The frequency distribution table below shows the recorded speeds (in \text{km/h}) of a sam… — JAMB UTME

The frequency distribution table below shows the recorded speeds (in $\text{km/h}$ ) of a sample of commercial buses passing through a highway toll checkpoint:

Speed ( $\text{km/h}$ )	Frequency
$40 - 49$	$6$
$50 - 59$	$10$
$60 - 69$	$f$
$70 - 79$	$12$
$80 - 89$	$8$

If the mean speed of the buses is $66.5\text{ km/h}$ , find the value of the missing frequency $f$ .

Answer: 6

Answer

The value of the missing frequency

f

6

Each class interval's midpoint is calculated by averaging its lower and upper limits. The total frequency is

\sum f = 36 + f

and the total sum of products is

\sum fx = 2382 + 64.5f

. Applying the grouped mean formula

\bar{x} = \frac{\sum fx}{\sum f} = 66.5

gives

2394 + 66.5f = 2382 + 64.5f

, which yields

2f = 12

, so

f = 6

Step-by-Step Solution

Find the class midpoints (

x

) for each grouped interval.

Midpoints are

44.5, 54.5, 64.5, 74.5,

and

84.5

Midpoints represent the central values of each interval for grouped mean calculations.

Calculate expressions for

\sum f

and

\sum fx

\sum f = 36 + f

and

\sum fx = 2382 + 64.5f$.

Summing the frequencies and the products of midpoints and frequencies gives the components required for the mean equation.

Substitute known values into the mean formula and solve for

f

66.5(36 + f) = 2382 + 64.5f \implies 2f = 12 \implies f = 6$.

Equating the formula expression to the given mean of

66.5\text{ km/h}

allows solving for the unknown frequency.

Key Concept

Grouped Mean with Missing Frequency

Question

Answer

Step-by-Step Solution

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