GTTH Question 1: What is statistics?
GTTH Question 2: What are the two major areas of statistical inference we have studied in this course?
Suppose a friend or colleague asked you "what is statistics?" What would you say?
The Textbook's Summary: A foundation for understanding statistics:
- The nature and purpose of data analysis.
- The central ideas of designs for data production.
- The reasoning behind confidence intervals and significance tests.
- Experience applying these ideas in practice.
Chapters 20,21,22,23
Chapter 25 The Chi-Square Test
Are two categorical variables independent?
Example $$ \begin{array}{c|c|c|} & \mbox{Drives a Big Truck} & \mbox{Doesn't Drive a Big Truck} \\ \hline \mbox{Female} & & \\ \hline \mbox{Male} & & \\ \hline \end{array} $$
$H_0: \mbox{GENDER and VEHICLE TYPE are independent}$.
For example, under the null hypothesis, knowing a person is female gives us no extra information about the kind of car they are likely to drive.
$H_a: \mbox{GENDER and VEHICLE TYPE are NOT independent}$
The alternative hypothesis says that knowing a person is female DOES give us extra information about the kind of car they are likely to drive.
Note: Chi-Square gives no indication of the nature of the relationship if we reject $H_0$. This requires a follow-up analysis.
Chapter 27 Analysis of Variance (a.k.a. ANOVA)
ANOVA allows us to compare any number of means.
$H_0: \mu_1=\mu_2=\mu_3=\cdots=\mu_I$
$H_a: \mbox{At least one of the means are not equal.}$
Compute the $F$-statistic, $F=\frac{\mbox{MSG}}{\mbox{MSE}}$ and determine its $p$-value from its $F$-distribution with $(I-1, N-I)$-degrees of freedom.
Example Suppose I'm a stats teacher and I want to test the hypothesis that all my final exams are equally as challenging from semester to semester.
Final Exam Scores in Universe A $$ \begin{array}{c|c|c} \mbox{Spring 2015} & \mbox{Fall 2015} & \mbox{Spring 2016} \\ \hline 77 & 89 & 83 \\ 76 & 86 & 62 \\ 59 & 68 & 57 \\ 62 & 84 & 83\\ 80 & 70 & 81\\ 68 & 68 & 80\\ 66 & 64 & 53\\ 84 & 70 & 72\\ 73 & 62 & 80\\ 74 & 67 & 65\\ 77 & 83 & 77\\ 80 & 75 & 66\\ 55 & 76 & 77\\ 71 & 88 & 66\\ 83 & 72 & 89\\ \end{array} $$
In Universe A, the $F$-statistic is 0.53 and the $p$-value is 0.593.
Our conclusion?
Final Exam Scores in Universe B $$ \begin{array}{c|c|c} \mbox{Spring 2015} & \mbox{Fall 2015} & \mbox{Spring 2016} \\ \hline 77 & 74 & 83 \\ 76 & 71 & 62 \\ 59 & 53 & 57 \\ 62 & 69 & 83\\ 80 & 55 & 81\\ 68 & 53 & 80\\ 66 & 49 & 53\\ 84 & 55 & 72\\ 73 & 48 & 80\\ 74 & 52 & 65\\ 77 & 68 & 77\\ 80 & 60 & 66\\ 55 & 61 & 77\\ 71 & 73 & 66\\ 83 & 57 & 89\\ \end{array} $$ In Universe B, the $F$-statistic is 8.039 and the $p$-value is 0.001.
Our conclusion?
Note: Like the Chi-Square Test, ANOVA gives no indication of the nature of the relationship if we reject $H_0$.
That is, we must do a follow-up analysis to determine how the various means relate to one another.
In the case of Universe B: $$ \begin{array}{lllll} \mbox{Sample} & n & \mbox{Mean} & \mbox{Standard Deviation} & \mbox{95% Conf. Interval}\\ \mbox{Spring 2015} & 15 & 72.333 & 8.756 & (67.4, 77.26)\\ \mbox{Fall 2015} & 15 & 59.867 & 8.935 & (54.94, 64.8)\\ \mbox{Spring 2016} & 15 & 71.333 & 10.588 & (66.4, 76.26)\\ \end{array} $$ Our conclusion?
Conditions for Inference What are the most important criteria for the use of any of the inference techniques we have used throughout this course?
Parting Words of Wisdom If you ever find yourself in a position where you are required to use statistics, please keep the following in mind.
Before Collecting Data: Plan your study well. No amount of analysis can compensate for bad data.
Make sure you can answer the following questions (which is neither a definitive nor exhaustive list, but a good start):
- What do you want to know?
- Can you define your population?
- Can you reasonably take a random sample from this population?
- What are you going to measure?
- How are you going to measure it?
- Can you measure it?
Before Collecting Data
When you can provide concrete answers to the above questions, you may then address the following:
- What kind of analysis will be appropriate for the data you want to collect?
- What sample size will you need?
There are various techniques for determining the sample size you will need to achieve your goal. If you do a test of significance, you'll likely do a power analysis.
Human Subjects: when trying to get information about human populations, remember
- don't hurt them (neither physically nor psychologically),
- they don't tell always tell the truth,
- some of the perils of telephone surveys and random digit dialing.
By the Way... what are the perils of telephone surveys and random digit dialing?
Another Question: Who remembers one of the worst sampling designs?
After Collecting Data:
Be Honest: Be sure to give your readers as much detail as possible about your sampling design, analysis, and any assumptions you made either in data collection or your analysis. Also, if you're ever caught fibbing, no one will ever listen to you again.
Any Result is a Revelation: Not rejecting $H_0$ is neither good nor bad. It's only a conclusion.
Practical Significance: Just because a result is statistically significant doesn't mean it's useful. Recall the SAT Coaching example.
It's Okay to Not Know: There's nothing wrong with asking someone for help. Most researchers find themselves in a position where they must seek advice from someone with expertise in another field.
Coming Full Circle
Some people hate the very name of statistics, but I find them full of beauty and interest. Whenever they are not brutalised, but delicately handled by the higher methods, and are warily interpreted, their power of dealing with complicated phenomena is extraordinary.
-Sir Francis Galton
Final Exam: Materials Allowed.
- Your Exam
- Your TI Graphing Calculator
- An Unmarked Table of Cumulative Standard Normal Probabilities
- An Unmarked Copy of t Distribution Critical Values (Table C)
- An Unmarked Copy of Chi-Square Distribution Critical Values (Table D)
- An Unmarked Copy of the formula sheet.
- Non-alcoholic beverage of choice.
Important: cell-phones and other technology ARE NOT allowed during the exam.
Practice Exams: You may use this practice exam.
OR
You may generate as many practice exams as you like using our course website.
NOTE: I will use our course website to generate the real in class exam.
If the "Download PDF Button" doesn't work, you'll need to use the snip tool and a word document to create your own paper exam.