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Magazine Chapter 11: Problem 13
The seven numbers \(7,10,12,17,21, x\) and \(y\) have a mean \(\mu=12\) and a
variance \(\sigma^{2}=\frac{172}{7}\). Find \(x\) and \(y\) given that \(x
Short Answer
Expert verified
The values are \(x = 5\) and \(y = 12\).
Step by step solution
01
Understand Mean Formula
The mean \( \mu \) of a set of numbers is calculated as the sum of all numbers divided by the count of numbers. For our set, the mean is given as \( \mu = 12 \). We have 7 numbers in the set \( (7, 10, 12, 17, 21, x, y) \). Therefore, we have the equation: \[ \frac{7 + 10 + 12 + 17 + 21 + x + y}{7} = 12. \]
02
Calculate Sums for Mean
Compute the sum of the known numbers: \( 7 + 10 + 12 + 17 + 21 = 67 \). Substitute this back into the mean equation: \[ \frac{67 + x + y}{7} = 12. \] Multiply through by 7 to eliminate the fraction: \[ 67 + x + y = 84. \] So, \[ x + y = 17. \]
03
Understand Variance Formula
The variance \( \sigma^2 \) is calculated as the average of the squared differences from the Mean: \[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (a_i - \mu)^2. \] We know that \( \sigma^2 = \frac{172}{7} \) for 7 numbers \( (7, 10, 12, 17, 21, x, y) \).
04
Calculate Known Sum of Squares
Using the formula for variance: \[ \frac{(7-12)^2 + (10-12)^2 + (12-12)^2 + (17-12)^2 + (21-12)^2 + (x-12)^2 + (y-12)^2}{7} = \frac{172}{7}. \] Calculate the known squared differences: \((7-12)^2 + (10-12)^2 + (12-12)^2 + (17-12)^2 + (21-12)^2 = 25 + 4 + 0 + 25 + 81 = 135.\)\[\frac{135 + (x-12)^2 + (y-12)^2}{7} = \frac{172}{7}. \] Multiply both sides by 7: \[ 135 + (x-12)^2 + (y-12)^2 = 172, \] and simplify to find: \[ (x-12)^2 + (y-12)^2 = 37. \]
05
Solve System of Equations
We now have two equations: 1) \( x + y = 17 \) and 2) \( (x-12)^2 + (y-12)^2 = 37 \). Substitute \( y = 17 - x \) into the second equation: \[ (x-12)^2 + ((17-x)-12)^2 = 37. \] Simplify: \[ (x-12)^2 + (5-x)^2 = 37. \] Expand: \[ (x-12)^2 = x^2 - 24x + 144 \text{ and } (5-x)^2 = x^2 - 10x + 25. \] Substitute back: \[ x^2 - 24x + 144 + x^2 - 10x + 25 = 37 \] Simplify: \[ 2x^2 - 34x + 169 = 37. \] Subtract 37 from both sides: \[ 2x^2 - 34x + 132 = 0. \] Divide through by 2: \[ x^2 - 17x + 66 = 0. \]
06
Solve Quadratic Equation
This is a quadratic equation: \( x^2 - 17x + 66 = 0 \). Use the quadratic formula \( \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), with \( a=1 \), \( b=-17 \), \( c=66 \). The discriminant \( b^2 - 4ac = (-17)^2 - 4 \cdot 1 \cdot 66 = 49 \) is positive, indicating two solutions. Calculate: \[ x = \frac{17 \pm 7}{2}. \] Thus, \( x = 12 \text{ or } x = 5. \)
07
Assign Values to x and y
Since \( x < y \), we assign \( x=5 \) and \( y=12 \). Therefore, the solution is \( x = 5 \) and \( y = 12 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equation
Understanding quadratic equations is crucial when solving problems involving two unknowns. A quadratic equation is of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( x \) is the unknown variable. In this exercise, we arrived at the quadratic equation \( x^2 - 17x + 66 = 0 \) by analyzing a system related to mean and variance. Quadratic equations can have two, one, or no real solutions, indicated by the discriminant \( b^2 - 4ac \). Here, the discriminant was found to be 49, a positive number, suggesting two real solutions. Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we calculated the potential values of \( x \) as 12 and 5. Having two possible values helps us understand different scenarios for the problem at hand.
System of Equations
A system of equations is a collection of two or more equations with the same set of variables. Solving such a system involves finding values for these variables that satisfy all equations simultaneously. In our exercise, we have a system formed by two equations: \( x + y = 17 \) and \( (x-12)^2 + (y-12)^2 = 37 \). The first equation deals with the sum of the numbers, while the second involves their variance. By substituting \( y = 17 - x \) into the second equation, we reduced a two-variable problem to a single-variable quadratic equation, which vastly simplifies solving. Understanding systems of equations allows us to uncover relationships between variables and solve complex real-world problems efficiently.
Variance Formula
Variance is a measure of how much the numbers in a data set differ from the mean, calculated using the formula: \( \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (a_i - \mu)^2 \). In this problem, we are given a variance of \( \frac{172}{7} \) for numbers including unknowns \( x \) and \( y \). This formula helps in quantifying the average spread of the data points. Similarly, the variance helped us form and ultimately solve the equations in conjunction with the mean. Once we computed the squared differences for the known numbers and substituted these into our variance equation, it allowed us to simplify and deduce the relationship \( (x-12)^2 + (y-12)^2 = 37 \). Recognizing the role of variance and its formulation was vital in establishing equations that facilitated finding the original problem's solution.
