Probablity
- 2. What is Probability? • A method for measuring and quantifying the likelihood of obtaining a specific sample from a specific population I want to pick a single marble (a sample) out of each of these jars (two populations) What are my chances of getting a black marble from both jars? Jar A Jar B50 white 50 black 10 white 90 black You have a 50-50 chance of getting either color Your marble (your sample) will probably be black
- 3. What is Probability? • A method for measuring and quantifying the likelihood of obtaining a specific sample from a specific population Now you are blindfolded when you pick your marble (your sample) so you do not know from which jar (population) it comes from. If you picked 4 black marbles in a row, which jar would say they came from? Jar A Jar B Very high probability they came from this one Very low probability they came from this one
- 4. Probability Defined • The probability (p) of a specific outcome for any given event is defined as the fraction or proportion of all the possible outcomes (A, B, C, D, etc.) p(A)= # of outcomes classified as A total # of possible outcomes A ratio that represents the likelihood that some event will occur relative to the total number of possible outcomes
- 5. Probability Defined • If an event can occur in ‘A’ ways • If an event can fail to occur in ‘B’ ways • If all possible ways are equally likely (we guarantee this through random selection) • The probability of the event occurring: • The probability of the event failing to occur: BA A BA B p(A)= # of outcomes classified as A total # of possible outcomes
- 6. A Demonstration Select one card from a deck of 52 cards What is the probability of selecting a king? • The probability of the event occurring is computed by: • A = 4 • there are 4 kings in the deck • B = 48 • there are 48 cards that aren’t kings in the deck BA A 52 4 )484( 4
- 7. A Demonstration Select one card from a deck of 52 cards What is the probability of selecting a king of diamonds? The probability of the event occurring is computed by: • A = 1 • There is 1 king of diamonds in the deck • B = 51 • There are 51 cards that are NOT the king of diamonds in the deck BA A 52 1 )511( 1
- 8. Another way to think of probability • Sample space: A set of possible outcomes • Rolling a die: {1, 2, 3, 4, 5, 6} 6 possibilities • Event: A subset of the sample space, the occurrence of the thing you are interested in • Die coming up odd: {1, 3, 5} 3 possibilities Let’s say we are interested in the probability a roll of a die will result in an odd number The probability a die will come up odd is the number of event possibilities over the number of sample space possibilities: 2 1 6 3 .# .# )( spacesample iespossibilitevent oddp
- 9. Interpreting Probability • For any event (A), we assign a number p(A) between 0 and 1 that is interpreted as: • The chance or likelihood that event A occurs • The proportion of times that event A occurs over a series of trials Continuing our rolling the die example… p(odd) = ½ = 0.50 = 50% We can express probability in a ratio, proportion, or %
- 10. Examples If we spin this wheel: • What is the p of landing on 7? • p(7) = 1/8 • p = 0.125 or 12.5% • What is the p of landing on an even number? • p(even) = 4/8 = 1/2 • p = 0.50 or 50% • What is the p of landing on a white space? • p(white) = 4/8 = 1/2 • p = 0.50 or 50%
- 11. More examples! What is the probability of: • Getting a correct answer on a multiple choice question with five answer choices (assuming you make a blind guess)? • 1/5 • Getting a tail in a regular coin toss? • 1/2 • Rolling a 3 with a regular die? • 1/6
- 12. When is probability reliable? • When you have a fair die, coin, roulette wheel, etc. • There is an equal chance of every outcome • If your coin is weighted to land on heads every time, the p(tail) is no longer ½ To use probability in statistics, we must ensure that our sample is chosen in a way to mirror this “fair chance”
- 13. Random Sampling To insure that the definition of probability is accurate, we must use random sampling 1. Every individual in the population of interest has an equal chance of being selected 1. If more than one individual is to be selected for the sample, there must be a constant probability for each and every selection • This may require sampling with replacement
- 14. An Example of Random Sampling Imagine a large bag filled with coins. The coins include half-dollars, quarters, pennies, nickels, and dimes. You are interested in estimating the weight of this bag, but may not actually weigh every coin. How would you estimate the weight?
- 15. Estimating the weight… • Take a random sample of coins • Every coin must have an equal chance of being selected • There must be a constant probability for each and every selection • Requires sampling with replacement • Use this information to generalize what we might expect the weight of the entire bag to be
- 16. The Role of Probability in Inferential Statistics • Probability is used to predict what kind of samples are likely to be obtained from a population • Establishes a connection between samples and populations • Inferential statistics rely on this connection when they use sample data as the basis for making conclusions about populations
- 17. Probability and Statistics This is how we use probability in statistics We think ‘backwards’ because we typically do not have the population information Based on the sample, what do we estimate the population to look like? vs. Based on the population’s characteristics, what do we estimate our sample to look like?
