Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Spring 2017 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays. Welcome http://www.youtube.com/watch?v=oSQJP40PcGI http://www.youtube.com/watch?v=oSQJP40PcGI

A note on doodling

Before next exam (March 3rd) Schedule of readings Before next exam (March 3rd) Please read chapters 1 - 8 in OpenStax textbook 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 2/15/17 Counting ‘standard deviationses’ – z scores Connecting raw scores, z scores and probability Connecting probability, proportion and area of curve Percentiles

Homework Assignment Assignment 10 Please complete the homework worksheet Finding z scores and areas under the curve. Due: Friday, February 17th Assignment 11 Please complete the homework modules on the D2L website HW10-Normal Curve, z scores and probabilities

Lab sessions Everyone will want to be enrolled in one of the lab sessions Labs continue With Project 2

Hand out z tables

z score = raw score - mean standard deviation If we go up one standard deviation z score = +1.0 and raw score = 105 z = -1 z = +1 68% If we go down one standard deviation z score = -1.0 and raw score = 95 85 90 95 100 105 110 115 If we go up two standard deviations z score = +2.0 and raw score = 110 z = -2 95% z = +2 If we go down two standard deviations z score = -2.0 and raw score = 90 85 90 95 100 105 110 115 If we go up three standard deviations z score = +3.0 and raw score = 115 99.7% z = -3 z = +3 If we go down three standard deviations z score = -3.0 and raw score = 85 85 90 95 100 105 110 115 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

If score is within 2 standard deviations (z < 2) “not unusual score” If score is beyond 2 standard deviations (z = 2 or up to 3) “is unusual score” If score is beyond 3 standard deviations (z = 3 or up to 4) “is an outlier” If score is beyond 4 standard deviations (z = 4 or beyond) “is an extreme outlier”

Scores, standard deviations, and probabilities What is total percent under curve? What proportion of curve is above the mean? .50 100% The normal curve always has the same shape. They differ only by having different means and standard deviation

Scores, standard deviations, and probabilities What percent of curve is below a score of 100? What score is associated with 50th percentile? 50% median Mean = 100 Standard deviation = 5

Raw scores, z scores & probabilities Distance from the mean (z scores) convert convert Raw Scores (actual data) Proportion of curve (area from mean) 68% z = -1 z = 1 We care about this! What is the actual number on this scale? “height” vs “weight” “pounds” vs “test score” We care about this! “percentiles” “percent of people” “proportion of curve” “relative position” 68% Raw Scores (actual data) Proportion of curve (area from mean) z = -1 z = 1 Distance from the mean (z scores) convert convert

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

Find z score for raw score of 60 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 60 50 10 z = 1 Mean = 50 Standard deviation = 10

Find the area under the curve that falls between 50 and 60 50 60 34.13% 1) Draw the picture 2) Find z score 3) Go to z table - find area under correct column 4) Report the area 60 50 10 z = 1

Find the area under the curve that falls between 40 and 60 Mean = 50 Standard deviation = 10 68.26% Find the area under the curve that falls between 40 and 60 34.13% 34.13% z score = raw score - mean standard deviation Hint always draw a picture! z score = 60 - 50 10 z score = 40 - 50 10 z score = 10 = 1.0 10 z score = 10 = -1.0 10 z table z table z score of 1 = area of .3413 z score of 1 = area of .3413 .3413 + .3413 = .6826

Find the area under the curve that falls between 30 and 50 Mean = 50 Standard deviation = 10 1) Draw the picture 2) Find z score 3) Go to z table - find area under correct column 4) Report the area Find the area under the curve that falls between 30 and 50 z score = raw score - mean standard deviation z score = 30 - 50 10 z score = - 20 = - 2.0 10 Hint always draw a picture!

Find the area under the curve that falls between 30 and 50 Mean = 50 Standard deviation = 10 1) Draw the picture 2) Find z score 3) Go to z table - find area under correct column 4) Report the area 47.72% Find the area under the curve that falls between 30 and 50 z score = raw score - mean standard deviation z score = 30 - 50 10 z score = - 20 = - 2.0 10 z table z score of - 2 = area of .4772 Hint always draw a picture! Hint always draw a picture!

