Lecturer’s desk INTEGRATED LEARNING CENTER ILC 120 Screen Row A Row B Row C Row D Row E Row F Row G Row H Row I Row J Row K Row L Computer Storage Cabinet Cabinet Table broken desk

Hand out z tables

Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Fall, 2014 Room 120 Integrated Learning Center (ILC) 10: :50 Mondays, Wednesdays & Fridays.

Reminder A note on doodling

Labs continue this week

One positive correlation One negative correlation One t-test

Schedule of readings Before next exam (October 17 th ) Please read chapters 5, 6, & 8 in Ha & Ha Please read Chapters 10, 11, 12 and 14 in Plous Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability and Risk Chapter 14: The Perception of Randomness

By the end of lecture today 10/6/14 Use this as your study guide Counting ‘standard deviationses’ – z scores Connecting raw scores, z scores and probability Connecting probability, proportion and area of curve Percentiles

Homework due – Wednesday (October 8 th ) On class website: Please print and complete homework worksheet #11 Calculating z-score, raw scores and areas under normal curve Deadline extended

Mean = 100 Standard deviation = 5 If we go up one standard deviation z score = +1.0 and raw score = 105 If we go down one standard deviation z score = -1.0 and raw score = 95 If we go up two standard deviations z score = +2.0 and raw score = 110 If we go down two standard deviations z score = -2.0 and raw score = 90 If we go up three standard deviations z score = +3.0 and raw score = 115 If we go down three standard deviations z score = -3.0 and raw score = 85 z score: A score that indicates how many standard deviations an observation is above or below the mean of the distribution z score = raw score - mean standard deviation z = -1 z = +1 z = -2 z = +2 z = -3 z = +3 68% 95% 99.7%

Raw scores, z scores & probabilities Raw Scores Area & Probability Z Scores Formula z table Have raw score Find z Have z Find raw score Have area Find z Have z Find area

Ties together z score with Draw picture of what you are looking for... Find z score (using formula)... Look up proportions on table probability proportion percent area under the curve 68% 34%

Writing Assignment Let’s do some problems Mean = 50 Standard deviation = 10

1) Find z score z score = Find the percentile rank for score of ? 2) Go to z table z score = = -0.5 Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Problem 3 Problems 1 & 2 were completed in lecture on Friday

Mean = 50 Standard deviation = 10 1) Find z score z score = Find the percentile rank for score of ? 2) Go to z table z score = = -0.5 ?.1915 Problem 3

Mean = 50 Standard deviation = 10 1) Find z score z score = Find the percentile rank for score of ) Go to z table z score = = ) Percentile rank or score of 45 = 30.85% 3) Look at your picture - subtract = ?.3085 Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Problem 3

Mean = 50 Standard deviation = 10 1) Find z score z score = Find the percentile rank for score of 55 ? 2) Go to z table z score = 5 10 = Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Problem 4

Mean = 50 Standard deviation = 10 1) Find z score z score = Find the percentile rank for score of 55 2) Go to z table z score = 5 10 = ? Problem 4

Mean = 50 Standard deviation = 10 1) Find z score z score = Find the percentile rank for score of 55 ? 2) Go to z table z score = 5 10 = 0.5 4) Percentile rank or score of 55 = 69.15% 3) Look at your picture - add = Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Problem 4

Find the score for z = -2 Mean = 50 Standard deviation = 10 raw score = mean + (z score)(standard deviation) Raw score = 50 + (-2)(10) Raw score = 50 + (-20) = 30 Hint always draw a picture! Find the score that is associated with a z score of -2 ? 30 Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion

Mean = 50 Standard deviation = 10 Find the score for percentile rank of 77%ile.7700 ? ? Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion Problem 5

Mean = 50 Standard deviation = 10 Find the score for percentile rank of 77%ile 1) Go to z table - find z score for for area.2700 ( ) = ? ? area =.2704 (closest I could find to.2700) z = 0.74 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion Problem =.77

Mean = 50 Standard deviation = 10 Find the score for percentile rank of 77%ile ? ) x = mean + (z)(standard deviation) x = 50 + (0.74)(10) x = 57.4 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion x = ?.5.27 Problem 5

Mean = 50 Standard deviation = 10 Find the score for percentile rank of 55%ile.5500 ? ? Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Problem 6 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion

Mean = 50 Standard deviation = 10 Find the score for percentile rank of 55%ile 1) Go to z table - find z score for for area.0500 ( ) = ? ? area =.0517 (closest I could find to.0500) z = 0.13 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion Problem =.55

Mean = 50 Standard deviation = 10 Find the score for percentile rank of 55%ile 1) Go to z table - find z score for for area.0500 ( ) = ? ? area =.0517 (closest I could find to.0500) z = 0.13 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion Problem 6

Mean = 50 Standard deviation = 10 Find the score for percentile rank of 55%ile 1) Go to z table - find z score for for area.0500 ( ) =.0500 area =.0517 (closest I could find to.0500) z = ? ? ) x = mean + (z)(standard deviation) x = 50 + (0.13)(10) x = 51.3 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion x = 51.3 Problem 6

nearest z = 1.64 Go to table.4500 Normal Distribution has a mean of 50 and standard deviation of 4. Determine value below which 95% of observations will occur. Note: sounds like a percentile rank problem x = mean + z σ = 50 + (1.64)(4) = Problem 7

nearest z = Go to table.4700 Normal Distribution has a mean of $2,100 and s.d. of $250. What is the operating cost for the lowest 3% of airplanes. Note: sounds like a percentile rank problem = find score for 3 rd percentile x = mean + z σ = (-1.88)(250) = 1,630 Problem 8

nearest z = 2.33 Go to table.4900 Normal Distribution has a mean of 195 and standard deviation of 8.5. Determine value for top 1% of hours listened. x = mean + z σ = (2.33)(8.5) = Problem 9

. 75 th percentile Go to table.2500 nearest z =.67 x = mean + z σ = 30 + (.67)(2) = z =.67 Problem 10

. 25 th percentile Go to table.2500 nearest z = -.67 x = mean + z σ = 30 + (-.67)(2) = z = -.67 Problem 11

. Try this one: Please find the (2) raw scores that border exactly the middle 95% of the curve Mean of 30 and standard deviation of 2 Go to table.4750 nearest z = 1.96 mean + z σ = 30 + (1.96)(2) = Go to table.4750 nearest z = mean + z σ = 30 + (-1.96)(2) = Problem 12

Notice: 3 types of numbers raw scores z scores probabilities Mean = 50 Standard deviation = 10 If we go up two standard deviations z score = +2.0 and raw score = 70 If we go down two standard deviations z score = -2.0 and raw score = 30 Raw scores, z scores & probabilities z = -2 z = +2

Raw Scores Area & Probability Z Scores Formula z table Have raw score Find z Have z Find raw score Have area Find z Have z Find area Normal distribution Raw scores z-scores probabilities

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Hint: Always draw a picture! Homework worksheet

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