2/14/11 Notes

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IT 223 -- 2/14/11

Midterm Exam

The Regression Fallacy

During their flight training, Air Force pilots make two practice landings with their instructor. The pilots that make a good first landing tend to do worse on the second landing; the pilots that make a poor first landing do better on the second landing. Conclusion: criticism helps the pilots, whereas praise hurts them. Policy change: the supervisor of the flight instructors tells them to criticise all landings, good or bad. Is this policy change warrented?
In fact, the criticism is not warrented. This is an example of the regression fallacy.
Look at this graph of pre-test and post-test scores marked by the dots.

The green line indicates where the dots would be if the pre and post scores were the same. The red line is the regression line, which shows where the pre-test score and its associated predicted post-test score would lie. As in the previous example of the Air Force pilots, a student with a high pre-test score is predicted to do worse on the post-test; a student with a low pre-test score is predicted to do better on the post-test.
Practice Problem Collect the following data for adult men: the subject's height, the height of the subject's father.
1. Would you expect a person with a very tall father to be shorter or taller than his father?
  Ans: A very tall person is expected to be shorter than his father (regression fallacy).
2. Would you expect a person with a very short father to be shorter or taller than his father?
  Ans: A very short person is expected to be taller than his father (regression fallacy).

Additional Regression Problem

Root Mean Square Error
A law school finds this relationship between LSAT scores (independent variable x) and first-year scores (dependent variable y). The data are bivariate normal. Here are the summary variables:
1. About what percentage of the students have first-year scores over 75? Because the data are bivariate normal, the first-year scores are normally distributed, so you can use the normal table.
  Ans: z = (y - y) / SD_y = (75 - 68) / 10 = 0.7. The area of the bin (-∞, 0.7] is 0.7580 = 76%. Therefore the percentage of students with first year score over 75 is 100% - 76% = 24%.
2. Of those students who scored 165 on the LSAT, about what percentage have first-year scores over 75? Visualize these scores as lying in a thin vertical rectangle centered at LSAT = 165.
  Ans: The regression equation is
  so the predicted value for the students in the thin vertical rectangle centered at x = 165 is y = x - 94 = 165 - 94 = 71.
  The RMSE for those students in the thin vertical rectangle centered at x = 165 is
  Then z = (y - y^{^}) / RMSE = (75 - 71) / 8 = 0.5. The area of the bin (-∞, 0.5] is 0.6915 = 69%. Therefore of the students having LSAT score equal to 165, the percentage of students having first year score over 75 is 100% - 69% = 31%.

Probability

The probability that an event occurs is a value between 0 and 1 that indicates how likely that event is to occur; 0 means that the event is impossible, 1 means that the event is certain to occur.
The sample space is the set of all possible outcomes in a probability situation.
Example 1: Flip a coin once. The sample space is {head, tail}.
Example 2: Roll a die once. The sample space is {1, 2, 3, 4, 5, 6}.
Example 3: An urn contains 3 red balls and 4 green balls. Choose a random ball from the urn. The sample space is {red, red, red, green, green, green, green}.
Example 4: Choose a random part on an assembly line and test it. The sample space is {defective, nondefective}.
Three common ways of determining probabilities:
1. Theoretical or A Priori Probability
  Assume that all outcomes are equally likely. Suppose you are interested in the probability that the event A occurs.
  P(A) = # ways A can occur / Total # of events
  Example 5: What is the probability of getting one head out of two coin flips?
  Sample space S = {HH, HT, TH, TT}
  A = Event of getting exactly one head = {HT, TH}
  P(A) = # of events in A / # events in S = 2 / 4 = 0.5
2. Empirical Probability
  Perform an experiment, repeated a large number of times. Count the number of times s the event A occurs and the total number of times n the experiment is repeated.
  P(A) = s / n
  Example 5: Flip a coin 1 million times. 499,523 heads were obtained. The probability of obtaining a head is 499,523 / 1,000,000 = 0.499523.
3. Subjective Probability
  Each person uses personal intuition or judgement.
  What is the probability that I will need an umbrella today?
  Las Vegas oddsmakers combine subjective probabilities for many sources when determining betting odds.
  Example 6: What is the probability that the Bears will win the Super Bowl in 2012?
4. A subjective probability can be defined as a fair bet, that is a bet for which you would be willing to take either side.
No matter how probabilities are chosen, a probability distribution is a table that assigns a probability to each outcome in the sample space:

Probability Distribution

Outcome Probability

HHH 0.125

HHT 0.125

HTH 0.125

HTT 0.125

THH 0.125

THT 0.125

TTH 0.125

TTT 0.125
Rules for Probabilities:
1. Each probability is between 0 and 1, inclusive.
2. The sum of all probabilities of events in the sample space is 1.
3. If p is the probability that the event A occurs, then 1 - p is the probability that A does not occur.

Probability Distribution
Outcome	Probability
HHH	0.125
HHT	0.125
HTH	0.125
HTT	0.125
THH	0.125
THT	0.125
TTH	0.125
TTT	0.125

Random Variables

A random variable is a probability distribution where the sample space consists of numbers.
Example 7: Pick a person at random. That person's height is a random variable.
Example 8: The random variable x is the number of heads obtained out of two coin flips.

x Probability

0 0.25

1 0.50

2 0.25
Example 9: The random variable x is the number of heads obtained out of three coin flips.

x Probability

0 0.125

1 0.375

2 0.375

3 0.125
A random variable that has only two outcomes 0 and 1 is called a Bernoulli random variable. Usually 1 denotes success and 0 denotes failure. Here is the probability table for a Bernoulli random variable:

x Probability

0 1-p

1 p
Some examples of Bernoulli random variables: flipping a coin (T = 0, H = 1), rolling an ace with a single die (non-ace = 0, ace), shooting a free throw (failure = 0, success = 1), choosing a part from an assembly line to inspect (non-defective = 0, defective = 1).

x	Probability
0	0.25
1	0.50
2	0.25

x	Probability
0	1-p
1	p