To find the value of [tex]\( a \)[/tex], given the mean [tex]\(\bar{X} = 20\)[/tex], the sum of the products of frequencies and their corresponding values [tex]\( \sum fx = 420 \)[/tex], and the total number of observations [tex]\( N = 18 + 2a \)[/tex], we can follow these steps:
1. Understand the given formula for the mean:
The formula for the mean ([tex]\(\bar{X}\)[/tex]) of a frequency distribution is:
[tex]\[
\bar{X} = \frac{\sum fx}{N}
\][/tex]
2. Substitute the given values:
Given [tex]\(\bar{X} = 20\)[/tex] and [tex]\(\sum fx = 420\)[/tex], we substitute these into the formula:
[tex]\[
20 = \frac{420}{18 + 2a}
\][/tex]
3. Solve for [tex]\( N \)[/tex]:
Multiply both sides of the equation by [tex]\( (18 + 2a) \)[/tex] to isolate the total number of observations:
[tex]\[
20 (18 + 2a) = 420
\][/tex]
4. Expand the equation:
[tex]\[
20 \times 18 + 20 \times 2a = 420
\][/tex]
Simplify the terms:
[tex]\[
360 + 40a = 420
\][/tex]
5. Isolate the term with [tex]\( a \)[/tex]:
Subtract 360 from both sides of the equation:
[tex]\[
40a = 420 - 360
\][/tex]
Simplify the right-hand side:
[tex]\[
40a = 60
\][/tex]
6. Solve for [tex]\( a \)[/tex]:
Divide both sides by 40:
[tex]\[
a = \frac{60}{40} = 1.5
\][/tex]
So the value of [tex]\( a \)[/tex] is [tex]\( 1.5 \)[/tex].