Find the area under the curve that falls between 70 and 50 Mean = 50 Standard deviation = 10 47.72% Find the area under the curve that falls between 70 and 50 z score = raw score - mean standard deviation z score = 70 - 50 10 z score = 20 = +2.0 10 z table z score of 2 = area of .4772 Hint always draw a picture!

Find the area under the curve that falls between 30 and 70 Mean = 50 Standard deviation = 10 .4772 .4772 95.44% z score of 2 = area of .4772 Find the area under the curve that falls between 30 and 70 .4772 + .4772 = .9544 Hint always draw a picture!

Scores, standard deviations, and probabilities Actually 68.26 Actually 95.44 To be exactly 95% we will use z = 1.96

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

Find the percentile rank for score of 60 Mean = 50 Standard deviation = 10 ? Let’s do some problems 60 Find the area under the curve that falls below 60 means the same thing as Find the percentile rank for score of 60 Problem 1

Find the percentile rank for score of 60 Mean = 50 Standard deviation = 10 ? Let’s do some problems Find the percentile rank for score of 60 60 .5000 .3413 1) Find z score z score = 60 - 50 10 = 1 2) Go to z table - find area under correct column (.3413) 3) Look at your picture - add .5000 to .3413 = .8413 4) Percentile rank or score of 60 = 84.13% Problem 1 Hint always draw a picture!

Find the percentile rank for score of 75 Mean = 50 Standard deviation = 10 ? Find the percentile rank for score of 75 75 .4938 1) Find z score z score = 75 - 50 10 z score = 25 10 = 2.5 2) Go to z table Problem 2 Hint always draw a picture!

Find the percentile rank for score of 75 Mean = 50 Standard deviation = 10 ? Find the percentile rank for score of 75 75 .4938 .5000 1) Find z score z score = 75 - 50 10 z score = 25 10 = 2.5 2) Go to z table 3) Look at your picture - add .5000 to .4938 = .9938 4) Percentile rank or score of 75 = 99.38% Problem 2 Hint always draw a picture!

Find the percentile rank for score of 45 Mean = 50 Standard deviation = 10 ? Find the percentile rank for score of 45 45 1) Find z score z score = 45 - 50 10 z score = - 5 10 = -0.5 2) Go to z table Problem 3

Find the percentile rank for score of 45 Mean = 50 Standard deviation = 10 ? Find the percentile rank for score of 45 .1915 45 ? 1) Find z score z score = 45 - 50 10 z score = - 5 10 = -0.5 2) Go to z table Problem 3

Find the percentile rank for score of 45 Mean = 50 Standard deviation = 10 ? Find the percentile rank for score of 45 .1915 45 .3085 1) Find z score z score = 45 - 50 10 z score = - 5 10 = -0.5 2) Go to z table 3) Look at your picture - subtract .5000 -.1915 = .3085 Problem 3 4) Percentile rank or score of 45 = 30.85%

Find the percentile rank for score of 55 Mean = 50 Standard deviation = 10 ? Find the percentile rank for score of 55 55 1) Find z score z score = 55 - 50 10 z score = 5 10 = 0.5 2) Go to z table Problem 4

Find the percentile rank for score of 55 Mean = 50 Standard deviation = 10 ? Find the percentile rank for score of 55 55 .1915 1) Find z score z score = 55 - 50 10 z score = 5 10 = 0.5 2) Go to z table Problem 4

Find the percentile rank for score of 55 Mean = 50 Standard deviation = 10 ? Find the percentile rank for score of 55 55 .1915 .5 1) Find z score z score = 55 - 50 10 z score = 5 10 = 0.5 2) Go to z table 3) Look at your picture - add .5000 +.1915 = .6915 4) Percentile rank or score of 55 = 69.15% Problem 4

Find the score that is associated Mean = 50 Standard deviation = 10 ? Find the score for z = -2 30 Hint always draw a picture! Find the score that is associated with a z score of -2 raw score = mean + (z score)(standard deviation) Raw score = 50 + (-2)(10) Raw score = 50 + (-20) = 30 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

percentile rank of 77%ile Mean = 50 Standard deviation = 10 ? Find the score for percentile rank of 77%ile .7700 ? 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

percentile rank of 77%ile Mean = 50 Standard deviation = 10 .27 ? Find the score for percentile rank of 77%ile .5 .5 + .27 = .77 .5 .27 .7700 ? 1) Go to z table - find z score for for area .2700 (.7700 - .5000) = .27 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 area = .2704 (closest I could find to .2700) z = 0.74 Problem 5

