0.1 - Review of Algebra

0.1 - Review of Algebra

Knowledge of the following mathematical operations is required for STAT 200:

  • Addition
  • Subtraction
  • Division
  • Multiplication
  • Radicals (square roots)
  • Exponents
  • Summations  \(\left( \sum \right) \)
  • Basic linear equations (i.e., \(y=mx+b\))

The ability to perform these operations in the appropriate order is necessary. Follow the links below to work through a quick review of order of operations, summations, factorials, and combinations. 

We want our students to be successful! In the past, students without knowledge of these topics have struggled to be successful in STAT 200. If you have any questions about these mathematical operations after reviewing these materials, please contact your instructor as soon as possible!


0.1.1 - Order of Operations

0.1.1 - Order of Operations

When performing a series of mathematical operations, begin with those inside parentheses or brackets. Next, calculate any exponents or square roots. This is followed by multiplication and division, and finally, addition and subtraction.

Order of Operations

Example: Standard Error for Two Proportions

This is a formula that you will see in Lesson 9: \( \sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}\)

\(p\) is a proportion and \(n\) is a sample size. Let's look at an example of working through this formula with the following values:

\(p_1=0.60\)

\(p_2=0.35\)

\(n_1=40\)

\(n_2=80\)

 

We can begin by plugging these values into the formula:

\( \sqrt{\frac{0.60(1-0.60)}{40}+\frac{0.35(1-0.35)}{80}}\)

The first operations that we perform are the ones in the parentheses:

\( \sqrt{\frac{0.60(0.40)}{40}+\frac{0.35(0.65)}{80}}\)

Though not typically shown, the values under the square root symbol in the fractions are treated as if they are in parentheses:

\(\sqrt{\left ( \frac{0.60(0.40)}{40}\right )+ \left (  \frac{0.35(0.65)}{80}\right ) } \)

Working within each set of parentheses, are next step is to multiply in the numerators:

\(\sqrt{\left (  \frac{0.24}{40} \right ) + \left (   \frac{0.2275}{80}\right ) } \)

Then, the division (i.e., the fractions):

\( \sqrt{0.006+0.00284375}\)

The addition underneath the square root:

\( \sqrt{0.00884375}\)

And finally, we take the square root:

\(0.0940\) 

Example: Pooled Proportion


0.1.2 - Summations

0.1.2 - Summations

\(\Sigma\)

This is the Greek capital letter "sigma." In math, this symbol is also known as a summation. This tells us that we should add a series of numbers (i.e., take the sum).

Example: Candy

Candy Corn

Four children are comparing how many pieces of candy they have:

ID Child Pieces of Candy
1 Marty 9
2 Harold 8
3 Eugenia 10
4 Kevi 8

We could say that: \(x_{1}=9\),\(x_{2}=8\), \(x_{3}=10\), and \(x_{4}=8\).

If we wanted to know how many total pieces of candy the group of children had, we could add the four numbers. The notation for this is:

\[\sum x_{i}\]

So, for this example, \(\sum x_{i}=9+8+10+8=35\)

To conclude, combined, the four children have \(35\) pieces of candy.

You will first see a summation in Lesson 2 when you learn to compute a sample mean (\(\overline{x}\)). This is also known as the average. You will learn that \(\overline{x}=\frac{\Sigma{X}}{n}\); in other words, the sum of all of the observations divided by the number of observations. 

In this example, \(\overline{x}=\frac{\Sigma{X}}{n}=\frac{9+8+10+8}{4}=\frac{35}{4}=8.75\)

In this sample of four children, the average number of pieces of candy is \(8.75\)


0.1.3 - Basic Linear Equations

0.1.3 - Basic Linear Equations

You may recall from an algebra class that the formula for a straight line is \(y=mx+b\) where \(m\) is the slope and \(b\) is the y-intercept. The x-axis is the horizontal axis and the y-axis is the vertical axis.

Axes

The y-intercept is where the line crosses the y-axis.  The slope is a measure of change in y over change in x, sometimes written as \(\frac{\Delta y}{\Delta x}\) where \(\Delta\) ("delta") means change.

Example

\(y=2x+1\)

Here, the y-intercept is 1 and the slope is 2.  This line will cross the y-axis at the point (0, 1). From there, for every one increase in \(x\), \(y\) will increase by 2. In other words, for every one unit we move to the right, we will move up 2. 

Slope

In statistics, the y-intercept is typically written before the slope. The equation for the line above would be written as \(y=1+2x\).  

Different notation may also be used:

\(\widehat y = a + bx\), where \(\widehat y\) is the predicted value of y, \(a\) is the y-intercept and \(b\) is the slope.

\(\widehat y = b_0 + b_1 x\), where \(\widehat y\) is the predicted value of y, \(b_0\) is the y-intercept, and \(b_1\) is the slope.


Legend
[1]Link
Has Tooltip/Popover
 Toggleable Visibility