Lecture Notes

Chapter 6 Theoretical Distributions Including the Normal Distribution

Theoretical distributions are useful because they model many situations in the real world. By knowing the properties of theoretical distributions, you can use them to predict the data of an empirical distribution. We will look at the properties of rectangular, binomial, and normal distributions. (Revised 2003-07-24)

Empirical Distributions
- Empirical distributions are representations of actual scores, and are based on samples
Theoretical Distributions
- Theoretical distributions are idealized representations based on logical or mathematical descriptions of populations
Probability
- Range = 0.00 (no chance of event occurring) to 1.00 (event will definitely occur)
- Can be used to predict the likelihood of an event occurring
- Successes and failures
  - in probability, a success is the event specified = p
  - in probability, a failure is the event specified not occurring = q
  - p + q = 1
    - probability = success / total of successes + failures

for a coin coming up heads:
probability = head (success) / head (success) + tail (failure)

or

probability = 1 / 1 + 1 = 1 / 2 = .50 (probabilities are expressed as decimals)

for a one showing up on a throw of a die:
probability = one (success) / one (success) + two (failure) + three (failure) + four (failure) + five (failure) + six (failure)

or

probability = 1 / 1 + 1 + 1 + 1 + 1 + 1 = 1 / 6 = .167 (probabilities are expressed as decimals)

what is the probability of drawing the Ace of Hearts? (1/1 + 51)

what is the probability of drawing the two of spades? (1/1 + 51)

Odds

odds are the probability expressed as a fraction and reversed

what are the odds of drawing the Ace of Hearts? (52:1)

Rectangular Distributions
- In a rectangular distribution the frequencies of each score or event are the same, hence the distribution will graph like a rectangle. (It is still called a curve, however.)
- For playing cards the distribution will have a frequency of 4, and thirteen categories (Ace--2)
- Show rectangular distribution
- Note that the probabilities for each card add up to 1.00 (0.077 * 13)
- Use rectangular distributions to calculate probabilities by adding or subtracting probabilities as necessary
- Problems
  - What is the probability that a diamond will be drawn?
    - probability = success / total of successes + failures
    - p = 13/13 + 39 = 1/4 or .25, odds 4:1
  - What is the probability that face card will be drawn?
    - p = 12/12 + 40 = 12/52 or .23, odds 4.33:1
  - What is the probability that a red king will be drawn?
    - p = 2/2 + 50 = 2/52 or .038, odds 26:1
  - What is the probability that a royal straight flush will be dealt?
    - 4 hands/2,598,960 hands or,
    - 0.000001539077
    - which works out to: 1 royal straight flush/every 649,740 hands
The Binomial Distribution
- The binomial distribution is another theoretical distribution that allows you to compute the probability of multiple probabilistic events
- Tossing coins theoretically
  - Recall that the probability of a head is 0.50
  - What is the probability of throwing three heads in a row?
    - (1/2) * (1/2) * (1/2) = 1/8 = 0.1250
- Tossing particular combinations of coins
- Probabilities of Outcomes When 8 Coins are Tossed (these values come from expansion of binomial equation, or (p + q)⁸
- You many also use Pascal's triangle (see below) to get the nominator and denominator or you may use a table.

H	T	p	q
8	0	1/256	.004
7	1	8/256	.031
6	2	28/256	.109
5	3	56/256	.219
4	4	70/256	.273
3	5	56/256	.219
2	6	28/256	.109
1	7	8/256	.031
0	8	1/256	.004
		256/256	1.000

Is this a theoretical or empirical distribution?

Tossing coins really

What happens if we throw a coin 10 times? How about 100 times? How about 1000 times?

Demo

The Normal (or Gaussian) Distribution
- Gauss, Quetelet, Pearson (named normal curve)
- Formula:
- Description:
  - Bell-shaped, symmetrical, asymptotic to x-axis, steepest at + or - 1 SD
  - Properties of normal curve
- Tables
  - Table C in text (p. 366)
  - Use of tables
    - because any empirical distribution that is normally distributed can be converted to z scores, you can then use the theoretical normal distribution and its associated tables to compute probabilities.
- Example (11, page 126) Proportion of the Population
  - Children with IQ scores of less that 70 are usually placed in special education. What proportion of the population should have scores below 70?
  - Solve by converting IQ = 70 to a standard score:
    z = (70 - 100) / 15 = (-30 / 15) = -2.00
  - Look up -2.00 in Table C (column C, proportion beyond z):
    - .0228 or 2.28%
    - To estimate how many, simply multiply %age and sample size
- Example (15, page 128) Separating Population into Two Proportions
  - To be eligible for membership in Mensa, one must have an IQ greater than that of 98% of the population. What is that IQ?
  - Solve by looking up 98% or .9800 in Table C as area beyond z, or 2% or .020
  - That point corresponds to a z score of: 2.06
  - Insert z into the equation for z scores and solve for the score
    X = 100 + (2.06)(15) = 100 + 30.9 = 130.9

Example (19, page 130) Proportion Between Two Scores

800 scores on a general psychology test are normally distributed with a mean of 35, and a standard deviation of 6. What proportion of students had scores between 30-40? What is the probability that a randomly selelected student would score between between 30-40?

Solve by finding the z scores for 30 and 40, they are - or + 0.83, respectively. -0.83 converts to a proportion of .2967, doubling that gives .5934, the answer (or 59.34%). The probability is the same.

Example (23, page 131) Finding Extreme Scores

What IQ scores are only achieved by 5% of the population (upper and lower 2.5%)?

Solve by looking up the z score for .0250 (the equivalent of 2.5%). It is + or - 1.96. Then solve for X in the equation for each value. (rounds to scores of 129 and 71)

URLs
- Pascal's Triangle-works 7/23/03
- Demo of binomial distribution with coin tosses (Rice)-works 7/23/03
- Definition of Normal Distribution--short (ptt Inc.)-works 7/23/03
- Standard Normal Distribution--two pages from HyperStat (Rice)-works 7/23/03
- Simulation of Normal Distribution--Java dropping balls (animated, Sweden)-works 7/23/03
- Poker: A Guaranteed Income for Life--NEW, tutorial, basic, long, links-works 7/23/03
  - Comprehensive look at the game of poker, includes extensive discussion of odds of ending up with particular hands in various scenarios. http://www.neo-tech.com/poker/contents.html
- Pascal's Triangle--NEW, interactive, basic, short-works 7/23/03
  - Users may view Pascal's triangle for specified depth. http://forum.swarthmore.edu/~ken/pascal.cgi
Quiz

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