Question

The following data gives the information on the observed lifetimes (in hours) of $225$ electrical components:

$\begin{array}{ccccccc}\text{Lifetimes(in hours)}& 0-20& 20-40& 40-60& 60-80& 80-100& 100-120\\ \text{No.of components}& 10& 35& 52& 61& 38& 29\end{array}$

Determine the modal lifetimes of the components.

Answer 1

$\begin{array}{ccccccc}\text{Lifetimes(in hours)}& 0-20& 20-40& 40-60& 60-80& 80-100& 100-120\\ \text{No.of components}& 10& 35& 52& 61& 38& 29\end{array}$

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

As frequency is maximum for the class $60-80,$ modal class is $60-80.$

where,

$l$ = lower limit of modal class $=60$

$f_1$ = frequency of modal class $=61$

$f_0$ = frequency of class preceding the modal class $=52$

$f_2$ = frequency of class succeding the modal class $=38$

$h$= higher limit - lower limit $=80-60=20$

Mode $=60+\frac{61-52}{2\times 61-(52+38)}\times 20$

$\Rightarrow$ Mode $=60+\frac{9}{32}\times 20=60+5.625=65.625$

Hence, the modal lifetimes of the components are $65.625$ hrs.