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Updated on: 30 Mar 2020, 12:33
Originally posted by MathRevolution on 27 Mar 2020, 02:07.
Last edited by MathRevolution on 30 Mar 2020, 12:33, edited 1 time in total.
Last edited by MathRevolution on 30 Mar 2020, 12:33, edited 1 time in total.
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D
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[GMAT math practice question]
The average and standard deviation of x1, x2, x3, x4 is 5 and \(\sqrt{2}\), respectively. What is the average of x1^2, x2^2, x3^2, x4^2?
A. 11
B. 17
C. 22
D. 27
E. 33
The average and standard deviation of x1, x2, x3, x4 is 5 and \(\sqrt{2}\), respectively. What is the average of x1^2, x2^2, x3^2, x4^2?
A. 11
B. 17
C. 22
D. 27
E. 33
ArunSharma12
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29 Mar 2020, 00:26
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MathRevolution
S.D = \(\sqrt{\frac{\sum(x-m)^2}{n}}\); where x is individual element, m is average/mean, n is the number of elements
2 = \(\sqrt{\frac{(x_1-5)^2+(x_2-5)^2+(x_3-5)^2+(x_4-5)^2}{4}}\)
squaring both sides,
4 = \(\frac{(x_1-5)^2+(x_2-5)^2+(x_3-5)^2+(x_4-5)^2}{4}\)
16 =\((x_1-5)^2+(x_2-5)^2+(x_3-5)^2+(x_4-5)^2\)
16 =\({x_1}^2+25-10*x_1+{x_2}^2+25-10*x_2+{x_3}^2+25-10*x_3+{x_4}^2+25-10*x_4\)
16 =\(({x_1}^2+{x_2}^2+{x_3}^2+{x_4}^2)+100-10*(x_1+x_2+x_3+x_4)\)
16 =\(({x_1}^2+{x_2}^2+{x_3}^2+{x_4}^2)+100-10*(20)\);sum of the four numbers is 20
116 = \({x_1}^2+{x_2}^2+{x_3}^2+{x_4}^2\)
\(\frac{{x_1}^2+{x_2}^2+{x_3}^2+{x_4}^2}{4}=\frac{116}{4}\)=29
I think option D needs to be changed to 29.
Updated on: 30 Mar 2020, 12:32
Originally posted by MathRevolution on 30 Mar 2020, 00:34.
Last edited by MathRevolution on 30 Mar 2020, 12:32, edited 1 time in total.
Last edited by MathRevolution on 30 Mar 2020, 12:32, edited 1 time in total.
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=>
We have (x1+x2+x3+x4)/4 = 5 or x1+x2+x3+x4 = 20.
Since we know the standard deviation of x1,x2,x3, and x4 is 2, we have
((x1 - 5)^2 + (x2 - 5)^2 + (x3 - 5)^2 + (x4 - 5)^2)/4 = \((\sqrt{2})^2\)
=> (x1 - 5)^2 + (x2 - 5)^2 + (x3 - 5)^2 + (x4 - 5)^2 = 8
=> (x1^2 - 10x1 + 25) + (x2^2 - 10x2 + 25) + (x3^2 - 10x3 + 25) + (x4^2 - 10x4 + 25) = 8
=> (x1^2 + x2^2 + x3^2 + x4^2) - 10(x1 + x2 + x3 + x4) + 25 + 25 + 25 + 25 = 8
=> (x1^2 + x2^2 + x3^2 + x4^2) - 10(20) + 100 = 8
=> (x1^2 + x2^2 + x3^2 + x4^2) = 108
=> (x1^2 + x2^2 + x3^2 + x4^2)/4 = 27
Therefore, D is the answer.
Answer: D
We have (x1+x2+x3+x4)/4 = 5 or x1+x2+x3+x4 = 20.
Since we know the standard deviation of x1,x2,x3, and x4 is 2, we have
((x1 - 5)^2 + (x2 - 5)^2 + (x3 - 5)^2 + (x4 - 5)^2)/4 = \((\sqrt{2})^2\)
=> (x1 - 5)^2 + (x2 - 5)^2 + (x3 - 5)^2 + (x4 - 5)^2 = 8
=> (x1^2 - 10x1 + 25) + (x2^2 - 10x2 + 25) + (x3^2 - 10x3 + 25) + (x4^2 - 10x4 + 25) = 8
=> (x1^2 + x2^2 + x3^2 + x4^2) - 10(x1 + x2 + x3 + x4) + 25 + 25 + 25 + 25 = 8
=> (x1^2 + x2^2 + x3^2 + x4^2) - 10(20) + 100 = 8
=> (x1^2 + x2^2 + x3^2 + x4^2) = 108
=> (x1^2 + x2^2 + x3^2 + x4^2)/4 = 27
Therefore, D is the answer.
Answer: D
