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Last Time • Notions of Center – Expected Value (probability distributions) • Interpretation • Binomial Expected Value • Properties – Median • Comparison to mean Administrative Matters Midterm I Results: • Check Scores – Points taken off in circles – Total points for each problem in [purple] – Total for each page, lower right corner – Total those to score on front – Check Blackboard entry Administrative Matters Midterm I Results: Interpretation of scores: Administrative Matters Midterm I Results: Interpretation of scores: 84 – 100 Very Pleased Administrative Matters Midterm I Results: Interpretation of scores: 84 – 100 Very Pleased (No letter grade yet, since to early, expect many to change a lot) Administrative Matters Midterm I Results: Interpretation of scores: 84 – 100 Very Pleased (Good to see many here, including #(100s) = 12) Administrative Matters Midterm I Results: Interpretation of scores: 84 – 100 Very Pleased 65 – 83 OK Administrative Matters Midterm I Results: Interpretation of scores: 84 – 100 Very Pleased 65 – 83 OK Administrative Matters Midterm I Results: Interpretation of scores: 84 – 100 Very Pleased 65 – 83 OK 0 – 64 Reco: Drop Course Administrative Matters Midterm I Results: Interpretation of scores: 84 – 100 Very Pleased 65 – 83 OK 0 – 64 Reco: Drop Course Don’t want to? Let’s talk… Reading In Textbook Approximate Reading for Today’s Material: Pages 55-68, 319-326 Approximate Reading for Next Class: Pages 59-62, 279-285, 62-64, 337-344 Big Picture • Margin of Error • Choose Sample Size Need better prob tools Start with visualizing probability distributions Big Picture • Margin of Error • Choose Sample Size Need better prob tools Start with visualizing probability distributions, Next exploit constant shape property of Bi Big Picture Start with visualizing probability distributions, Next exploit constant shape property of Binom’l Big Picture Start with visualizing probability distributions, Next exploit constant shape property of Binom’l Centerpoint feels p Big Picture Start with visualizing probability distributions, Next exploit constant shape property of Binom’l Centerpoint feels p Spread feels n Big Picture Start with visualizing probability distributions, Next exploit constant shape property of Binom’l Centerpoint feels p Spread feels n Now quantify these ideas, to put them to work Notions of Center Will later study “notions of spread” Notions of Center Textbook: Sections 4.4 and 1.2 Recall parallel development: (a) Probability Distributions (b) Lists of Numbers Study 1st, since easier Notions of Center Caution about mean: Works well for ~symmetric distributions E.g. Buffalo Snowfalls Manual bins Small Binwidth 8 7 6 3 2 1 130 125 120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 0 30 Using AVERAGE) 4 25 (from Excel, 5 20 Mean = 80.3 Notions of Center Caution about mean: Works well for ~symmetric distributions E.g. Buffalo Snowfalls Manual bins Small Binwidth 8 7 6 3 2 1 130 125 120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 0 30 Notion of “Center” 4 25 Visually sensible 5 20 Mean = 80.3 Notions of Center Caution about mean: But poorly for asymmetric distributions Notions of Center Caution about mean: But poorly for asymmetric distributions E.g. British Suicides Data • Time (in days) to suicide attempt • Of Suicide Patients • After Initial Treatment Notions of Center Caution about mean: But poorly for asymmetric distributions E.g. British Suicides Data Analyzed in: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg6.xls Notions of Center Caution about mean: But poorly for asymmetric distributions E.g. British Suicides Data British Suicides Clearly not mound shaped Very asymmetric 20 18 16 14 12 8 6 4 2 0 0 40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 Called “right skewed” 10 Notions of Center Caution about mean: But poorly for asymmetric distributions E.g. British Suicides Data British Suicides 20 Mean = 122.3 18 Sensible as “center”?? 14 16 12 10 Too Small… 8 6 4 2 0 0 40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 %(data ≥) = 30.2% Notions of Center Perhaps better notion of “center”: • Take center to be point in middle • I.e. have 50% of data smaller • And 50% of data larger This is called the “median” Notions of Center Median: = Value in middle (of sorted list) Unsorted E.g: Sorted E.g: 3 0 1 1 27 2 2 3 0 27 Notions of Center Median: = Value in middle (of sorted list) Unsorted E.g: Sorted E.g: 3 0 1 1 27 2 2 3 0 27 One in middle??? NO, must sort Notions of Center Median: = Value in middle (of sorted list) Unsorted E.g: Sorted E.g: 3 0 1 1 27 2 2 3 0 27 Sensible version of “middle” Notions of Center What about ties? Tie for point in middle Sorted E.g: 0 1 2 3 Notions of Center What about ties? Sorted E.g: 0 Tie for point in 1 middle 2 3 Break by taking average (of two tied values): e.g. Median = 1.5 Notions of Center Median: = Value in middle (of sorted list) Unsorted E.g: Sorted E.g: 3 0 1 1 27 2 2 3 0 27 EXCEL: use function “MEDIAN” Notions of Center EXCEL: use function “MEDIAN” Very similar to other functions E.g. see: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg6.xls Notions of Center E.g. Buffalo Snowfalls Mean = 80.3 Manual bins Small Binwidth 8 Median = 79.6 7 6 3 2 1 130 125 120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 0 30 symmetry) 4 25 (expected from 5 20 Very similar Notions of Center E.g. British Suicides Data Mean = 122.3 Median = 77.5 British Suicides 20 Substantially different But which is better? 