The distribution of daily solar energy generation (in kWh) recorded at an agricultural res… — JAMB UTME

The distribution of daily solar energy generation (in kWh) recorded at an agricultural research station over a period of days is summarized in the table below:

Daily Generation (kWh)	Frequency ( $f$ )
$10 - 19$	$3$
$20 - 29$	$x$
$30 - 39$	$8$
$40 - 49$	$5$
$50 - 59$	$x + 2$

If the mean daily solar energy generation for the recorded period is $37.5\text{ kWh}$ , what is the value of $x$ ?

A
$1$
B
$3$
$6$ Answer
D
$11$

Answer

6

The exact class marks are

14.5, 24.5, 34.5, 44.5,

and

54.5

. The total frequency is

\sum f = 18 + 2x

and the sum of products is

\sum f x = 651 + 79x

. Substituting these into

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

with

\bar{x} = 37.5

gives

675 + 75x = 651 + 79x

, which solves to

4x = 24

, giving the value

6

Step-by-Step Solution

Determine the class mark (midpoint

x_i

) for each class interval

The midpoints are

14.5, 24.5, 34.5, 44.5,

and

54.5

The class mark is calculated as

\frac{\text{Lower Limit} + \text{Upper Limit}}{2}

Calculate total frequency

\sum f

and total product sum

\sum f x_i

\sum f = 3 + x + 8 + 5 + (x + 2) = 18 + 2x

\sum f x_i = 3(14.5) + x(24.5) + 8(34.5) + 5(44.5) + (x+2)(54.5) = 43.5 + 24.5x + 276 + 222.5 + 54.5x + 109 = 651 + 79x

To compute the mean of grouped data, sum the products of each frequency and its corresponding class mark.

Set up the mean formula equation and solve for

x

\bar{x} = \frac{\sum f x_i}{\sum f} \implies 37.5 = \frac{651 + 79x}{18 + 2x}

37.5(18 + 2x) = 651 + 79x

675 + 75x = 651 + 79x

79x - 75x = 675 - 651

4x = 24 \implies x = 6

Equating the expression for mean to the given mean value

37.5\text{ kWh}

yields a linear equation in

x

Key Concept

Measures of Central Tendency for Grouped Data - Finding Unknown Frequencies from Grouped Mean

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