Hand out z tables

Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Spring 2016 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays Welcome

Before next exam (March 4th) Schedule of readings Before next exam (March 4th) 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 2/17/16 Counting ‘standard deviationses’ – z scores Connecting raw scores, z scores and probability Connecting probability, proportion and area of curve Percentiles

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

Why do we concern ourselves about research? – Five objectives 1. To explore potential phenomena explore whether phenomenon is present explore a phenomenon with a fresh take generate new ideas and discover relationships . e.g. business socks – how might you explore this? e.g. dubstep – what is it – how might we reach out to this community?

Why do we concern ourselves about research? – Five objectives Yo, you wanna meet up, have a seizure whilst listening to the noise of a wampwampwampwamp wampwampwampwampwamp until your ears bleed?" Why do we concern ourselves about research? – Five objectives 1. To explore potential phenomena explore whether phenomenon is present explore a phenomenon with a fresh take generate new ideas and discover relationships e.g. business socks – how might you market this? e.g. dubstep – what is it – how might we reach out to this community?

Why do we concern ourselves about research? – Five objectives 2. To describe phenomena build a vocabulary of constructs and make distinctions between similar constructs (how is dubstep different from techno or house?) cluster similar characteristics into related constructs . - Types of management style - Strategies for quality control

Why do we concern ourselves about research? – Five objectives 3. To explain and model phenomena explanation: find cause and effect relationships propose mechanisms that determine outcomes show how and why a phenomenon operates as it does

Why do we concern ourselves about research? – Five objectives 4. To predict future behavior what characteristics are likely to result in worker productivity, consumer behavior, etc... explanations can help with predictions, but being able to predict an outcome doesn’t necessarily provide a good explanation

Why do we concern ourselves about research? – Five objectives 5. To influence behavior how can we use what we know about human behavior to affect how people around us react and behave (and do what we want) increase number of volunteers supervisors increasing probability of happy employees parent increasing probability of child taking out the trash to advance better practices

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”

Raw scores, z scores & probabilities Please note spatially where 1 standard deviation falls on the curve 68% 95% 99.7%

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

Price per Movie Package Movie Packages We sampled 100 movie theaters (Two tickets, large popcorn and 2 drinks) What is the most common “deviation score”? 12 Deviation = 0 10 Frequency 8 6 4 2 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 Price per Movie Package This distance is about 3 - 4 points on number line What’s the ‘typical’ or standard deviation? Mean = $37 Range = $27 - $47 Standard Deviation = 3.5

Amount of Bonuses (based on commission) We sampled 100 retail workers 68% 95% 99.7% What’s the ‘typical’ or standard deviation? This distance is about 10 points on number line What’s the largest possible deviations? Mean = $50 Range = $25 - $75 $75 – $50= $25 $25 – $50= -$25 Standard Deviation = 10

Pounds of pressure to break casing on an insulator (We applied pressure until the insulator casing broke) What’s the ‘typical’ or standard deviation? This distance is about 200 points on number line Mean = 1700 pounds Range = 1200 – 2100 What’s the largest possible deviation? 2100-1700 = 400 Standard Deviation = 200 1200-1700 = -500

What’s the ‘typical’ or standard deviation? Waiting time for service at bank We sampled 100 banks (From time entering line to time reaching teller) What’s the ‘typical’ or standard deviation? This distance is about .2 -.3 points on number line Mean = 3 minutes Range = 2.2- 3.8 What’s the largest possible deviation? 3.8-3.0=.8 2.2-3.0=-.8 Standard Deviation = 0.30

Number correct on exam (counted number of correct on 100 point test) Mean = 50 Standard Deviation = 10

Monthly electric bills for 50 apartments (amount of dollars charged for the month) Mean = $150 Range = 95 - 205 The best estimate of the population standard deviation is a. $150 b. $25 c. $50 d. $63 Standard Deviation = 25

Scores on Art History Exam Quiz: Just for fun Scores on Art History Exam One way to estimate standard deviation* σ≈ range / 5 Mean = 50 Range = 25 - 70 Range 70 – 25 = 45 45 / 5 = 9 The best estimate of the population standard deviation is a. 50 b. 25 c. 10 d. .5 Standard Deviation = 10

Scores, standard deviations, and probabilities The normal curve always has the same shape. They differ only by having different means and standard deviation

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 Review

Find z score for raw score of 30 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 30 50 10 z = - 2 Mean = 50 Standard deviation = 10

Find z score for raw score of 70 Raw scores, z scores & probabilities If we go up to score of 70 we are going up 2.0 standard deviations Then, z score = +2.0 z score = raw score - mean standard deviation z score = 70 – 50 . 10 = 20. 10 = 2 Mean = 50 Standard deviation = 10

Find z score for raw score of 80 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 80 50 10 z = 3 Mean = 50 Standard deviation = 10

Raw scores, z scores & probabilities Find z score for raw score of 20 Raw scores, z scores & probabilities If we go down to score of 20 we are going down 3.0 standard deviations Then, z score = -3.0 z score = raw score - mean standard deviation z score = 20 – 50 10 = - 30 . 10 = - 3 Mean = 50 Standard deviation = 10

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

Thank you! See you next time!!