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1. Screen Cabinet Cabinet Lecturer’s desk Table Computer Storage Cabinet Row A 3 4 5 19 6 18 7 17 16 8 15 9 10 11 14 13 12 Row B 1 2 3 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 Row C 1 2 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 Row D 1 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 Row E 1 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 Row F 27 1 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 28 Row G 27 1 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 29 10 19 11 18 16 15 13 12 17 14 28 Row H 27 1 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 Row I 1 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 1 Row J 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 28 27 1 Row K 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 Row L 20 1 19 2 18 3 17 4 16 5 15 6 7 14 13 INTEGRATED LEARNING CENTER ILC 120 9 8 10 12 11 broken desk
2. Please click in My last name starts with a letter somewhere between A. A – D B. E – L C. M – R D. S – Z
3. Introduction to Statistics for the Social SciencesSBS200, COMM200, GEOG200, PA200, POL200, SOC200Lecture Section 001, Spring, 2012Room 120 Integrated Learning Center (ILC)9:00 - 9:50 Mondays, Wednesdays & Fridays+ Lab Session. Welcome http://www.youtube.com/watch?v=oSQJP40PcGI
4. Use this as your study guide By the end of lecture today2/20/12 Measures of variability Standard deviation and Variance Estimating standard deviation Exploring relationship between mean and variability Probability of an event Complement of an event; Union of two events Intersection of two events; Mutually exclusive events Collectively exhaustive events Conditional probability Confidence Intervals
5. Homework Assignment #11 Dan Gilbert Reading – Law of Large Numbers Due Wednesday (February 22nd) Please double check – Allcell phones other electronic devices are turned off and stowed away
6. If P(A) = 0, then the event cannot occur. If P(A) = 1, then the event is certain to occur. The probability of an event is the relative likelihood that the event will occur. The probability of event A [denoted P(A)], must lie within the interval from 0 to 1: 0 <P(A) < 1
7. Probability The probabilities of all simple events must sum to 1 P(S) = P(E1) + P(E2) + … + P(En) = 1 For example, if the following number of purchases were made by
8. What is the complement of the probability of an event The probability of event A = P(A). The probability of the complement of the event A’ = P(A’) A’ is called “A prime” Complement of A just means probability of “not A” P(A) + P(A’) = 100% P(A) = 100% - P(A’) P(A’) = 100% - P(A) Probability of getting a rotten apple 5% chance of “rotten apple” 95% chance of “not rotten apple” 100% chance of rotten or not Probability of getting into an educational program 66% chance of “admitted” 34% chance of “not admitted” 100% chance of admitted or not
9. Two mutually exclusive characteristics: if the occurrence of any one of them automatically implies the non-occurrence of the remaining characteristic Two events are mutually exclusive if they cannot occur at the same time (i.e. they have no outcomes in common). Two propositions that logically cannot both be true. NoWarranty Warranty For example, a car repair is either covered by the warranty (A) or not (B). http://www.thedailyshow.com/video/index.jhtml?videoId=188474&title=an-arab-family-man
10. Collectively Exhaustive Events Events are collectively exhaustive if their union isthe entire sample space S. Two mutually exclusive, collectively exhaustive events are dichotomous (or binary) events. For example, a car repair is either covered by the warranty (A) or not (B). NoWarranty Warranty
11. NoWarranty Satirical take on being “mutually exclusive” Warranty Recently a public figure in the heat of the moment inadvertently made a statement that reflected extreme stereotyping that many would find highly offensive. It is within this context that comical satirists have used the concept of being “mutually exclusive” to have fun with the statement. Decent , family man Arab Transcript: Speaker 1: “He’s an Arab” Speaker 2: “No ma’am, no ma’am. He’s a decent, family man, citizen…” http://www.thedailyshow.com/video/index.jhtml?videoId=188474&title=an-arab-family-man
12. Union versus Intersection ∩ P(A B) Union of two events means Event A or Event B will happen Intersection of two events means Event A and Event B will happen Also called a “joint probability” P(A ∩ B)
13. The union of two events: all outcomes in the sample space S that are contained either in event Aor in event Bor both (denoted A  B or “A or B”).  may be read as “or” since one or the other or both events may occur.
14. The union of two events: all outcomes contained either in event Aor in event Bor both (denoted A  B or “A or B”). What is probability of drawing a red card or a queen? what is Q  R? It is the possibility of drawing either a queen (4 ways) or a red card (26 ways) or both (2 ways).
15. Probability of picking a Queen Probability of picking a Red 26/52 4/52 P(Q) = 4/52(4 queens in a deck) 2/52 P(R) = 26/52 (26 red cards in a deck) P(Q  R) = 2/52 (2 red queens in a deck) Probability of picking both R and Q When you add the P(A) and P(B) together, you count the P(A and B) twice. So, you have to subtract P(A  B) to avoid over-stating the probability. P(Q  R) = P(Q) + P(R) – P(Q  R) = 4/52 + 26/52 – 2/52 = 28/52 = .5385 or 53.85%
16. Union versus Intersection ∩ P(A B) Union of two events means Event A or Event B will happen Intersection of two events means Event A and Event B will happen Also called a “joint probability” P(A ∩ B)
17. The intersection of two events: all outcomes contained in both event A and event B(denoted A  B or “A and B”) What is probability of drawing red queen? what is Q R? It is the possibility of drawing both a queen and a red card (2 ways).