percentile rank of 77%ile Mean = 50 Standard deviation = 10 .27 ? Find the score for percentile rank of 77%ile .5 x = 57.4 .5 .27 .7700 ? 2) x = mean + (z)(standard deviation) x = 50 + (0.74)(10) x = 57.4 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 Problem 5

percentile rank of 55%ile Mean = 50 Standard deviation = 10 ? Find the score for percentile rank of 55%ile .5500 ? 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

percentile rank of 55%ile Mean = 50 Standard deviation = 10 .05 ? Find the score for percentile rank of 55%ile .5 .5 + .05 = .55 .5 .05 .5500 ? 1) Go to z table - find z score for for area .0500 (.5500 - .5000) = .05 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 area = .0517 (closest I could find to .0500) z = 0.13 Problem 7

percentile rank of 55%ile Mean = 50 Standard deviation = 10 .05 ? Find the score for percentile rank of 55%ile .5 .5 .05 .5500 ? 1) Go to z table - find z score for for area .0500 (.5500 - .5000) = .05 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 area = .0517 (closest I could find to .0500) z = 0.13 Problem 7

percentile rank of 55%ile Mean = 50 Standard deviation = 10 .05 ? Find the score for percentile rank of 55%ile .5 x = 51.3 .5 .05 .5500 ? 1) Go to z table - find z score for for area .0500 (.5500 - .5000) = .0500 area = .0517 (closest I could find to .0500) z = 0.13 2) 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 7

Not included in class lecture 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 Go to table .4500 nearest z = 1.64 x = mean + z σ = 50 + (1.64)(4) = 56.56 .9500 .4500 .5000 Additional practice Problem 8 38 42 46 50 54 56.60 ? 58 62 Not included in class lecture

Not included in class lecture 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 3rd percentile Go to table .4700 nearest z = - 1.88 x = mean + z σ = 2100 + (-1.88)(250) = 1,630 .0300 .4700 Additional practice Problem 9 1,630 ? 2100 Not included in class lecture

Not included in class lecture Normal Distribution has a mean of 195 and standard deviation of 8.5. Determine value for top 1% of hours listened. Go to table .4900 nearest z = 2.33 x = mean + z σ = 195 + (2.33)(8.5) = 214.805 .4900 .5000 .0100 Additional practice Problem 10 195 214.8 ? Not included in class lecture

Not included in class lecture . Find score associated with the 75th percentile 75th percentile Go to table nearest z = .67 .2500 x = mean + z σ = 30 + (.67)(2) = 31.34 .7500 .25 .5000 24 26 28 30 ? 34 36 31.34 Additional practice Problem 11 z = .67 Not included in class lecture

Not included in class lecture . Find the score associated with the 25th percentile 25th percentile Go to table nearest z = -.67 .2500 x = mean + z σ = 30 + (-.67)(2) = 28.66 .2500 .25 .25 24 26 28.66 ? 28 30 34 36 Additional practice Problem 12 z = -.67 Not included in class lecture

. 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) = 33.92 Go to table .4750 nearest z = -1.96 mean + z σ = 30 + (-1.96)(2) = 26.08 .9500 .475 .475 Additional practice Problem 13 26.08 33.92 24 ? 28 32 ? 30 36 Not included in class lecture

. Try this one: Please find the (2) raw scores that border exactly the middle 95% of the curve Mean of 100 and standard deviation of 5 Go to table .4750 nearest z = 1.96 mean + z σ = 100 + (1.96)(5) = 109.80 Go to table .4750 nearest z = -1.96 mean + z σ = 100 + (-1.96)(5) = 90.20 .9500 .475 .475 Additional practice Problem 14 90.2 109.8 85 ? 95 105 ? 100 115 Not included in class lecture

. Try this one: Please find the (2) raw scores that border exactly the middle 99% of the curve Mean of 30 and standard deviation of 2 Go to table .4750 nearest z = 1.96 mean + z σ = 30 + (2.58)(2) = 35.16 Go to table .4750 nearest z = -1.96 mean + z σ = 30 + (-2.58)(2) = 24.84 .9900 .495 .495 Additional practice Problem 15 24.84 ? 35.16 28 32 ? 30 Not included in class lecture

Thank you! See you next time!!