1 / 29

Probability: The Foundation of Inferential Statistics

Probability: The Foundation of Inferential Statistics. October 14, 2009. Subjective Probability. Throughout the course I have used the word probability. Yet I have not defined it. Instead I have relied on the assumption that you all have a sense of probability.

wayde
Download Presentation

Probability: The Foundation of Inferential Statistics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Probability: The Foundation of Inferential Statistics October 14, 2009

  2. Subjective Probability • Throughout the course I have used the word probability. • Yet I have not defined it. Instead I have relied on the assumption that you all have a sense of probability. • The book calls this sense of probability “subjective probability”.

  3. Classical Approach to Probability • The mathematical Definition of Probability or

  4. Empirical Approach to Probability • In classical approach, the parameters are known • The numbers of cards in a deck. (e.g. the probability of drawing a king from a full deck of cards) • In the empirical approach the parameters are not known. • Instead we use samples to calculate estimates of relative probabilities.

  5. Foundations of Empirical Probability • Discrete vs. Continuous Variables • Discrete- a variable that is represented in whole numbers. • Decimals don’t make sense • Continuous- a variable where intermediate or fractional values are valid. • Sample Space • All possible outcomes that can occur • In Mendelian genetics, dominant recessive allele Aalocated at one locus may have the following sample space: AA, aa, Aa. • If Hardy-Weinberg Equilibrium is not violated the relative frequencies in the population should be: • f(AA)= 1, p=.25 • f(Aa)= 2, p=.5 • f(aa)=1, p=.25

  6. Mutually vs. Nonmutually Exclusive Events • Mutually Exclusive • Events in our sample that cannot occur together or overlap. • Nonmutually exclusive events • Events in our sample that can occur together. • A joint probability is the degree in which a set of events do occur together in a sample.

  7. Calculating Probability • Probability can be expressed • in a percentage or relative frequency in 100. • As a decimal. • p= .5 is the same as, 50% chance, is the same as saying 50 in one hundred. • The addition rule: p(A or B) • It is used when we want to calculate the probability of selecting an element that has one or more conditions.

  8. Calculating Probability • The addition rule: p(A or B) continued • p(A and B) is the joint probability. • When the events are mutually exclusive p(A and B) = 0

  9. The addition rule and odds of dying • The odds of dying p=1.0 • Usually you die of one cause. • What are the odds of dying of either a plane accident or a bicycling accident. • Bicycling accident= 1 in 4919 • Air/space accident= 1 in 5051

  10. The addition rule and odds of dying • p(A or B)= .0002 + .0002 – 0= .0004 • -0 because these are mutually exclusive causes of death. • .4% chance of dying in a bicycling accident or a air/space accident.

  11. Multiplication Rule for Independent Events: p(A) x p(B) • Used to determine the probability of two or more events occurring at the same time that are mutually exclusive. • Example: You want to know what the odds are that you will win the lottery. You have to match all five numbers. The choices range from 1:40. • The probability for choosing the first number is 1 in 40, the second number 1 in 39, third number 1 in 38…

  12. Joint and Marginal Probabilities • Joint and marginal probabilities refer to the proportion of an event as a fraction of the total. • To calculate these probabilities we divide the frequency of the joint or marginal probability of two or more events by the total frequency.

  13. Calculating Probabilities • The previous graph just gave you where the different types of probabilities are located on the chart. • This chart gives you the way you would calculate these probabilites. • You will be given a frequency for each cell (e.g. B= 20, not B=80, A= 48, not A= 52) • With this information you should be able to create a similar chart.

  14. Frequency Table Example

  15. Frequency Table Example

  16. Frequency Table Example

  17. Calculating Probabilities

  18. Calculating Probabilities

  19. Conditional Probabilities • Conditional probabilities are used when categories are not mutually exclusive. • The “|” symbol means given. • Therefore the first cell p(A|B) reads the probability of picking A given B • example from book A= Alcohol Abuse B= drug abuse. p(A|B) means the probability of picking an alcohol abuser among drug abusers.

  20. Conditional Probabilities

  21. p(B|A)≠p(A|B)

  22. p(B|A)≠p(A|B)

  23. Determining Joint Probabilites when Conditional and Marginal Probabilities are given

  24. The Binomial Distribution • Probability of Discrete Sequences • Lets say you want to know what the probability is that by chance you can guess 8 out of 10 of a true false exam. • For this you would use the formula that describes the binomial distribution is: • ! is the symbol for factorial. Example the 4!=4*3*2*1=24 • Don’t worry, I won’t make you calculate these.

  25. Mean and Standard Deviation for a Binomial Distribution. • When p=.5 the binomial distribution is symmetrical and approximates a bell curve. • This approximation becomes more accurate as N increases. • When p<.5 the binomial distribution is positively skewed. • When p>.5 it will have a negative skew.

  26. The Binomial Distribution Continued. • The binomial distribution has all the same descriptive statistics we already know. • Mean, standard deviation, and z scores. • We can use what we already know to relate this distribution to the normal curve.

  27. Example • Take the midterms I haven’t passed out yet. • Let’s say that the mean on this test Is around .9 and has a standard deviation of .2. • What is the probability of picking a person at random who actually failed the test?

  28. Example Continued • z=(.6-.9)/.2 • z=-1.5 • Go to the back of the book and see • area beyond z=.0668 • Only a 6.6% chance that you failed the test.

  29. Quiz time.

More Related