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IT 223 -- Feb 11, 2026

Review Exercises

1. What is the difference between the true regression equation and the estimated regression equation?
   Answer: recall the ideal measurement model: x_i = μ + e_i (actual measurement = true value + random error). μ is usually unknown and estimated by μ^{^}= x.
   
   The true regression equation is y_i = a * x_i + b, where the slope a and the intercept b are unknown and must be estimated by a^{^} and b^{^}, which are determined from the estimated regression equation:
         y - y = (r^{^} * SD+_y / SD+_x)(x - x)
   which is rewritten as
         y = a^{^} * x + b^{^}.
2. Look at the Regression Document. What is the RMSE for a regression model?
   Ans: The root mean squared error (RMSE is the standard deviation of the residuals. In particular, RMSE is the SD of the residuals in a thin vertical rectangle that contains a specific x value. RMSE is defined as SD+_y √(1 - r²).
3. 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:
          x = 162    SD_x = 6
          y = 68    SD_y = 10
          r = 0.6
   
   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 we can use the normal table.
      Answer: 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. The observations in the thin vertical rectangle centered over x=165 are normally distributed.
      Answer: The regression equation is
            y - 68 = (0.6 * 10 / 6) (x - 162)
            y - 68 = 1 (x - 162)
            y = x - 94
      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
            RMSE = √1 - r² SD_y = √1 - 0.6² * 10 = 0.8 * 10 = 8
      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%.

Project 3

• Look at the project description for Project 3 .

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 criticize 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?
     Answer: A very tall person is expected to be shorter than his father.
  2. Would you expect a person with a very short father to be shorter or taller than his father?
     Answer: A very short person is expected to be taller than his father.

Learning Outcomes for Probability

• Know the definitions of these terms:

  Sample Space, A Priori Probability, Empirical, Subjective Probability, Sample Space, Probability Distribution, Random Variable, Expected Value, Risk.

• Be able to:
  
  1. Define a probability distribution in three ways: a priori, empirical and subjective.
  2. Define a probability distribution, given a sample space.
  3. Compute the expected value of a random variable.
  4. Understand the difference between independent and mutually exclusive events.
  5. Use the multiplication rule for independent events.
  6. Use the addition rule for mutually exclusive events.
  7. Find the expected value of a multiple of a random variable.
  8. Find the expected value of a sum of random variables.

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.
• We will use S to represent the sample space. Some authors working in theoretical probability theory use Ω (Greek letter uppercase omega) to represent the sample space.
• Example 1:   Flip a coin once. The sample space is S = {head, tail}.
• Example 2:   Roll a die once. The sample space is S = {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 {red₁, red₂, red₃, green₁, green₂, green₃, green₄}.
• Example 4:   Choose a random part on an assembly line and test it. The sample space is {defective, nondefective}.
• An event is a subset of thee sample space.
• 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 2027?
     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
  THH	0.125
  TTH	0.125
  THT	0.125
  HTT	0.125
  TTT	0.125

  
  When flipping 3 coins, what is the probability of obtaining 1 head?
  Answer: There are three ways in the sample space of obtaining one head: HTT, THT, TTH. Therefore the probability of obtaining one head is 0.125 + 0.125 + 0.125 = 0.375. The event of obtaining 1 head is {TTH, THT, HTT}
• 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.

Practice Problems

1. What is wrong with this argument? Either the Bears will win the Super Bowl in 2027 or they won't. Therefore the probability that the Bears will win the Superbowl in 2027 is 50%.
   Ans: Just because there are two outcomes doesn't mean they are equally likely. Some outcomes are very likely; they have probabilities close to 1. Some outcomes are unlikely; they have outcomes close to 0. Other outcomes have probabilities close to 0.5. In this case it does not make sense to use the a priori probabilities of 0.5 for winning the superbowl or not winning it in 2027.
2. What is wrong with this strategy? Double down after each loss. Eventually you win and recoup your losses. For example:
   -1 - 2 - 4 - 8 - 16 + 32 = 1.
   Now start over with 1 and repeat the double down strategy.
   Ans: The problem with this strategy is that, eventually, you will either reach the casino betting limit or you will run out of money.
3. A bookmaker offers 20 to 1 odds that the Bears will win the Super Bowl in 2026. If this is a fair bet, what is the probability p that the Bears will win the Super Bowl?
   Answer: The expected amount that you win is 20p + (-1)(1 - p); this expression is 0 because we are assuming that is a fair bet. Now solve for p:
        20p + (-1)(1 - p) = 0
        20p - 1 + p = 0
        21p = 1
        p = 1 / 21 = 0.0476 = 4.8%.

Random Variables

• A random variable is a function from the sample space to the set of real 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; 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 basketball free throw (failure = 0, success = 1), choosing a part from an assembly line to inspect (non-defective = 0, defective = 1).