To Notes

IT 403 -- Oct 5, 2016

Review Exercises

1. In the Correlation document, look at the scatterplots of bivariate datasets with various correlations.
2. Estimate the correlation r in these situations:
   
   1. Height of father, height of son.
      i. -0.30    ii. 0.05    iii. 0.70    iv. 0.99    Ans: 0.70
   2. IQ of husband, IQ of wife.
      i. -0.70    ii. 0.00    iii. 0.60    iv. 1.00    Ans: 0.60
   3. Height of husband, height of wife if men always married women that were exactly 6 inches shorter.
      i. -0.60    ii. 0.60    iii. 0.99    iv. 1.00    Ans: 1.00
   4. Weight of husband, weight of wife if men always married women that weighed 70% of their husbands weight.
      i. 0.00    ii. 0.50    iii. 0.70    iv. 1.00     Ans: 1.00
3. Match the correlation to the dataset:
   
   1. GPA in freshman year, GPA in sophomore year.   Ans: 0.70
   2. GPA in freshman year, GPA in senior year.    Ans: 0.30
   3. Length and weight of 2 by 4 boards.   Ans: 0.99
   -0.50   0.005   0.30   0.70   0.99
4. What would happen to the correlation r if
   
   1. x were replaced by x + 10.
   2. y were replaced by 2 times y + 8.
   3. x and y were interchanged.
   Ans: in all three cases, the correlation would remain the same.
5. How large must r be to be considered meaningful?
   Ans: See the table in the Correlation document.
6. Why is the computed value of r the same whether the SD or the SD+ is used for the x and y standard deviations?
7. Use SPSS to compute the pairwise correlations of the variables in the Nielsen Dataset. The rows of this dataset are the ratings for various television shows. Interpret the correlations.
8. What is a bivariate normal dataset? Which statistics parsimoniously describe a bivariate normal dataset?
   Ans: x   y   SD_x   SD_y   r_xy.
9. How many statistics would you need to parsimoniously describe a multivariate normal dataset with three variables x, y, and z? a multivariate normal dataset with k variables?
   Ans: For three variables: three sample means, three sample SDs, three sample correlations r_xy, r_xz, r_yz: 9 total summary statistics. For k variables: k sample means, k SDs, k(k-1)/2 sample correlations: (k² + 3k) / 2 total summary statistics.
10. Compute the correlation r of this dataset "by hand" using SPSS  but not using Analyze >> Correlate >> Bivariate. If you use SD+ for x and y, don't forget the correction factor n / (n - 1). Check your answer with Analyze >> Correlate >> Bivariate.
    

    x	y
    1	1
    2	3
    3	2
    4	4

    
    Ans: r = 0.8.

Linear Correlation

• Look at Cautions section of the Linear Correlation document.

Linear Regression

• With the Bears Roster 1985 Dataset, use SPSS to:
  
  1. Compute the height in meters and the weight in kilos of the Bears players. The conversion rates are 0.3048 meters per foot and 0.453592 kilos per pound.
  2. Use SPSS to create a scatter plot of weight (y-axis) vs. height (x-axis).
  3. Find the simple linear regression equation for predicting weight in kilos from height in meters.
  4. Create the residual plot, which is the residuals (y-axis) vs. the predicted values (x-axis).
  5. For the residual plot, SPSS plots the standardized residuals (ZRESID) vs. the standardized predicted values (ZPRED).

Project 3

• Look at the project descriptions 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 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:
  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 you can use the normal table.
  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.

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?
     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
  TTH	0.125
  TTT	0.125

  
  When flipping 3 coins, what is the probability of obtaining 1 head?
  Ans: 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.
• 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.

Random Variables

We will discuss random variables next time on Oct 12.

• 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).