Screen Stage Lecturer’s desk Gallagher Theater Row A Row A Row A Row B 17 16 15 14 13 12 11 10 9 8 7 6 5 4 Row A 3 2 1 Row A Left handed Row B 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row B 4 3 2 1 Row B Row C 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row C 4 3 2 1 Row C Row D 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row D 4 3 2 1 Row D Row E 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row E 4 3 2 1 Row E Row F 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row F 4 3 2 1 Row F Row G 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row G 4 3 2 1 Row G Row H 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row H 4 3 2 1 Row H Row I 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row I 4 3 2 1 Row I Row J 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row J 4 3 2 1 Row J Row K 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row K 4 3 2 1 Row K Row L 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row L 4 3 2 1 Row L Row M 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row M 4 3 2 1 Row M Row N 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row N 4 3 2 1 Row N Row O 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row O 4 3 2 1 Row O Need Labels B5, E1, I16, J17, K8, M4, O1, P16 Row P 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row P 4 3 2 1 Row P Row Q 16 15 14 13 12 11 10 9 8 7 6 5 4 Row Q 3 2 1 Row Q Row R Gallagher Theater 4 3 2 Row R 26Left-Handed Desks A14, B16, B20, C19, D16, D20, E15, E19, F16, F20, G19, H16, H20, I15, J16, J20, K19, L16, L20, M15, M19, N16, P20, Q13, Q16, S4 5 Broken Desks B9, E12, G9, H3, M17 Row S 10 9 8 7 4 3 2 1 Row S
Screen Stage Social Sciences 100 Lecturer’s desk broken desk R/L handed Row A 17 16 15 14 13 12 Row B 27 26 25 24 23 Row B 22 21 20 19 18 17 16 15 14 13 12 11 10 Row C 28 27 26 25 24 23 Row C 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 Row C Row D 30 29 28 27 26 25 24 23 Row D 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 Row D Row E 31 30 29 28 27 26 25 24 23 Row E 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row E Row F 31 30 29 28 27 26 25 24 23 Row F 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row F Row G 31 30 29 28 27 26 25 24 23 Row G 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row G Row H 31 30 29 28 27 26 25 24 23 Row H 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row H Row I 31 30 29 28 27 26 25 24 23 Row I 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row I Row J 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row J Row J 31 30 29 28 27 26 25 24 23 23 Row K 22 13 12 11 10 9 8 7 6 5 2 1 Row K 31 30 29 28 27 26 25 24 21 20 19 18 17 16 15 14 4 3 Row K Row L 31 30 29 28 27 26 25 24 23 Row L 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row L Row M 31 30 29 28 27 26 25 24 23 Row M 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row M Row N 31 30 29 28 27 26 25 24 23 Row N 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row N Row O 31 30 29 28 27 26 25 24 23 Row O 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row O 23 Row P 9 8 7 6 5 4 3 2 1 Row P 31 30 29 28 27 26 25 24 22 21 20 19 18 17 16 15 14 13 12 11 10 Row P Row Q 31 30 29 28 27 26 25 24 23 Row Q 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row Q Row R 31 30 29 28 27 26 25 24 23 Row R 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row R table broken desk 9 8 7 6 5 4 3 2 1 Projection Booth
MGMT 276: Statistical Inference in Management Fall, 2014 Welcome Green sheets
Reminder A note on doodling Talking or whispering to your neighbor can be a problem for us – please consider writing short notes.
