Algebra Review

Algebra Review

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

  • Addition
  • Subtraction
  • Division
  • Multiplication
  • Radicals (i.e., square roots)
  • Exponents
  • Summations  \(\left( \sum \right) \)
  • Factorials (!)

Additionally, the ability to perform these operations in the appropriate order is necessary. Use these materials to check your understanding and preparation for taking STAT 200.

We want our students to be successful! And we know that students that do not possess a working knowledge of these topics will struggle to participate successfully in STAT 200. 

Review Materials

Are you ready? As a means of helping students assess whether or not what they currently know and can do meets the expectations of instructors of STAT 200, the online program has put together a brief review of these concepts and methods. This is then followed by a short self-assessment exam that will help you determine if this prerequisite knowledge is readily available for you to apply.

Self-Assessment Procedure

  1. Review the concepts and methods on the pages in this section of this website. 
  2. Download and complete the Self-Assessment Exam.
  3. Review the Self-Assessment Exam Solutions and determine your score.

Your score on this self-assessment should be 100%! If your score is below this you should consider further review of these materials and are strongly encouraged to take MATH 021 or an equivalent course.

If you have struggled with the methods that are presented in the self-assessment, you will indeed struggle in the courses that expect this foundation.

A.1 Order of Operations

A.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.

  1. Parentheses
  2. Exponents & Square Roots
  3. Multiplication and Division
  4. Addition and Subtraction

Example A.1

Simplify: $(5+\dfrac{9}{3})^{2}$

Answer
\begin{align} (5+\frac{9}{3})^{2} &=(5+3)^{2} &&\text{(Parentheses first, 9/3)}\\ &=8^{2}&&\text{(Add inside parentheses)}\\ &=64 &&\text{(Square 8)}\\
\end{align}

Example A.2

Simplify: $\dfrac{5+6+7}{3}$

Answer
\begin{align} \frac{5+6+7}{3} &=\frac{18}{3}  &&\text{(Simplify the numerator)}\\ &=6 &&\text{(Divide)}\\
\end{align}

Example A.3

Simplify: $\dfrac{2^{2}+3^{2}+4^{2}}{3-1}$

Answer
\begin{align} \frac{2^{2}+3^{2}+4^{2}}{3-1} &=\frac{4+9+16}{2} &&\text{(Exponents in the numerator)}\\ &=\frac{29}{2} &&\text{(Simplify the numerator)}\\ &=14.5 &&\text{(Divide)}\\\end{align}

A.2 Summations

A.2 Summations

This is the upper-case Greek letter sigma. A sigma tells us that we need to sum (i.e., add) a series of numbers.

\[\sum\]

For example, 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.

In statistics, some equations include the sum of all of the squared values (i.e., square each item, then add). The notation is:

\[\sum x_{i}^{2}\]

or

\[\sum (x_{i}^{2})\]

Here, \(\sum x_{i}^{2}=9^{2}+8^{2}+10^{2}+8^{2}=81+64+100+64=309\).

Sometimes we want to square a series of numbers that have already been added. The notation for this is:

\[(\sum x_{i})^{2}\]

Here,\( (\sum x_{i})^{2}=(9+8+10+8)^{2}=35^{2}=1225\)

Note that \(\sum x_{i}^{2}\) and \((\sum x_{i})^{2}\) are different.

Summations 

Here is a brief review of summations as they will be applied in STAT 200:

A.3 Factorials

A.3 Factorials

Factorials are symbolized by exclamation points (!).

A factorial is a mathematical operation in which you multiply the given number by all of the positive whole numbers less than it. In other words \(n!=n \times (n-1) \times … \times 2 \times 1\).

For example,

“Four factorial” = \(4!=4\times3\times2\times1=24\)

“Six factorial” = \(6!=6\times5\times4\times3\times2\times1)=720\)

When we discuss probability distributions in STAT 200 we will see a formula that involves dividing factorials. For example,

\[\frac{3!}{2!}=\frac{3\times2\times1}{2\times1}=3\]

Here is another example,

\[\frac{6!}{2!(6-2)!}=\frac{6\times5\times4\times3\times2\times1}{(2\times1)(4\times3\times2\times1)}=\frac{6\times5}{2}=\frac{30}{2}=15\]

Also, note that 0! = 1

Factorials 

Here is a brief review of factorials as they will be applied in STAT 200:

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A.4 Self-Assess

A.4 Self-Assess

Self-Assessment Procedure

  1. Review the concepts and methods on the pages in this section of this website.
  2. Download and Complete the STAT 200 Algebra Self-Assessment
  3. Determine your Score by Reviewing the STAT 200 Algebra Self-Assessment: Solutions.

Your score on this self-assessment should be 100%!  If your score is below this you should consider further review of these materials and are strongly encouraged to take MATH 021 or an equivalent course.

If you have struggled with the methods that are presented in the self assessment, you will indeed struggle in the courses above that expect this foundation.

Note: These materials are NOT intended to be a complete treatment of the ideas and methods used in these algebra methods. These materials and the accompanying self-assessment are simply intended as simply an 'early warning signal' for students.  Also, please note that completing the self-assessment successfully does not automatically ensure success in any of the courses that use this foundation. 

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