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

BNAD 276: Statistical Inference in Management Spring 2016 Green sheets

By the end of lecture today 2/11/16 Characteristics of a distribution Central Tendency Dispersion The Normal Curve Raw scores, probability and z scores

Schedule of readings Before next exam: February 18 th Please read Chapters in OpenStax Supplemental reading (Appendix D) Supplemental reading (Appendix E) Supplemental reading (Appendix F) Please read Chapters 1, 5, 6 and 13 in Plous Chapter 1: Selective Perception Chapter 5: Plasticity Chapter 6: Effects of Question Wording and Framing Chapter 13: Anchoring and Adjustment

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 Actually Actually To be exactly 95% we will use z = 1.96

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%

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

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 Find z score for raw score of 60

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

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 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, 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 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, z scores & probabilities Find z score for raw score of 20 Mean = 50 Standard deviation = 10

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

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 %

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 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

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 Problem 3

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 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 = 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 = 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 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 ? ? 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 ? ? 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 7

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 7

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) = Additional practice Problem 8

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 Additional practice Problem 9

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) = Additional practice Problem 10

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

. 25 th percentile Go to table.2500 nearest z = -.67 x = mean + z σ = 30 + (-.67)(2) = z = -.67 Additional practice Problem 12

. 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) = Additional practice Problem 13

. 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 = (1.96)(5) = Go to table.4750 nearest z = mean + z σ = (-1.96)(5) = Additional practice Problem 14

. 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) = Go to table.4750 nearest z = mean + z σ = 30 + (-2.58)(2) = Additional practice Problem 15

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

Today I want to present some “typical designs”. We will spend the next couple weeks filling in the details. Connecting intentions of studies with Experimental Methodologies Appropriate statistical analyses Appropriate graphs We’ll come back to these distinctions over and over again, and build on them for the rest of the semester. Let’s get this overview well! Not worry about calculation details for now

Study Type 5: Correlation Study Type 2: t-test Study Type 3: One-way Analysis of Variance (ANOVA) Study Type 4: Two-way Analysis of Variance (ANOVA) Study Type 1: Confidence Intervals Study Type 6: Simple and Multiple regression Study Type 7: Chi Square Create example of each type Identify IV (one or two) Identify DV (one or two) Draw possible graph for each Homework Assignment Think about this as we work through each type of study We’ll come back to these distinctions over and over again, and build on them for the rest of the class. Let’s get this overview well! Not worry about calculation details for now

On average newborns weigh 7 pounds, and are 20 inches long. My sister just had a baby - guess how much it weighs? Point estimate versus confidence interval: Guessing a single number versus a range of numbers Makes sense, right?!? Guess the mean. On average you would be right most often if you always guessed the mean Study Type 1: Confidence Intervals What if you really needed to be right?!!? You could guess a range with smallest and largest possible scores. (how wide a range to be completely sure? Confidence interval: Guessing a range (max and min) and assigning a level of confidence that the score falls in that range Remember, this is just introduction to the idea Not worry about calculation details for now, we will get to those soon

95% Confidence Interval: We can be 95% confident that our population mean falls between these two scores 99% Confidence Interval: We can be 99% confident that our population mean falls between these two scores Which has a wider interval relative to raw scores 95% or 99%? 100% Confidence Interval: We can be 100% confident that our population mean falls between these two scores (Guess absurdly large and small values) Confidence Intervals: A range of values that, with a known degree of certainty, includes an unknown population characteristic, such as a population mean Study Type 1: Confidence Intervals Remember, this is just introduction to the idea Not worry about calculation details for now, we will get to those soon

This sample of 1000 flights, the mean number of empty seats is 12. This sample of 10,000 newborns a mean weight is 7 pounds. What do you think the minimum and maximum weights would be to capture 95% of all newborns? Confidence Intervals: A range of values that, with a known degree of certainty, includes an unknown population characteristic, such as a population mean Study Type 1: Confidence Intervals Remember, this is just introduction to the idea Not worry about calculation details for now, we will get to those soon You can use a mean of a sample to guess the mean of population mean of a smaller sample most likely score for an individual What do you think the minimum and maximum number of empty seats are likely to be in the flights today with a 95% level of certainty? This sample of 500 households produced a mean income of $35,000 a year. What do you think the minimum and maximum income levels are so that we are 95% confident that we captured Mabel’s?