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Mean of Data Set of Integers - Statistics










































If the mean of the data set of integers {3x+7, 5x+11< 4x-2, 8x+2, x+2} is x squared, what is x, then find the data values and the mean.

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Mar 22, 2012
Mean of Data Set of Integers
by: Staff

Question:

If the mean of the data set of integers {3x+7, 5x+11< 4x-2, 8x+2, x+2} is x squared, what is x, then find the data values and the mean.


Answer:

{3x+7, 5x+11, 4x-2, 8x+2, x+2}

The find the mean, add up the five data points and divide by 5

Mean = [(3x+7) + (5x+11) + (4x-2) + (8x+2) + (x+2)] / 5


Group like terms

Mean = [(3x+ 5x + 4x + 8x + x) + (7 +11 - 2 + 2 + 2)] / 5

Mean = (21x +20) / 5


The problem states that the mean of the data set is x²

(21x +20)/5 = x²


Solve for x

Multiply each side of the equation by 5

(21x + 20)/5 = x²

5*(21x + 20)/5 = 5*x²

(21x + 20) * (5/5) = 5*x²

(21x + 20) * (1) = 5*x²

21x + 20 = 5x²


Subtract 21x from each side of the equation

21x + 20 - 21x = 5x² - 21x

21x - 21x + 20 = 5x² - 21x

(21x - 21x) + 20 = 5x² - 21x

0 + 20 = 5x² - 21x

20 = 5x² - 21x



Subtract 20 from each side of the equation

20 = 5x² - 21x

20 - 20 = 5x² - 21x - 20

0 = 5x² - 21x - 20

5x² - 21x - 20 = 0


This is a quadratic equation. The quadratic formula can be used to solve for x.

ax² + bx + c = 0

x = [-b ± sqrt(b² - 4ac)]/2a

Using the quadratic formula, the roots to your quadratic equation are:

5x² - 21x - 20 = 0

x = [-(-21) ± sqrt((-21)² - 4*5*(-20)]/(2*5)

x = [21 ± sqrt(441 + 400)]/(10)

x = [21 ± sqrt(841)]/(10)

x = [21 ± 29]/(10)

x₁ = [21 - 29]/(10)

x₁ = -8/10

x₁ = -4/5

since the problem statement specifies that x must be an integer, x₁ is not a valid solution

x₂ = [21 + 29]/(10)

x₂ = 50/10

x₂ = 5

>>> the final answer is: x = 5


The data values:

The data values can be calculated by substituting 5 for x in all of the original data points.

{3x+7, 5x+11, 4x-2, 8x+2, x+2}

{3*5+7, 5*5+11, 4*5-2, 8*5+2, 5+2}

{22, 36, 18, 42, 7}


>>> the final answer is: data values = {22, 36, 18, 42, 7}


The mean:

The mean can be calculated by adding up all the data values and dividing by 5:

Mean = [(3x+7) + (5x+11) + (4x-2) + (8x+2) + (x+2)] / 5

Mean = [(22) + (36) + (18) + (42) + (7)] / 5

Mean = 125 / 5

Mean = 25


>>> the final answer is: mean = 25


------------------------------------------

Check the answer to see if x² equals the mean

25 = x²

25 = 5², OK → x = 5 is a valid solution




Thanks for writing.

Staff
www.solving-math-problems.com



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