Explanation
Solution
1. Find the Mean (xˉ)
The given observations are x1=3,x2=7,x3=12,x4=α,x5=43−α.
The number of observations n=5.
2. Apply the Variance Formula
The formula for variance (σ2) is:
Substitute the values:
σ2=532+72+122+α2+(43−α)2−(13)2
σ2=59+49+144+α2+(1849+α2−86α)−169
3. Simplify the Equation
To combine the terms, take a common denominator:
4. Determine the condition for Natural Number
For σ2 to be a natural number, the numerator must be divisible by 5.
Simplify the coefficients modulo 5:
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2≡2(mod5)
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−86≡−1≡4(mod5)
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1206≡1(mod5)
The congruence becomes:
Let's test values of α(mod5):
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If α≡0:2(0)2+4(0)+1=1≡0
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If α≡1:2(1)2+4(1)+1=7≡2≡0
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If α≡2:2(2)2+4(2)+1=8+8+1=17≡2≡0
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If α≡3:2(3)2+4(3)+1=18+12+1=31≡1≡0
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If α≡4:2(4)2+4(4)+1=32+16+1=49≡4≡0
Conclusion
Since no value of α∈{0,1,2,3,4} satisfies the condition, there are no natural numbers α that make the variance a natural number.
Correct Option: (A) 0
