In a grouped data set, the mean [tex](\bar{X}) = 40[/tex], the number of terms [tex](N) = 20 + 3n[/tex], and [tex]\Sigma fm = 1400[/tex]. Find the exact value of [tex]N[/tex] after finding the value of [tex]n[/tex].
To solve the problem, we need to find the exact value of [tex]\( N \)[/tex] by first determining the value of [tex]\( n \)[/tex]. Here’s the step-by-step solution:
1. Given Information: - Mean [tex]\( \bar{X} = 40 \)[/tex] - Sum of the product of frequency and mid-point values [tex]\( \Sigma fm = 1400 \)[/tex] - Number of terms [tex]\( N = 20 + 3n \)[/tex]
2. Formula for Mean: The formula for the mean in grouped data is: [tex]\[
\bar{X} = \frac{\Sigma fm}{N}
\][/tex]
3. Substitute Given Values: We can substitute the given values into the mean formula: [tex]\[
40 = \frac{1400}{20 + 3n}
\][/tex]
4. Solving for [tex]\( n \)[/tex]: - Multiply both sides of the equation by [tex]\( (20 + 3n) \)[/tex] to remove the denominator: [tex]\[
40 \times (20 + 3n) = 1400
\][/tex] - Simplify the left-hand side: [tex]\[
800 + 120n = 1400
\][/tex] - Isolate the term with [tex]\( n \)[/tex]: [tex]\[
120n = 1400 - 800
\][/tex] - Simplify the right-hand side: [tex]\[
120n = 600
\][/tex] - Solve for [tex]\( n \)[/tex]: [tex]\[
n = \frac{600}{120}
\][/tex] [tex]\[
n = 5
\][/tex]
5. Find the Exact Value of [tex]\( N \)[/tex]: - Substitute [tex]\( n = 5 \)[/tex] back into the equation for [tex]\( N \)[/tex]: [tex]\[
N = 20 + 3n
\][/tex] [tex]\[
N = 20 + 3 \times 5
\][/tex] [tex]\[
N = 20 + 15
\][/tex] [tex]\[
N = 35
\][/tex]
6. Conclusion: The exact value of [tex]\( N \)[/tex] is 35.
Therefore, the solution correctly determines that [tex]\( n = 5 \)[/tex] and the number of terms [tex]\( N = 35 \)[/tex].
