1 / 42

Probability

Probability. Why we care about probability?. Inferences about populations are based on probability We will use probability to judge the likelihood of events (e.g. a rat weighing 450mg) Z-scores can tell us something about the probability of a specific score. Probability

arios
Download Presentation

Probability

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

  2. Why we care about probability? Inferences about populations are based on probability • We will use probability to judge the likelihood of events (e.g. a rat weighing 450mg) • Z-scores can tell us something about the probability of a specific score

  3. Probability • P(A) = Total number of outcomes classified as A divided by the total number of possible outcomes. • EX: • P(heads) = 1 / 2 = .5 • P(Heart) = 13 / 52 = .25 • P(Queen) = 4 / 52 = .077 • Probability can range between 0 and 1.

  4. Random sampling: all elements or individuals in the population have an equal chance of being selected • like blindly drawing names from a hat • particular event occurs or doesn’t based on luck of the draw • The probability of an event is based on how often it occurs “over the long run” • (relative to other events that can occur)

  5. Independence: the probability of an event is not influenced by the occurrence of another event • Probability stays constant across samplings • Sampling with replacement • e.g., draw a card and replace it before you draw another

  6. Probability and inferential procedures • If we know the characteristics of the population we can use probability to describe the samples we could obtain. What are the chances of getting a white marble? What are the chances of getting a white marble?

  7. We can also make inferences about the population based on the samples we draw. We blindfold you • You draw n=4 marbles • You get 4 white marbles • Which jar are they most likely from?

  8. Probability: samples and the population The collection of all possible events that could occur = population All marbles in jar #2 Total deck of card All adult rats Event that does occur= sample • Pick a white marble • Draw a queen of hearts • Rat weight of 450g • Or… sample mean of 450g

  9. The relative freq of a score in the population = its probability if draw a sample of 1 P (hearts) = = .25

  10. Using frequency distributions to obtain probabilities If we take a random sample of 1 kid, what’s the probability they got more than 4 valentines? p (Valentines > 4)

  11. Using frequency distributions to obtain probabilities If we take a random sample of 1 kid, what’s the probability they got more than 4 valentines? p (Valentines > 4) = ==.50

  12. First-day anxiety ( 1 to 5 scale) What is the probability that if I randomly select one student from this course, their first-day anxiety will be 3? p (anxiety = 3) What is the probability that if I randomly select one student from this course, their first-day anxiety will be less than 3? p (anxiety < 3)

  13. First-day anxiety ( 1 to 5 scale) p (anxiety = 3) = .30 p (anxiety < 3) = p (2) + p (1) = .50 + .17 = .67

  14. Probability and z-scores….its all connected With Normally Distributed Data: 0.15% 0.15% Probability = Proportion = Relative Frequency

  15. Probability and z-scores….its all connected With Normally Distributed Data: 0.15% 0.15% What’s the probability of drawing a score within one SD of the mean? P (within +/- 1 SD) = .68

  16. Probability and z-scores….its all connected With Normally Distributed Data: 0.15% 0.15% What’s the probability of drawing a score more than +3 SD above the mean? P (> +3 SD ) = .0015

  17. Using z-scores to find exact probability in a normal distribution(….remember z scores are in SD units!)

  18. The population of biological male heights is normally distributed with μ =70 inches (equivalent of 5’10”) and σ = 6 inches. If we randomly select a person from the population of men, how likely is it that the man we select will be taller than 82 inches (6’10”)? P (x > 82 inches) • Sketch a distribution of the population and label 82 inches • Shade in the area of the distribution we are interested in 70 82

  19. The population of male heights is normally distributed with μ =70 inches (equivalent of 5’10”) and σ = 6 inches. If we randomly select a person from the population of men, how likely is it that the man will be taller than 82 inches (6’10”)? P (x>86 inches) • Sketch a distribution of the population and label 82 inches • Shade in the area of the distribution we are interested in • Identify the exact position of 82 by calculating a z-score = 2 • Find the proportion of the distribution that corresponds to the area above the z-score

  20. Probability and z-scores….its all connected 0.15% 0.15% -3 -2 -1 0 1 2 3 p(x > 82)= .0235 +.0015 = .025

  21. What happens when a z-score is not equal to 0, 1, 2, or 3? The Unit Normal Table!

  22. The Unit Normal Table Proportion in Tail Proportion in Body Proportion between Z and mean

  23. Example Cat lengths are normally distributed in the population: μ = 20 inches and σ = 8 inches. I have a cat that is 22 inches long. What percent of cats in the population are longer than my cat? = .25

