Question

Let m be the mean and σ be the standard deviation of the distribution

xi	0	1	2	3	4	5
fi	k+2	2k	k2-1	k2-1	k2+1	k-3

where ∑f_i = 62. if [x] denotes the greatest integer ≤ x, then [μ² + σ²] is equal

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

The Correct Option is C

To solve this problem, we need to find the greatest integer value of \( \mu^2 + \sigma^2 \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation of the given frequency distribution.

Let's denote \( \text{Sum of frequencies} \, \sum f_i = 62 \). 

1. Calculate the value of \( k \) by using the sum of the frequencies: \((k+2) + 2k + (k^2-1) + (k^2-1) + (k^2+1) + (k-3) = 62\).

Simplify this equation:

• Combine like terms: \(3k^2 + 4k - 2 = 62\)
• Simplify further: \(3k^2 + 4k - 64 = 0\)

1. We solve for \( k \) using the quadratic formula \( ax^2 + bx + c = 0 \), where \( a = 3 \), \( b = 4 \), \( c = -64 \). \(k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
   • Calculate the discriminant: \(b^2 - 4ac = 4^2 - 4 \times 3 \times (-64) = 16 + 768 = 784\)
   • The square root of 784 is 28.
   • Therefore, \(k = \frac{-4 \pm 28}{6}\). The possible solutions are \( k = 4 \) or \( k = -\frac{32}{6} \) (not valid since \( k \) must be positive).
2. Take \( k = 4 \). Calculate the distribution:
   • Substitute \( k = 4 \) into the frequency terms:
   • \(f_0 = (4+2) = 6$\), \( f_1 = 2 \times 4 = 8 \), \( f_2 = (4^2 - 1) = 15 \), \( f_3 = (4^2 - 1) = 15 \), \( f_4 = (4^2 + 1) = 17 \), \( f_5 = (4-3) = 1\)
   • Verify: \( 6 + 8 + 15 + 15 + 17 + 1 = 62 \)
3. Compute the mean:
   • \( \mu = \frac{\sum(f_i \cdot x_i)}{n} = \frac{6 \cdot 0 + 8 \cdot 1 + 15 \cdot 2 + 15 \cdot 3 + 17 \cdot 4 + 1 \cdot 5}{62} \)
   • \( \mu = \frac{177}{62} = 2.85 \) approximately
4. Compute the variance, \( \sigma^2 \):
   • \( \sigma^2 = \frac{\sum(f_i \cdot x_i^2)}{n} - \mu^2 \)
   • \( \sum(f_i \cdot x_i^2) = 6 \cdot 0^2 + 8 \cdot 1^2 + 15 \cdot 2^2 + 15 \cdot 3^2 + 17 \cdot 4^2 + 1 \cdot 5^2 \)
   • \( = 0 + 8 + 60 + 135 + 272 + 25 = 500 \)
   • \( \sigma^2 = \frac{500}{62} - (2.85)^2 \approx 1.47 \)
5. Calculate \( \mu^2 + \sigma^2 \):
   • \( \mu^2 + \sigma^2 = (2.85)^2 + 1.47 = 8.60 \)
   • The greatest integer less than or equal to \( 8.60 \) is 8.

Therefore, \([ \mu^2 + \sigma^2 ] = \) 8.