- 18. Probability and Frequency Distributions • X: 1, 1, 2, 3, 3, 4, 4, 4, 5, 6 • N = 10 If I sample n = 1: • p(X < 2)? • 2/10 = 1/5 • p(X > 3)? • 5/10 = 1/2 • p(X = 3)? • 2/10 = 1/5
- 19. PROBABILITY AND THE NORMAL DISTRIBUTION
- 20. 68.26% 94.46% 99.73% Probability and the Normal Distribution We can describe a normal distribution by the proportions of area contained in each section • Symmetric: Proportions on the right side will correspond to the mirror image on the left • Since we can identify sections with z-scores, these proportions correspond to any normal distribution (regardless of actual mean and standard deviation values)
- 21. μ 34.13% 13.59% 2.14% Because the z-distribution is normally distributed we can make certain assumptions about probabilities The z-score allows us to draw a vertical line through the distribution, dividing the distribution into two sections The Body The Tail
- 22. μ 34.13% 13.59% 2.14% Some probabilities associated with the normal distribution • 34.13% of the distribution falls between the mean and the first standard deviation (z = +1.00; z = -1.00) • 13.59% of the distribution falls between the 1st and 2nd standard deviation • 2.14% of the distribution falls between the 2nd and 3rd standard deviations
- 23. Probability and the Normal Distribution It is good to have a general idea of proportions within the curve What if you need to find the proportion below a z- score of -0.85? The Unit Normal Table (Appendix B, pp.699- 702) lists z-scores and their corresponding proportions Note: The Unit Normal Table lists positive z-scores only because the curve is symmetric; values can be used for negative z-scores as well.
- 24. The Unit Normal Table • A negative z-score: tail is on the left side • A positive z-score: tail is on the right side Because the Unit Normal Table only lists positive numbers, it is helpful to draw the curve and the area you want before looking up the proportion so you know which value (body or tail) you want
- 25. Finding the probability of a z-score What is the proportion of the normal distribution associated with the following sections of a graph: • z > 1.25 Interpreted as: • The probability of selecting a participant from this population with a z-score greater than 1.25 is 0.1056 • The proportion of the distribution with a z-score greater than 1.25 is 10.56% • z < 0.00 Interpreted as: • The probability of selecting a participant from this population with a z-score less than 0.00 is 0.5000 • The proportion of the distribution with a z-score less than 0 is 50% p = 0.1056 p = 0.5000
- 26. PROBABILITIES AND PROPORTIONS FOR SCORES from a Normal Distribution
- 27. Finding the probability of a raw score from a normal distribution What is the probability of randomly selecting a person with an IQ greater than 115 if the population μ=100 and σ=10? 1st step: Calculate the z-score 2nd step: Find the proportion of the curve greater than the z-score. • z = 1.50 • p(>1.50) = p in the tail (Column C) for 1.50 = 0.0668 50.1 10 )100115( z
- 28. You Try It: For a distribution with a μ = 100 and σ = 10: What is the probability of randomly selecting a person with an IQ less than 90? Greater than 125? 00.1 10 )10090( z 50.2 10 )100125( z Proportion in the tail above z = 2.50: p = 0.0062 This very high z-score indicates it is at the extreme high end of the distribution, with few scores above Proportion in the tail below z = -1.00: p = 0.1587
- 29. Finding the Probability for a Range of Scores What is the probability of randomly selecting a person with an IQ between 95 and 110? (μ=100 σ=10) • Find the two z-scores, and then find the proportion that covers the space between these two scores 00.1 10 )100110( 50.0 10 )10095( z z Add the distances of each z-score from the mean (Column D): (0.1915 + 0.3413) = 0.533 Or, subtract the proportion in the tail of one score from the proportion in the body of the other score (e.g., Body of 110 – Tail of 95) (0.8413 - 0.3085) = 0.533 p = 95< IQ <110( )
- 30. Finding the z-score corresponding to a certain proportion • If you know a specific location (z-score) in a normal distribution, you can use the table to look up corresponding proportions. • If you know a specific proportion(s), you can use the table to look up the exact z-score location in the distribution
- 31. Example (1) • What z-score separates the lowest 5% from the remainder of the distribution? • Find the z-score with p = 0.05 in the tail • z-table shows us z = 1.65 has p = 0.0495 (closest to p = 0.05 without exceeding it) • Remember we want the lowest 5%, and the table only shows us positive z-scores • Since the curve is symmetric: a z-score of z = -1.65 separates the lowest 5% from the remainder of the distribution
- 32. Example (2) • What z-scores separate the extreme 5% scores from the rest of the distribution? • We want the lowest and highest in this 5% • Lowest score separates (5%/2) = 2.5% lowest and highest score separates (5%/2) = 2.5% highest • We want to find the z-scores with p = 0.025 in tails • z-scores of z = +/- 1.96 separate these scores from the rest of the distribution
- 33. Solving for a specific score: • If I want the top 10%, what is the cutoff score? (μ = 100 σ = 15) • z-score that separates p = .0100 closest without going over is z = 1.29 • Now I want to solve for X to find the cutoff score 2.1191002.19100)15(28.1 zX
- 34. Recall from earlier, we can express cumulative percentages as percentile ranks. To find the 84th percentile we would find the score that separates p = 0.8400 below (body) and p = 0.1600 above (tail). (approximately z = 0.99)
- 35. Looking ahead to inferential statistics • Based on our observations of a sample, we make inferences about the population. • It is with statistics based on probability that we determine how likely our sample represents our population of interest • or, how unlikely our sample is compared to the population after some treatment/intervention/etc.