  24. X = 22 M = 20 SD = 8 Z = .25 Proportion in Tail Proportion in Body 40% of cats

  25. X = 22 M = 20 SD = 8 Z = .25 Proportion in Tail Proportion in Body Interpretations: 40% of cats in the population are longer than my 22 inch cat If I randomly select a cat from the population of cats, the probability that the cat will be longer than 22 inches is 40%

  26. Give it a whirl…. What proportion of scores are less than a z score of 1.29? What proportion of scores are more than a z-score of 1.96? What proportion of scores fall between a z-score of .71 and the mean (0)? .9015 .025 .2611

  27. A little more advanced What proportion of scores are greater than or less than a z of 1.96?

  28. A little more advanced What proportion of scores are greater than or less than a z of 1.96? • proportion in tail for -1.96 = .025 • proportion in tail for 1.96 = .025 • total proportion: .025 + .025 = .05 .475 .475

  29. A little more advanced What proportion of scores are greater than or less than a z of 1.96? • proportion in body for -1.96 = .475 • proportion in body for 1.96 = .475 • total proportion: 1 - (.475 + .475) = 1 – (.95) = .05 .475 .475

  30. A little more advanced What proportion of scores are between a z score of -.80 and 1.20?

  31. A little more advanced What proportion of scores are between a z score of -.80 and 1.20? • proportion between -.80 and mean = .2881 • proportion between 1.20 and mean = .3849 • Total proportion: .2881 + .3849 = .6730 .2881 .3849

  32. Finding probabilities between two scores The amount of Hershey kisses consumed on V-day is normally distributed. The mean is 25 the SD is 10. What is the likelihood that if we randomly poled one person, they would report consuming between 22 and 32 kisses? 1.) get the z-scores 2.) find the proportion between that score and the mean 3.) add the two proportions together .1179 +.2580 = .3759 = .70 = -.30

  33. You try it Police are out setting speed traps on Saturday night. They’ll pull you over if you go too fast (you might be drunk) or if you go too slow (you might be really drunk). The distribution of speeds on the highway is normally distributed with a mean of 58 mph and a SD of 10 mph. Police won’t pull you over if you go between 40 and 80 mph. If we randomly select a car that drives past, what’s the likelihood that they would be driving a speed that won’t get them pulled over?

  34. Finding an x value that corresponds to a proportion The amount of Hershey kisses consumed on V-day is normally distributed. The mean is 25 the SD is 10. How many kisses do you need to consume to be in the highest 5% of kisses consumers? • Find the z-score corresponding to .05 in the tail • Do we want a positive or a negative z-score? • Transform the z-score into an x value 1.65 X = μ + zσ X = 25 + 1.65 (10)= 41.5 kisses

  35. You try it The chief of police wants to increase speeding citations. She decides to set a policy to pull over any person driving above the 90th percentile for highway speeds. As you recall, the mean speed is 58 mph (SD = 10 mph) and speeds are normally distributed. What speed must someone now be over to get pulled over?

  36. Z scores and the Rare Event I bought a new Subaru. They told me that the average gas mileage is 32 mpg. I drive one day and get 25 mpg. If the standard deviation is 4, how likely is it that I would get such a low gas mileage? Assume gas mileage is normally distributed. How rare is it to have a Subaru with gas mileage this low? In other words, could my car be a lemon???

  37. My Car: 25 mpg z=-1.75 P > 25mpg = .96 P < 25mpg = .04 Is this a rare enough event to conclude my Subaru is a Lemon?

  38. Looking ahead to inferential statistics The Original Whole Population T R E A T M E N T Is the treated sample noticeably different from the population? Sample Treated Sample

  39. Looking ahead to inferential statistics μ = 400g σ = 20g Adult Rats H O R M O N E Sample of Adult Rats Treated Sample of Rats

  40. Treated rat: 418g μ = 400g σ = 20g Z= Z = 0 1.96 -1.96 Raw = 360.8g 400g 439.2g

  41. μ = 400g σ = 20g Treated rat: 450g Z= Z = 0 1.96 -1.96 Raw = 360.8g 400g 439.2g

More Related