Probability z scores Lesson 6 Science Probability Learn

Probability & z scores Lesson 6

Science & Probability Learn about populations by studying samples l Introduction of error n Drawing conclusions l Cannot make statements with certainty l Probability statements n Use of normal distribution l Can calculate probability of a result l Natural variables ≈ normal ~ n

Probability: Definitions n Probability(P) of an event (A) l Assuming each outcome equally likely P(A) = # outcomes Classified as A total # possible outcomes P(drawing ♥) = P(7 of ♥) = P(15 of ♥) = P(♥ or ♦ or ♣ or ♠) = ~

Probability & the Normal Distribution f IQ 70 85 100 115 130

z scores AKA standard scores l distance from mean in units of s n Uses l Determining probabilities l Percentile rank or scores l Compare scores from different distributions ~ n

z Score Equation z = X-m s

Standard Normal Distribution AKA Unit Normal Distribution n Parameters l m = 0, s = 1 n Distribution of z scores l Distance & direction from m in units of s ~ n

Standard Normal Distribution f -2 -1 0 Z scores (s) 1 2

Other Standardized Distributions Many natural variables ≈ normal n Standardized distributions l Have defined or set parameters l IQ: m = 100, s = 15 l ACT: m = 18, s = 6 l SAT: m = 500, s = 100 ~ n

IQ: m = 100, s = 15 ACT: m = 18, s = 6 f z scores -2 -1 0 1 2 IQ 70 85 100 115 130 ACT 6 18 24 30 12

Using z scores Distance and direction relative to mean l Standard Normal Distribution l m = 0, s = 1 n Answer questions by 1 st finding z score l What proportion of population have IQ scores greater than 115? l What is the percentile rank for IQ score of 70? l What percentage of people have IQ scores between 70 and 115? n

The Normal Distribution & Probability Area under curve = frequency l Area under curve represents all data n Proportion (p) including all scores = 1 l p for any area under curve can be calculated l Proportion = probability that a score(s) is in distribution l *Table B. 1, pg. 351 ~ n

Probability of obtaining IQ score below the median? Greater than 115? Percentile rank of 70? Use z scores. f IQ 70 85 100 115 130

IQ Scores m = 100 s = 15 f z scores -2 -1 0 1 2 IQ 70 85 100 115 130

z score for 115? P (Xi > 115) z score for 70? P (Xi < 70) P (70 < Xi < 115) f IQ 70 85 100 IQ Score 115 130

z score for 115? z score for 70? f IQ 70 85 100 IQ Score 115 130

z score for 115? P (Xi > 115) z score for 70? P (Xi < 70) P (70 < Xi < 115) f IQ 70 85 100 IQ Score 115 130

z score for 115? P (Xi > 115) =. 1587 z score for 70? P (Xi < 70) P (70 < Xi < 115) f IQ 70 85 100 IQ Score 115 130

z score for 115? P (Xi > 115) =. 1587 z score for 70? P (Xi < 70) =. 0228 P (70 < Xi < 115) f IQ 70 85 100 IQ Score 115 130

z score for 115? P (Xi > 115) =. 1587 z score for 70? P (Xi < 70) =. 0228 P (70 < Xi < 115) =. 7885. 4472 +. 3413 =. 7885 f. 4472 IQ 70 85 . 3413 100 IQ Score 115 130

z score for 115? P (Xi > 115) =. 1587 z score for 70? P (Xi < 70) =. 0228 P (115 < Xi < 130) =. 1359. 1587 -. 0228 =. 1359 f IQ 70 85 100 IQ Score 115 130

Handy Numbers n Standard Normal Distribution l z scores l Proportions of distribution u n i. e. , area under curve, table A. 1 3 handy proportions l Same for all normal distributions l Between z = 0 and ± 1 l Between z = 1 and 2 (also -1 & -2) l Beyond z = ± 2 (area in tails) ~

Areas Under Normal Curves f. 34. 02. 14 -2 . 14 -1 0 Z scores (s) 1 2

What % of students scored b/n 18 and 24? % greater than 30? % less than 30? ACT Scores m = 18 s=6 f z scores ACT -2 6 -1 12 0 1 2 18 24 30

Comparing Scores from Different Distributions How to compare ACT to SAT? l Use z scores 1. Raw ACT score z score n 2. Use z score to compute Raw SAT score

Percentile Rank & Percentile rank l % of scores ≤ a particular score (Xi) th percentile: 84% of IQ scores ≤ 115 l 84 n Percentile l Raw score (Xi) associated with a particular percentile rank th percentile l IQ score of 100 is the 50 n Use z scores & table to determine ~ n

IQ Scores f. 34. 02. 14 IQ 70 85 100 115 130 z scores -2 -1 0 1 2 84 th 98 th percentile rank 2 d 16 th 50 th

Using z scores Table: large/smaller portion column Raw Score (X) z score area under curve or proportion Or probability or percentage Table: z column