Probability &Using Frequency Distributions

Chapters 1 & 6

Homework: Ch 1: 9-12

Ch 6: 1, 2, 3, 8, 9, 14

Probability: Definitions

Chapter 1, pp. 8-10 Experiment:

controlled operation yields 1 of several possible outcomes e.g., drawing a card from deck

Event a set of possible outcomes

e.g. draw a heart

13 possible outcomes ~

Probability: Definitions Probability(P) of an event (E)

Assuming each outcome equally likely

P(E) = # outcomes favorable to E total # possible outcomes

P(drawing ) =

P(7 of ) =

P(15 of ) =

P(or or or ) = ~

Probability: 3 important characteristics

1. Probability event cannot occur is 0

2. P(E) that must occur = 1

3. 0 P(E) 1, probabilities lie b/n 0 & 1 ~

Determining Probabilities

Must count ALL possible outcomes A fair die: P(1) = P(2) = … = P(6)

P(4) = Event = sum of two fair dice

P(4) = 36 possible outcomes of rolling 2 dice Sum to 4: __ possible outcomes favorable to E ~

Determining Probabilities

Single fair die Addition rule

keyword: OR P(1 or 3) =

Multiplication rule keyword AND P(1 on first roll and 3 on second roll) = dependent events ~

Conditional Probabilities

Put restrictions on range of possible outcomes P(heart) given that card is Red P(Heart | red card) =

P(5 on 2d roll | 5 on 1st roll)? P = 1st & 2d roll independent events ~

Points in Distributions

Up to now describing distributions Comparing scores from different

distributions Need to make equivalent comparisons Percentile rank & standard scores z scores ~

Percentiles & Percentile Rank Percentile

score below which a specified percentage of scores in the distribution fall

start with percentage ---> score Percentile rank

Per cent of scores a given score start with score ---> percentage

Score: a value of any variable ~

Percentiles E.g., test scores

30th percentile = (A) 46; (B) 22

90th percentile = (A) 56; (B) 46 ~

A58565454525048464442

B50463230302323222120

Percentile Rank

e.g., Percentile rank for score of 46 (A) 30%; (B) = 90%

Problem: equal differences in % DO NOT reflect equal distance between values ~

A58565454525048464442

B50463230302323222120

Standard Scores

Convert raw scores to z scores raw score: value using original scale of

measurement z scores: # of standard deviations score is from

mean e.g., z = 2

= 2 std. deviations from mean z = 0 = mean ~

z Score Equations

Sample: z =

Population: z =

X - Xs

X -

z Score Computation

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

Areas Under Distributions

Area = frequency Relative area

total area = ____

= proportion of individual values in area under curve ~

Total area under curve = 1.0

10 20 30 40 50 60 70 80 90

Using Areas Under Distributions

Relative area is independent of shape of distribution

Given value, what is relative frequency? Question: what % of days is the

temperature over 60 o? Or P(temperature > 60o) ~

10 20 30 40 50 60 70 80 90

Average Daily Temperature (oF)

% of days the temperature is above 60 o

10 20 30 40 50 60 70 80 90

Average Daily Temperature (oF)

% of days temperature is between 30 & 50o?

Using Areas Under Distributions

Given relative frequency, what is value? e.g., What is temperature on the

hottest 10% of days find value of X at border ~

10 20 30 40 50 60 70 80 90

Average Daily Temperature (oF)

temperature on the hottest 10% of days

Areas Under Normal Curves

Many variables normal distribution Normal distribution completely

specified by 2 numbers mean & standard deviation

Many other normal distributions have different & ~

Areas Under Normal Curves Unit Normal Distribution

based on z scores

= 0

= 1 e.g., z = -2

relative areas under normal distribution always the same precise areas from Table A.1 ~

Areas Under Normal Curves

1 20-1-2

f

standard deviations

Calculating Areas from Tables

Table A.1 (in our text)

“Proportions of areas under the normal curve”

3 columns z (A) Area between mean and z (B) Area beyond z (in tail)

Negative z: area same as positive ~

Calculating Areas from Tables

Area between mean and z=1 0 < z < 1 = (from A) beyond z=1: (from B) A + B = .5

Area: 1 < z < 2 find z=2; 0 < z < 2 = subtract area for z=1 ~

Calculating Areas from Tables

Area between z=-2 and z=1 add areas for z=-2 and z=1 -2 < z < 0 = 0 < z < 1 = ~

Calculating Areas from Tables

Area between ...

z=0 and z=1.34 0 < z < 1.34

z=1.5 and z=1.92 1.5 < z < 1.92

z=-1.37 and z=.23 1.37 < z < .23

Other Standardized Distributions

Normal distributions, but not unit normal distribution

Standardized variables normally distributed specify and inadvance

e.g., IQ test = 100; = 15 ~

Other Standardized Distributions

115 1301008570

f

IQ Scores

1 20-1-2z scores

Transforming to & from z scores

From z score to standardized score in population X = z +

Standardized score ---> z score

z = X -

z = X - Xs

Samples: 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

Unit normal distributions 50th percentile = 0 = z = 1 is 84th percentile

50% + 34% Relationships

z score & standard score linear z score & percentile rank nonlinear ~

115 1301008570

f

IQ Scores

1 20-1-2z scores

percentile rank

IQ