Problem Statement
The following data gives the information on the observed lifetimes (in hours) of 225 electrical components. Determine the modal lifetimes of the components:
$\begin{array}{|c|c|c|c|c|c|c|} \hline \text{Lifetimes (hours)} & 0-20 & 20-40 & 40-60 & 60-80 & 80-100 & 100-120 \\ \hline \text{Frequency} & 10 & 35 & 52 & 61 & 38 & 29 \\ \hline \end{array}$
Verified Solution & Marking Scheme
Identify Modal Class
Maximum frequency is 61, corresponding to class $60 - 80$. Thus modal class is $60 - 80$.
Lower limit $L = 60$, class size $h = 20$.
Modal frequency $f_1 = 61$.
Preceding frequency $f_0 = 52$.
Succeeding frequency $f_2 = 38$.
Apply Mode Formula
$\text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$
$\text{Mode} = 60 + \left(\frac{61 - 52}{2(61) - 52 - 38}\right) \times 20 = 60 + \left(\frac{9}{122 - 90}\right) \times 20 = 60 + \left(\frac{9}{32}\right) \times 20$
$= 60 + \frac{45}{8} = 60 + 5.625 = 65.625\text{ hours}$