Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Fall 2015 Room 150 Harvill Building 10: :50 Mondays, Wednesdays & Fridays.

Homework Assignment Go to D2L - Click on “Interactive Online Homework Assignments” Assignment 8 HW8-Normal Curve, z scores and probabilities Due: Monday, October 2 nd

Hand out z tables

Everyone will want to be enrolled in one of the lab sessions Labs continue this week, Project 1

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By the end of lecture today 9/30/15 Counting ‘standard deviationses’ – z scores Connecting raw scores, z scores and probability Connecting probability, proportion and area of curve Percentiles

Before next exam (October 16th) Please read chapters 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 Schedule of readings

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%

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

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

Scores, standard deviations, and probabilities Mean = 50 Standard deviation = 10 What percent of curve is below a score of 50? 50% What score is associated with 50 th percentile? median

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

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

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 Mean = 50 Standard deviation = z = 1 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) 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 z = - 2 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) Find z score for raw score of 30 Mean = 50 Standard deviation = 10

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 – = = 2 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) Raw scores, z scores & probabilities Find z score for raw score of 70 Mean = 50 Standard deviation = 10

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

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 – = = - 3 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) Raw scores, z scores & probabilities Find z score for raw score of 20 Mean = 50 Standard deviation = 10

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

Hand out z tables

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%

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

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

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

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

Let’s do some problems z score = raw score - mean standard deviation z score = z score = 20 = z table z score of 2 = area of.4772 Hint always draw a picture! Find the area under the curve that falls between 70 and 50 Mean = 50 Standard deviation = %

Let’s do some problems =.9544 Hint always draw a picture! Find the area under the curve that falls between 30 and 70 Mean = 50 Standard deviation = 10 z score of 2 = area of % 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)

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

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

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

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

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

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

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

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