Intro to Statistics

1
Intro to Statistics

• Lecture Notes
• Dr. Sinn
• November 29, 2005

2
Chi-Square Tests

• Joker and Jody
• The notorious river boat gambler Joker McGants
• was plying his trade a bit too well in the ATL
  one
• April night. Sitting amidst a pile of chips,
  bills and
• booze, Joker realized that Bugsy Haygood was
• downright ill-tempered, ill-mannered and ill-
• equipped to restrain his erupting anger.
• Youre using loaded dice, mug, Bugsy accused.

3
Joker and Jody

• Joker donned a look so innocent it was worthy
  of
• the seraphim, as well it should have been. Joker
• had practiced in front of a mirror for hours upon
• hours for just such occasions. Given that
  Bugsys
• stock and trade involved 9mm projectiles aimed by
• the hoarse whispers of his family, the sweat
  that
• rolled icily down Jokers spine was
  understandable.
• Youre welcome to perform any test you wish
• on these dice, sir, Joker agreed.

4
Joker and Jody

• Bugsy performed the various test gamblers use
  to ascertain
• that the dice are true. No shaved corners, no
  weighted
• imbalances, no signs of any alterations. Joker
  smiled his
• winsome best. Bugsy was not amused.
• Just as Joker was judging the possibilities of
  a live-to-
• gamble-another-day escape, the cocktail waitress
  Jody
• DeLong offered a suggestion. Im taking a
  statistics course
• at North Georgia, and I know a way to tell if
  those dice are
• loaded. We can use chi-square.

5
Joker and Jody

• Both Joker and Bugsy agreed, although a debate
• quickly ensued. I prefer using a .01, Joker
• suggested. Dont want any Type I errors, you
• know.
• I disagree, Bugsy told him. I think an a
  .1
• would better control that insidious Type II
  error.
• We do want as much power as possible, dont we?
• Jody did not like the evil glint in Bugsys
  eye nor
• the way his hand inched toward the coat pocket
• that bulged noticeably.

6
Chi-Square Tests

• Jody, as bright as she was beautiful, just
• rolled her eyes, Everyone knows that a has
• to be .05. Duh.

7
Chi-Square Goodness of Fit

• Compare Expected vs. Actual Counts
• Expected Counts
• Proportions predicted by probability theory or a
  hypothesis stated by the researcher(s)
• Actual Counts
• Proportions that actually occurred during data
  collection

8
Were the Jokers Dice Wild?

• The die was cast 600 times, and the following
  frequencies were recorded

9
Chi-Square 2-Way Tables

• Comparison of 2 Categorical Variables
• Simplest Case (only one used in this course)
• One is an Either/Or Variable
• One is a Multi-Category Variable
• Analysis of 2-Way Tables
• Expected vs. Actual Counts
• Choose one row as actual, other as expected
• Choice is arbitrary
• Same results either way

10
Religion Versus Divorce Rates

• A study was done to determine if divorce
• vary based upon whether or not someone is
• religious. Respondents represented 4
• denominations and the non-religious. Is
• religion a factor in divorce rates?

11
Two Types of ?2 Tests

• Goodness of Fit
• Probability-Based Investigations
• Las Vegas Games
• Must know theoretical counts
• MMs Problem
• Prototype
• Are there enough RED MMs in my bag?

• 2 Categorical Variables
• 2-Way Table
• Compare Rows
• Do columns (groups) influence row totals?
• Religious Affiliations Problem
• Prototype
• Does sect or denomination influence drinking,
  drug use or divorce?

12
Examples of ?2 Two Categorical Variables

• Course Grades one math professor gives only 5
  As and 25 Bs. The rest of the department
  gives 15 As and 30 Bs. Is this professor
  off the reservation?
• Preventing Colds does vitamin C prevent colds?
  A control group takes no additional vitamin C.
  One research group drinks 3 6-oz. glasses of
  orange juice daily. A second research group
  takes 1000 mg vitamin C supplement daily. During
  one cold season, the percentages of colds in each
  group is calculated.

13
Examples of ?2 Goodness of Fit

• Genetics the probability of offspring having
  certain inherited traits are well known. If two
  Dobermans are bred, we can verify hypotheses
  about dominant vs. recessive traits by comparing
  expected counts (based on the model) to actual
  counts (based on counting the pups).
• Fair Casino? a roulette wheel has three colors
  red, black and green. Compare 500 spins of the
  wheel to see if wheel is fair given that
  Black-Red-Green theoretically occur in a 18-18-2
  ratio.

14
Example

• A certain color of cars stolen more often than
  others? Of all cars manufactured, 15 are white,
  15 blue, 35 red, 30 black and 5 other, then
  test the hypothesis that a certain color is
  stolen more often at an appropriate level of
  significance. Of 830 stolen vehicle reports, 140
  were white, 100 blue, 270 red, 230 black and 90
  others.