# 0: Prerequisite Skills

0: Prerequisite Skills

## What is Statistics?

Welcome to STAT 200! In this Lesson, you will be briefly introduced to the field of statistics and we will review some of the prerequisite skills that are necessary for success in this course. Let's start by looking at some general definitions.

Statistics
The art and science of answering questions and exploring ideas through the processes of gathering data, describing data, and making inferences about a population on the basis of a smaller sample.
Statistical literacy

“People’s ability to interpret and critically evaluate statistical information and data-based arguments appearing in diverse media channels, and their ability to discuss their opinions regarding such statistical information” (Gal, as cited by Rumsey, 2002)

Rumsey, D. J. (2002). Statistical literacy as a goal for introductory statistics courses. Journal of Statistics Education, 10(3). http://www.amstat.org/publications/jse/v10n3/rumsey2.html

One of the primary goals of this course is to improve your statistical literacy. Statistical literacy is important because you are faced with statistics problems in your personal, academic, and professional lives. After completing this course, we hope that you are better able to interpret and evaluate statistics in any setting and that you are able to identify and conduct some of the most commonly used analyses.

## Examples

Choosing a Medication
Your doctor gives you the option to choose one of two different medications. She provides you with research studies comparing the two medications. How can you use those research studies to inform your decision?

Fantasy Football
In planning for your fantasy football team you come across many tables of statistics. How can you synthesize all of those numbers to best inform your fantasy football draft?

School Curriculum
Your child's school is selecting a new science curriculum. The administration has narrowed it down to three different curricula and is asking parents to vote. What information would you ask for to inform your vote?

Driving to Work
There are two routes that you could take to get to work in the morning. If you go through town, it usually takes between 6 and 14 minutes, depending on the traffic and red lights. If you take the interstate, it consistently takes 10 minutes. Which route will you take to work this morning?

Marketing Decisions
Your company has put you in charge of making a decision between two marketing campaigns. How can you design a research study to collect data to inform your decision?

This is not a mathematics course. While you will need to be able to apply some elementary mathematical procedures (e.g., multiplication, division, exponents, summations), the emphasis of this course is on improving your statistical literacy and using technology to answer questions. On the next page you will review the mathematical operations that we will use this semester.

# 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. The following sections will review order of operations, summations, and basic linear equations.

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. We want our students to be successful!

# 0.1.1 - Order of Operations

0.1.1 - Order of Operations

The acronym PEMDAS, or the mnemonic "please excuse my dear aunt Sally," are sometimes used to help students remember the basic order of operations, where P = parentheses, E = exponents (and square roots), M = multiplication, D = division, A = addition, and S = subtraction.

When performing a series of mathematical operations, begin inside the parentheses. Next, calculate any exponents or square roots. Then, multiplication and division. And finally, addition and subtraction. For a more in-depth review, we recommend the Khan Academy's Order of Operations lesson.

In this course, we will be using fractions often. When working with fractions, you can imagine that the operations in the numerator are within parentheses and the operations in the denominator are in parentheses. Below are a few examples of mathematical operations that will be applied in this course. We'll learn about the applications of these operations later in the course, here, we're focusing only on the mathematical operations.

## Example: Confidence Interval for a Mean

A confidence interval for a mean can be computed using the equation $$\bar x \pm SE (t^*)$$. Let's construct a confidence interval given $$\bar x = 5.770$$, $$SE = 0.335$$, and $$t^* = 2.080$$.

First, we'll plug in the given values.

$$5.770 \pm 0.355(2.080)$$

There are no operations within parentheses and no exponents or square roots, so our next step will be to multiply.

$$5.770 \pm 0.697$$

The symbol ± tells use to both subtract and add.

$$5.770 - 0.697 = 5.073$$

$$5.770 + 0.697 = 6.467$$

The confidence interval is (5.073, 6.467).

## Example: Test Statistic for a Proportion

A test statistic for a one sample proportion hypothesis test can be computed using the equation $$z = \frac {\hat p - p_0}{\sqrt{\frac{p_0 (1-p_0)}{n}}}$$. Let's work through this formula using the following values: $$\hat p = 0.87$$, $$p_0 = 0.8$$, and $$n=100$$.

First, we'll plug in the given values.

$$z = \frac {0.87 - 0.8}{\sqrt{\frac{0.8 (1- 0.8)}{100}}}$$

In the first few steps we'll work in the numerator and denominator separately. Let's start in the numerator, which contains only subtraction.

$$z = \frac {0.07}{\sqrt{\frac{0.8 (1- 0.8)}{100}}}$$

Next, let's focus on the denominator. The operation within the parentheses should be conducted first.

$$z = \frac {0.07}{\sqrt{\frac{0.8 (0.2)}{100}}}$$

Within the denominator, we can work within the top of the fraction first,

$$z = \frac {0.07}{\sqrt{\frac{0.16}{100}}}$$

Solve for fraction under the square root in the denominator.

$$z = \frac {0.07}{\sqrt{0.0016}}$$

Next, take the square root in the denominator.

$$z = \frac {0.07}{0.04}$$

And finally, divide the numerator by the denominator.

$$z = 1.75$$

# 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

Four children are counting how many pieces of candy they each have. They want to know how many pieces of candy they have all together.

Child Pieces of Candy
Jenny 9
Carrie 8
Eugenia 10
Kevin 8

If we wanted to know how many total pieces of candy the group of children had, we could add these four numbers. The notation for this is $$\sum x_{i}$$.

$$\sum x_{i}=9+8+10+8=35$$

Combined, the four children have a total of 35 pieces of candy.

# 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.

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.

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.

# 0.2 - Introduction to Minitab

0.2 - Introduction to Minitab

## Minitab®

Starting in Lesson 1 we will be using Minitab 20 Cloud app on a regular basis. Take time this week to make sure that you have access to Minitab.

#### Minitab™

Minitab is a statistical software application commonly used by statisticians to summarize, visualize and analyze data. It is required software for STAT 200 online.

##### A Quick Introduction to Minitab Statistical Software

This introduction to Minitab is intended to provide you with enough information to get you started using the basic functionality of Minitab. Of course, you will learn more about Minitab and its capabilities as you proceed through the course you are taking.

##### Obtaining a Copy of Minitab

If you are a Penn State student, information regarding how to obtain the software will be provided by your instructor.

##### Getting Started with Minitab

The following resources from Minitab will familiarize you with the basic components of the application and how they are used in analysis or homework situations.

# 0.3 - Word's Equation Editor

0.3 - Word's Equation Editor

For many of your lab assignments you will be asked to make calculations showing all work. When showing your work you should use the equation editor in Word. The equation editor will look slightly different depending on the version of Word that you are using. The video below walks through a few examples using Microsoft® Word 2016 and Microsoft® Word for Mac 2011.

Word 2016 Equation Editor
Word 2011 Equation Editor

# 0.4 - Canvas' Equation Editor

0.4 - Canvas' Equation Editor

On Canvas discussion boards and essay quiz questions you have the option to "Insert Math Equation"

When you click on this button you have the option of switching between "Basic" and "Advanced" views. Most of what you will need to do in STAT 200 can be accomplished under the basic view, with the exception of $$\widehat{p}$$. The video below will walk you through a few examples.

## Canvas Equation Editor

The table below contains some of the symbols and functions that you will use most often in this course. Note that \widehat will only work in the advanced view.

 Symbol Code $$\overline x$$ \overline x $$\widehat p$$ \widehat p $$\sum$$ \sum $$\mu$$ \mu $$H_0:$$ H_0 \$: $$s^2$$ s^2

 [1] Link ↥ Has Tooltip/Popover Toggleable Visibility