We’ll be jumping around some…we will start with chapter 7 Schedule of readings We’ll be jumping around some…we will start with chapter 7 Before our next exam (October 21st) Lind (5 – 11) Chapter 5: Survey of Probability Concepts Chapter 6: Discrete Probability Distributions Chapter 7: Continuous Probability Distributions Chapter 8: Sampling Methods and CLT Chapter 9: Estimation and Confidence Interval Chapter 10: One sample Tests of Hypothesis Chapter 11: Two sample Tests of Hypothesis Plous (10, 11, 12 & 14) Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability and Risk Chapter 14: The Perception of Randomness
Homework due – Thursday (October 9th) On class website: Please print and complete homework worksheet #9 Calculating z-score, raw scores and probabilities using the normal curve
By the end of lecture today 10/7/14 Use this as your study guide By the end of lecture today 10/7/14 Counting ‘standard deviationses’ – z scores Connecting raw scores, z scores and probability Connecting probability, proportion and area of curve Percentiles
Study Type 5: Correlation Study Type 1: Confidence Intervals Connecting intentions of studies with Experimental Methodologies Appropriate statistical analyses Appropriate graphs Study Type 5: Correlation Study Type 1: Confidence Intervals Study Type 6: Regression Study Type 2: t-test Study Type 3: One-way ANOVA Study Type 7: Chi-squared Study Type 4: Two-way ANOVA Remember when p < 0.05 we say: - results are statistically significant there is “real” difference (not just due to chance)
Class standing If there is a difference that was statistically significant, then both of these answers will be “yes” 4 Ordinal Quasi # Bags Sold # of bags of peanuts sold Ratio Between One-way ANOVA Fr So Jr Sr Class Standing
Homework Review Average # of bags of peanuts sold Frequency 95% Confidence Interval Average # of bags of peanuts sold
Homework Review Type of Diet 2 Weight Loss Nominal True Experiment Ratio Between Regular New t-test Type of Diet
Homework Review Type of Diet Male 2 Weight Loss Gender 2 Female Mixed Regular New Between Two-way ANOVA Type of Diet
Homework Review Distance Strong, positive +1.0 Time Correlation Time
Examples of the seven prototypical designs Homework Worksheet Examples of the seven prototypical designs
You already know this by heart 1 sd above and below mean 68% 2 sd above and below mean 95% 3 sd above and below mean 99.7% 68% Confidence interval 95% Confidence interval 99% Confidence interval You already know this by heart
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”
z scores 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 Review
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
Scores, standard deviations, and probabilities What is total percent under curve? What proportion of curve is above the mean? .50 100% Given any of these values (score, probability of occurrence, or distance from the mean) and you can figure out the other two.
Scores, standard deviations, and probabilities What percent of curve is below a score of 50? What score is associated with 50th percentile? 50% median Mean = 50 S = 10 (Note S = standard deviation)
Raw scores, z scores & probabilities Distance from the mean (z scores) convert convert Raw Scores (actual data) Proportion of curve (area from mean) 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” Raw Scores (actual data) Proportion of curve (area from mean) Distance from the mean (z scores) convert convert
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 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 60 50 10 z = 1 Mean = 50 Standard deviation = 10
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 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 30 50 10 z = - 2 Mean = 50 Standard deviation = 10
Find z score for raw score of 70 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 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 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 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 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) 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
Raw scores, z scores & probabilities Have z Find area Have z Find raw score Z Scores z table Formula Have area Find z Area & Probability Have raw score Find z Raw Scores
Ties together z score with Draw picture of what you are looking for... Find z score (using formula)... Look up proportions on table Ties together z score with probability proportion percent area under the curve 68% 34% 34%
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 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 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 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 = 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 Let’s do some problems 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 Let’s do some problems .4772 .4772 95.44% z score of 2 = area of .4772 Find the area under the curve that falls between 30 and 70 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) .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 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) .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 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) 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 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) 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 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) 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 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) .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 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 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 Problem 5
percentile rank of 77%ile Mean = 50 Standard deviation = 10 ? Find the score for percentile rank of 77%ile 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) .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 6
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 6
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 6
percentile rank of 55%ile Mean = 50 Standard deviation = 10 ? Find the score for percentile rank of 55%ile 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) .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 7
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
? .9500 .4500 .5000 x = mean + z σ = 50 + (1.64)(4) = 56.56 Problem 8 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 Problem 8 38 42 46 50 54 56.60 ? 58 62
? .4700 .0300 x = mean + z σ = 2100 + (-1.88)(250) = 1,630 Problem 9 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 Problem 9 1,630 ? 2100
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 Problem 10 195 214.8 ?
? .7500 .25 .5000 Find score associated with the 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 z = .67 Problem 11
? .2500 .25 .25 Find the score associated with the 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 z = -.67 Problem 12
Mean of 30 and standard deviation of 2 . 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 Problem 13 26.08 33.92 24 ? 28 30 32 ? 36
Mean of 100 and standard deviation of 5 . 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 Problem 14 90.2 109.8 85 ? 95 100 105 ? 115
Mean of 30 and standard deviation of 2 . 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 Problem 13 24.84 ? 35.16 28 32 ? 30
Remember Confidence Intervals . Remember Confidence Intervals 95% Confidence Interval: We can be 95% confident that the estimated score really does fall between these two scores Please find the raw scores that border the middle 95% of the curve 99% Confidence Interval: We can be 99% confident that the estimated score really does fall between these two scores Please find the raw scores that border the middle 99% of the curve
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
Raw scores, z scores & probabilities Notice: 3 types of numbers raw scores z scores probabilities Mean = 50 Standard deviation = 10 z = -2 z = +2 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
Thank you! See you next time!!