To find the value of [tex]\(x\)[/tex] given the data and their frequencies, and knowing that the mean is [tex]\(\frac{43}{4}\)[/tex], we first write down the information:
### Data and Frequencies:
- Data: [tex]\(\{1, 2, 3, 4, 5\}\)[/tex]
- Frequencies: [tex]\(\{x+2, x-1, x-3, x+4, 3x-4\}\)[/tex]
### Mean Calculation:
The formula for the weighted mean is given by:
[tex]\[
\bar{x} = \frac{\sum (data_i \times Frequency_i)}{\sum Frequency_i}
\][/tex]
Given the mean:
[tex]\[
\bar{x} = \frac{43}{4}
\][/tex]
### Step-by-Step Solution:
1. Calculate the sum of frequencies:
[tex]\[
\sum Frequency_i = (x + 2) + (x - 1) + (x - 3) + (x + 4) + (3x - 4)
\][/tex]
Combine like terms:
[tex]\[
\sum Frequency_i = x + 2 + x - 1 + x - 3 + x + 4 + 3x - 4 = 7x - 2
\][/tex]
2. Calculate the sum of the product of data and frequencies:
[tex]\[
\sum (data_i \times Frequency_i) = 1(x+2) + 2(x-1) + 3(x-3) + 4(x+4) + 5(3x-4)
\][/tex]
Expand each term:
[tex]\[
1(x + 2) = x + 2
\][/tex]
[tex]\[
2(x - 1) = 2x - 2
\][/tex]
[tex]\[
3(x - 3) = 3x - 9
\][/tex]
[tex]\[
4(x + 4) = 4x + 16
\][/tex]
[tex]\[
5(3x - 4) = 15x - 20
\][/tex]
Sum these expressions:
[tex]\[
x + 2 + 2x - 2 + 3x - 9 + 4x + 16 + 15x - 20 = 25x - 13
\][/tex]
3. Set up the mean equation:
[tex]\[
\frac{25x - 13}{7x - 2} = \frac{43}{4}
\][/tex]
4. Cross-multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
4(25x - 13) = 43(7x - 2)
\][/tex]
Expand both sides:
[tex]\[
100x - 52 = 301x - 86
\][/tex]
5. Isolate [tex]\(x\)[/tex]:
[tex]\[
100x - 301x = -86 + 52
\][/tex]
Combine like terms:
[tex]\[
-201x = -34
\][/tex]
Divide both sides by -201:
[tex]\[
x = \frac{34}{201}
\][/tex]
So, the value of [tex]\(x\)[/tex] that satisfies the given conditions is:
[tex]\[
x = \frac{34}{201}
\][/tex]
