Question

1. automobile fuel efficiency thirty automobiles were tested for fuel efficiency (in miles per gallon). this frequency distribution was obtained. find the mean and modal class for the data. class boundaries frequency 7.5 - 12.5 3 12.5 - 17.5 5 17.5 - 22.5 15 22.5 - 27.5 5 27.5 - 32.5 2

Explanation:

Step1: Find the mid - points of each class

For the class $7.5 - 12.5$, the mid - point $x_1=\frac{7.5 + 12.5}{2}=10$.
For the class $12.5 - 17.5$, the mid - point $x_2=\frac{12.5+17.5}{2}=15$.
For the class $17.5 - 22.5$, the mid - point $x_3=\frac{17.5 + 22.5}{2}=20$.
For the class $22.5 - 27.5$, the mid - point $x_4=\frac{22.5+27.5}{2}=25$.
For the class $27.5 - 32.5$, the mid - point $x_5=\frac{27.5 + 32.5}{2}=30$.

Step2: Calculate the product of mid - points and frequencies

$x_1f_1=10\times3 = 30$.
$x_2f_2=15\times5 = 75$.
$x_3f_3=20\times15=300$.
$x_4f_4=25\times5 = 125$.
$x_5f_5=30\times2=60$.

Step3: Calculate the sum of frequencies and the sum of products

$\sum f_i=3 + 5+15 + 5+2=30$.
$\sum x_if_i=30 + 75+300+125+60=590$.

Step4: Calculate the mean

The mean $\bar{x}=\frac{\sum x_if_i}{\sum f_i}=\frac{590}{30}=\frac{59}{3}\approx19.67$.

Step5: Find the modal class

The class with the highest frequency is $17.5 - 22.5$ (frequency = 15).

Answer:

Mean: $\frac{59}{3}\approx19.67$; Modal class: $17.5 - 22.5$