18. If two events are mutually exclusive (or disjoint) their intersection is a null set (and we can use the “Special Law of Addition”) P(A ∩ B) = 0 Intersection of two events means Event A and Event B will happen Examples: mutually exclusive If A = Poodles If B = Labradors Poodles and Labs:Mutually Exclusive (assuming purebred)
19. If two events are mutually exclusive (or disjoint) their intersection is a null set (and we can use the “Special Law of Addition”) P(A ∩ B) = 0 ∩ Dog Pound P(A B) = P(A) +P(B) Intersection of two events means Event A and Event B will happen Examples: If A = Poodles If B = Labradors (let’s say 10% of dogs are poodles) (let’s say 15% of dogs are labs) What’s the probability of picking a poodle or a lab at random from pound? P(poodle or lab) = P(poodle) + P(lab) P(poodle or lab) = (.10) + (.15) = (.25) Poodles and Labs:Mutually Exclusive (assuming purebred)
20. Conditional Probabilities Probability that A has occurred given that B has occurred Denoted P(A | B): The vertical line “ | ” is read as “given.” P(A ∩ B) P(A | B) = P(B) The sample space is restricted to B, an event that has occurred. A  B is the part of B that is also in A. The ratio of the relative size of A  B to B is P(A | B).
21. Conditional Probabilities Probability that A has occurred given that B has occurred Of the population aged 16 – 21 and not in college: P(U) = .1350 P(ND) = .2905 P(UND) = .0532 What is the conditional probability that a member of this population is unemployed, given that the person has no diploma? .0532 P(A ∩ B) .1831 = P(A | B) = = .2905 P(B) or 18.31%
22. Conditional Probabilities Probability that A has occurred given that B has occurred Of the population aged 16 – 21 and not in college: P(U) = .1350 P(ND) = .2905 P(UND) = .0532 What is the conditional probability that a member of this population is unemployed, given that the person has no diploma? .0532 P(A ∩ B) .1831 = P(A | B) = = .2905 P(B) or 18.31%
23. Estimating the mean of a population On average newborns weigh 7 pounds, and are 20 inches long.My sister just had a baby - guess how much it weighs? - guess how long it is Point estimate: a single number that represents your best guess at a single value in an unknown population (use measure of central tendency, like the mean) Makes sense, right?!? Guess the mean. On average you would be right most often if you always guessed the mean
24. Point estimate: a single number that represents your best guess at a single value in an unknown population This sample of 10,000 newborns a mean weight is 7 pounds. What do you think the mean weight of a random sample of 10 newborns would be? Without more info, on average you would be right most often if you always guessed the mean This sample of 500 households produced a mean income of $35,000 a year. What do you think the mean income of Mabel is? This sample of 1000 kids had a mean IQ of 100. What do you think the mean of all kids is? If you didn’t know Miguel, what would you guess his IQ to be?
25. Problem with point estimate Mean kids IQ of 100. Mean income of $35,000 a year. Mean weight 7 pounds. Are we right always? - no How close is our estimation? - what other information about these distributions would we want to know? Variability! Which of these distributions would allow our guess to be closest to what’s right?
26. Estimating the mean of a population 95% 2.5% 2.5% On average newborns weigh 7 pounds, and are 20 inches long.My sister just had a baby - guess how much it weighs? - guess how long it is What if you really needed to be right?!!? You could guess a range with min and max scores. (how wide a range to be completely sure? 7 pounds
27. Point estimate: a single number that represents your best guess at a single value in an unknown population 95% Confidence Interval: We can be 95% confident that our population mean falls between these two scores 99% Confidence Interval: We can be 99% confident that our population mean falls between these two scores Which has a wider interval relative to raw scores?
28. Confidence intervals Can actually generate CI for any confidence level you want – these are just the most common Confidence Intervals (based on z): A range of values that, with a known degree of certainty, includes an unknown population characteristic, such as a population mean The interval refers to possible values of the population mean. We can be reasonably confident that the population mean falls in this range (90%, 95%, or 99% confident) In the long run, series of intervals, like the one we figured out will describe the population mean about 95% of the time.
29. Confidence Intervals (based on z): A range of values that, with a known degree of certainty, includes an unknown population characteristic, such as a population mean How can we make our confidence interval smaller? Decrease level of confidence Decrease variability through more careful assessment and measurement practices (minimize noise) Increase sample size (This will decrease variability too) . 95% 95%
30. Confidence Intervals (based on z): A range of values that, with a known degree of certainty, includes an unknown population characteristic, such as a population mean Choosing a Confidence Level A higher confidence level leads to a wider confidence interval. Greater confidence implies loss of precision.(95% confidence is most often used) . 95% 95%
31. Thank you! See you next time!!