Understanding Mean and Variance for Finding Observations

This problem involves using statistical concepts, specifically the mean and variance, to determine two unknown values within a set of observations. We are given the mean and variance of five observations, along with the values of three of these observations. Let's break down the problem and use the formulas for mean and variance to find the missing numbers.

Identifying the Knowns and Unknowns

We have five observations in total. Let the two unknown observations be \(a\) and \(b\). The five observations are: 11, 16, 20, \(a\), \(b\).

• Total number of observations, \(n = 5\).
• Mean (\(\bar{x}\)) of the five observations = 14.
• Variance (\(\sigma^2\)) of the five observations = 13.2.

Step 1: Using the Mean to Find the Sum of Unknown Observations

The mean of a set of observations is calculated as the sum of all observations divided by the number of observations.

The formula for the mean is:

\[ \bar{x} = \frac{\sum x_i}{n} \]

We know \(\bar{x} = 14\) and \(n = 5\). The sum of observations is \(11 + 16 + 20 + a + b\).

\[ 14 = \frac{11 + 16 + 20 + a + b}{5} \]

\[ 14 = \frac{47 + a + b}{5} \]

Multiply both sides by 5:

\[ 14 \times 5 = 47 + a + b \]

\[ 70 = 47 + a + b \]

Subtract 47 from both sides to find the sum of \(a\) and \(b\):

\[ a + b = 70 - 47 \]

\[ a + b = 23 \]

This gives us the sum of the two missing observations.

Step 2: Using the Variance to Form Another Equation

The variance measures how spread out the observations are from the mean. The formula for population variance is:

\[ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \]

We know \(\sigma^2 = 13.2\), \(\bar{x} = 14\), and \(n = 5\). Let's calculate the squared differences from the mean for the known observations:

• For 11: \((11 - 14)^2 = (-3)^2 = 9\)
• For 16: \((16 - 14)^2 = (2)^2 = 4\)
• For 20: \((20 - 14)^2 = (6)^2 = 36\)

The squared differences for the unknown observations are \((a - 14)^2\) and \((b - 14)^2\).

Substitute these values into the variance formula:

\[ 13.2 = \frac{9 + 4 + 36 + (a - 14)^2 + (b - 14)^2}{5} \]

\[ 13.2 = \frac{49 + (a - 14)^2 + (b - 14)^2}{5} \]

Multiply both sides by 5:

\[ 13.2 \times 5 = 49 + (a - 14)^2 + (b - 14)^2 \]

\[ 66 = 49 + (a - 14)^2 + (b - 14)^2 \]

Subtract 49 from both sides:

\[ 66 - 49 = (a - 14)^2 + (b - 14)^2 \]

\[ 17 = (a - 14)^2 + (b - 14)^2 \]

This is our second equation.

Step 3: Solving the System of Equations

We have two equations with two variables \(a\) and \(b\):

1. \(a + b = 23\)
2. \((a - 14)^2 + (b - 14)^2 = 17\)

From equation 1, we can express \(b\) in terms of \(a\): \(b = 23 - a\). Substitute this into equation 2:

\[ (a - 14)^2 + ((23 - a) - 14)^2 = 17 \]

\[ (a - 14)^2 + (9 - a)^2 = 17 \]

Expand the squared terms:

\[ (a^2 - 28a + 196) + (81 - 18a + a^2) = 17 \]

Combine like terms:

\[ a^2 + a^2 - 28a - 18a + 196 + 81 = 17 \]

\[ 2a^2 - 46a + 277 = 17 \]

Move all terms to one side to form a quadratic equation:

\[ 2a^2 - 46a + 277 - 17 = 0 \]

\[ 2a^2 - 46a + 260 = 0 \]

Divide the entire equation by 2:

\[ a^2 - 23a + 130 = 0 \]

This is a quadratic equation in the form \(ax^2 + bx + c = 0\). We can solve this by factoring. We need two numbers that multiply to 130 and add up to -23. These numbers are -10 and -13.

So, the equation can be factored as:

\[ (a - 10)(a - 13) = 0 \]

This gives two possible values for \(a\):

• \(a - 10 = 0 \implies a = 10\)
• \(a - 13 = 0 \implies a = 13\)

If \(a = 10\), substitute into \(b = 23 - a\) to find \(b\): \(b = 23 - 10 = 13\). If \(a = 13\), substitute into \(b = 23 - a\) to find \(b\): \(b = 23 - 13 = 10\).

In both cases, the other two observations are 10 and 13.

Verification of the Observations

Let's verify if the observations 11, 16, 20, 10, and 13 give the specified mean and variance.

• Sum = \(11 + 16 + 20 + 10 + 13 = 70\)
• Mean = \(70 / 5 = 14\). This matches the given mean.

Calculate the variance:

Sum of squared deviations = \(9 + 4 + 36 + 16 + 1 = 66\).

Variance = \(\frac{\text{Sum of squared deviations}}{\text{Number of observations}} = \frac{66}{5} = 13.2\). This matches the given variance.

Thus, the two other observations are 10 and 13.

Revision Table: Key Statistical Concepts

Additional Information: Solving Quadratic Equations

In this problem, finding the unknown observations led us to a quadratic equation of the form \(a^2 - 23a + 130 = 0\). A quadratic equation is a polynomial equation of the second degree.

There are several ways to solve quadratic equations:

• Factoring: This method involves expressing the quadratic as a product of two linear factors, as we did in this solution \(((a-10)(a-13)=0)\). This works when the roots are rational.
• Using the Quadratic Formula: For any quadratic equation \(Ax^2 + Bx + C = 0\), the solutions for \(x\) are given by:

  \[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

  This formula can solve any quadratic equation, regardless of whether it's easily factorable.
• Completing the Square: This method involves rearranging the equation so that one side is a perfect square trinomial.

Understanding how to solve quadratic equations is a useful skill in many areas of mathematics and problem-solving, including statistics.