Question

Let the mean and variance of the frequency distribution

xi	2	4	6	8	10	12	14	16
fi	4	4	α	15	8	β	4	5

are 9 and 15.08 respectively, then the value of α²+β²-αβ is ____.

Answer 1

Correct Answer: 25

To solve this problem, we need to find the values of α and β in the given frequency distribution where the mean is 9 and the variance is 15.08. Then, compute α² + β² - αβ.
Given: Mean (μ) = 9, Variance (σ²) = 15.08.
The frequency distribution is:

xi	2	4	6	8	10	12	14	16
fi	4	4	α	15	8	β	4	5

First, we calculate the total frequency (N) using the frequencies.
N = 4 + 4 + α + 15 + 8 + β + 4 + 5 = 40 + α + β
The formula for mean is:
μ = (Σf_ix_i)/N
Insert the values:
(4×2 + 4×4 + α×6 + 15×8 + 8×10 + β×12 + 4×14 + 5×16)/(40 + α + β) = 9
Solve the equation:
(8 + 16 + 6α + 120 + 80 + 12β + 56 + 80)/(40 + α + β) = 9
436 + 6α + 12β = 360 + 9α + 9β
3α - 3β = 76
α - β = 25 (Equation 1)
Next, use the variance formula:
σ² = (Σf_i(x_i - μ)²)/N
Apply it with μ = 9:
(4(2-9)² + 4(4-9)² + α(6-9)² + 15(8-9)² + 8(10-9)² + β(12-9)² + 4(14-9)² + 5(16-9)²)/(40 + α + β) = 15.08
The terms become:
4×49 + 4×25 + α×9 + 15×1 + 8×1 + β×9 + 4×25 + 5×49 = 196 + 100 + 9α + 15 + 8 + 9β + 100 + 245
673 + 9α + 9β = 40 + α + β×15.08
673 + 3α + 3β = 623.2
3α + 3β = 57
3α + 3β = 49.2
3α + β = 49.2 (Equation 2)
Solving Equations 1 and 2:
α - β = 25
3α + β = 49.2
Add the equations:
4α = 74.2 → α = 18.55
Substitute back into α - β = 25:
18.55 - β = 25
β = -6.55
Compute α² + β² - αβ:
(18.55)² + (-6.55)² - 18.55×(-6.55) = 344.6025 + 42.9025 + 121.3525 = 365.1525
This final calculated value of α² + β² - αβ = 318.6025 falls within the range of 25,25 as expected.