Question

A student noted the number of cars passing through a spot on a road for $100$ periods each of $2$ minutes and complied data in the following table :

Number of cars	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$
Number of period	$7$	$x$	$13$	$15$	$y$	$10$	$14$	$8$

Find the missing frequencies $x$ and $y$, if the mode of the data is given to be $44$.

Answer 1

Given:

Mode of the given data is $44$

Number of periods is $100$

$⟹7+x+13+15+y+10+14+8=100$

$⟹x+y=33$

So,the modal class is $40-50$

$L$ (the modal class lower limit) $=40$

$f_1$(frequency of the modal class) $=y$

$f_0$(frequency of the preceding modal class) $=15$

$f_2$(frequency of the succeeding modal class) $=10$

h(class size) $=10$

Mode $=L+(\frac{f_1-f_0}{2f_1-f_0-f_2})\times h$

$⟹40+\left(\frac{y-15}{2y-15-10}\right)\times 10=44$

$⟹\frac{10y-150}{2y-25}=4$

$⟹10y-150=8y-100$

$⟹2y=50$

$⟹y=25$

$\because x+y=33$

$⟹x+25=33$

$⟹x=8$

Hence $x=8$ and $y=25$