Probability & Using Frequency Distributions Chapters 1 & 6 Homework: Ch 1: 9-12 Ch 6: 1, 2, 3, 8, 9, 14

Probability: Definitions n Chapter 1, pp n Experiment: l controlled operation l yields 1 of several possible outcomes l e.g., drawing a card from deck n Event l a set of possible outcomes e.g. draw a heart 13 possible outcomes ~

Probability: Definitions n Probability(P) of an event (E) l Assuming each outcome equally likely P(E) = # outcomes favorable to E total # possible outcomes P(drawing ª ) = P(7 of ª ) = P(15 of ª ) = P( « or ª or © or ¨ ) = ~

Probability: 3 important characteristics 1. Probability event cannot occur is 0 2. P(E) that must occur =  P(E)  1, probabilities lie b/n 0 & 1 ~

Determining Probabilities n Must count ALL possible outcomes n A fair die: P(1) = P(2) = … = P(6) l P(4) = n Event = sum of two fair dice l P(4) = l 36 possible outcomes of rolling 2 dice l Sum to 4: l __ possible outcomes favorable to E ~

Determining Probabilities n Single fair die n Addition rule l keyword: OR l P(1 or 3) = n Multiplication rule l keyword AND l P(1 on first roll and 3 on second roll) = l dependent events ~

Conditional Probabilities n Put restrictions on range of possible outcomes l P(heart) given that card is Red l P(Heart | red card) = n P(5 on 2d roll | 5 on 1st roll)? l P = l 1st & 2d roll independent events ~

Points in Distributions n Up to now describing distributions n Comparing scores from different distributions l Need to make equivalent comparisons l Percentile rank & standard scores l z scores ~

Percentiles & Percentile Rank n Percentile l score below which a specified percentage of scores in the distribution fall l start with percentage ---> score n Percentile rank Per cent of scores  a given score l start with score ---> percentage n Score: a value of any variable ~

Percentiles n E.g., test scores l 30 th percentile = (A) 46; (B) 22 l 90 th percentile = (A) 56; (B) 46 ~ A B

Percentile Rank n e.g., Percentile rank for score of 46 l (A) 30%; (B) = 90% n Problem: equal differences in % DO NOT reflect equal distance between values ~ A B

Standard Scores n Convert raw scores to z scores n raw score: value using original scale of measurement n z scores: # of standard deviations score is from mean l e.g., z = 2 = 2 std. deviations from mean l z = 0 = mean ~

z Score Equations Sample: z = Population: z = X - X s X -  

z Score Computation n e.g., 90 th percentile = (A) 56; (B) 46 l convert to z scores A:  = 5;  = 50 B:  = 10;  = 29

Areas Under Distributions n Area = frequency n Relative area l total area = ____ = proportion of individual values in area under curve ~

Total area under curve =

Using Areas Under Distributions n Relative area is independent of shape of distribution n Given value, what is relative frequency? n Question: what % of days is the temperature over 60 o ? l Or P(temperature > 60 o ) ~

Average Daily Temperature ( o F) % of days the temperature is above 60 o

Average Daily Temperature ( o F) % of days temperature is between 30 & 50 o ?

Using Areas Under Distributions n Given relative frequency, what is value? l e.g., What is temperature on the hottest 10% of days l find value of X at border ~

Average Daily Temperature ( o F) temperature on the hottest 10% of days

Areas Under Normal Curves Many variables  normal distribution l Normal distribution completely specified by 2 numbers l mean & standard deviation n Many other normal distributions have different  &  ~

Areas Under Normal Curves n Unit Normal Distribution l based on z scores  = 0  = 1 l e.g., z = -2 n relative areas under normal distribution always the same l precise areas from Table A.1 ~

Areas Under Normal Curves f standard deviations

Calculating Areas from Tables n Table A.1 (in our text) “Proportions of areas under the normal curve” n 3 columns l z l (A) Area between mean and z l (B) Area beyond z (in tail) n Negative z: area same as positive ~

Calculating Areas from Tables n Area between mean and z=1 l 0 < z < 1 = (from A) l beyond z=1: (from B) l A + B =.5 n Area: 1 < z < 2 l find z=2; 0 < z < 2 = l subtract area for z=1 ~

Calculating Areas from Tables n Area between z=-2 and z=1 l add areas for z=-2 and z=1 l -2 < z < 0 = l 0 < z < 1 = ~

Calculating Areas from Tables n Area between... z=0 and z=1.34 l 0 < z < 1.34 z=1.5 and z=1.92 l 1.5 < z < 1.92 z=-1.37 and z=.23 l 1.37 < z <.23

Other Standardized Distributions n Normal distributions, l but not unit normal distribution n Standardized variables l normally distributed specify  and  in  advance n e.g., IQ test  = 100;  = 15 ~

Other Standardized Distributions f IQ Scores 120-2z scores

Transforming to & from z scores n From z score to standardized score in population X = z  +  n Standardized score ---> z score z = X -   z = X - X  s n Samples: l X = zs + X

Know/want Diagram Raw Score (X) z score area under distribution z = X -   X = z  +  Table: column A or B Table: z column

Normal Distributions: Percentiles/Percentile Rank n Unit normal distributions 50th percentile = 0 =  l z = 1 is 84th percentile 50% + 34% n Relationships l z score & standard score linear l z score & percentile rank nonlinear ~

f IQ Scores 120-2z scores percentile rank IQ