}+\mathrm{q}\) \(\displaystyle 15 p+45 q+1765\) Z \(\displaystyle \mathrm{f}_{\mathrm{i}}=100=65+\mathrm{p}+\mathrm{q} \Rightarrow \mathrm{p}+\mathrm{q}=35\) Mean \(\displaystyle =28=\frac{15 \mathrm{p}+45 \mathrm{q}+1765}{100}\) \(\displaystyle \Rightarrow \mathrm{p}+3 \mathrm{q}=69\) On solving, we get \(\displaystyle \mathrm{p}=18, \mathrm{q}=17\) \(\displaystyle \frac{70}{2}=35\), Median class is \(\displaystyle 30-45\) Median \(\displaystyle =30+\frac{35-25}{12} \times 15\) \[=\frac{85}{2}=42.5 \] Modal class is \(\displaystyle 0-15\) \[\begin{aligned} \text { Mode } & =0+\frac{15-0}{30-0-10} \times 15 \\ & =\frac{45}{4}=11.25 \end{aligned} \]