18 16 14 12 10 8 6 4 2 0 0 40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 Goal 1: ½ - ½ middle Notions of Center E.g. British Suicides Data Mean = 122.3 Median = 77.5 British Suicides 20 Substantially different But which is better? 18 16 14 12 10 Goal 2: long run average 8 6 4 2 0 0 40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 Goal 1: ½ - ½ middle Notions of Center HW: 1.63 a (median only), c 1.65 (hint: use histogram) Research Corner Recall Hidalgo StampData & Movie over binwidth Main point: Binwidth drives histogram performance Research Corner Less known fact: Bin location also has Serious effect (even for fixed width) Research Corner How many bumps? ~2? Research Corner How many bumps? ~3? Research Corner How many bumps? ~7? Research Corner Compare with “smoothed version” called “Kernel Density Estimate” Peaks appear: when entirely in a bin Research Corner Compare with “smoothed version” called “Kernel Density Estimate” Peaks disappear: when split between two bins bin Research Corner Question: What is a Kernel Density Estimate? E.g. Chondrite Data • Type of Meteor (never part of planet) Research Corner Question: What is a Kernel Density Estimate? E.g. Chondrite Data • Type of Meteor (never part of planet) • From how many sources do they come? Research Corner Question: What is a Kernel Density Estimate? E.g. Chondrite Data • Type of Meteor (never part of planet) • From how many sources do they come? (Current Issue: Meteors from Mars show life?) Research Corner Question: What is a Kernel Density Estimate? E.g. Chondrite Data • Type of Meteor (never part of planet) • From how many sources do they come? Research Corner Question: What is a Kernel Density Estimate? E.g. Chondrite Data • Type of Meteor (never part of planet) • From how many sources do they come? • Data Set 22 Chondrites Research Corner Question: What is a Kernel Density Estimate? E.g. Chondrite Data • Type of Meteor (never part of planet) • From how many sources do they come? • Data Set 22 Chondrites (Histograms as slippery) Research Corner Question: What is a Kernel Density Estimate? E.g. Chondrite Data • Type of Meteor (never part of planet) • From how many sources do they come? • Data Set • Interesting measurement: 22 Chondrites % Silica Research Corner Question: What is a Kernel Density Estimate? Chondrite Data % Silica Research Corner Question: What is a Kernel Density Estimate? Chondrite Data % Silica Research Corner Question: What is a Kernel Density Estimate? Chondrite Data % Silica Area = 1/n for each data point Research Corner Question: What is a Kernel Density Estimate? Chondrite Data % Silica Area = 1/n for each data point Sum for Total Area = 1 Research Corner Question: What is a Kernel Density Estimate? Smooth Histogram Research Corner Question: What is a Kernel Density Estimate? Smooth Histogram (Areas correspond to frequencies) Research Corner Question: What is a Kernel Density Estimate? Smooth Histogram Shows 3 clear bumps Research Corner Question: What is a Kernel Density Estimate? Smooth Histogram Shows 3 clear bumps Suggests 3 sources of chondrites Research Corner Kernel Density Estimate For Hidalgo Stamps Research Corner Kernel Density Estimate For Hidalgo Stamps Kernel width is crucial Research Corner Kernel Density Estimate For Hidalgo Stamps Kernel width is crucial Deep question: How to choose? Notions of Center Another view of mean vs. median: Notions of Center Another view of mean vs. median: Use e-textbook Applet: “Mean and Median” Notions of Center Another view of mean vs. median: Use e-textbook Applet: “Mean and Median” Interactive Construction of Toy Examples Mean and Median Applet On Stats Portal: http://courses.bfwpub.com/ips6e • Login • Resources • Student Resources – Statistical Applets • Mean and Median Mean and Median Applet Toy Example 1: Understand Applet • Click below line to add data pt. Mean and Median Applet Toy Example 1: Understand Applet • Click below line to add data pt. • Add another Mean and Median Applet Toy Example 1: Understand Applet • Click below line to add data pt. • Add another • And one more Mean and Median Applet Toy Example 1: Understand Applet • Show mean (recall balance point) in green Mean and Median Applet Toy Example 1: Understand Applet • Show median (recall ½ -way point) in red Mean and Median Applet Toy Example 1: Understand Applet • Big difference, since skewed data set Mean and Median Applet Toy Example 2: Effect of Single Outlier Mean and Median Applet Toy Example 2: Effect of Single Outlier • Note mean outside range of other data points Mean and Median Applet Toy Example 2: Effect of Single Outlier • Note mean outside range of other data points • While median is inside Mean and Median Applet Toy Example 2: Effect of Single Outlier • Fun to “grab” outlier and move around Mean and Median Applet Toy Example 2: Effect of Single Outlier • Fun to “grab” outlier and move around Mean and Median Applet Toy Example 2: Effect of Single Outlier • Fun to “grab” outlier and move around • Median stable • Mean “feels” outlier Mean and Median Applet Robust Statistics: • Mean is sensitive to outliers • Median is not sensitive Called a “robust notion of center” • Studied as part of statistical research • There are many others (beyond scope of this course) Mean and Median Applet Toy Example 3: Bimodal distribution • Two lumps of data Mean and Median Applet Toy Example 3: Bimodal distribution • Two lumps of data • Where is center of population??? Mean and Median Applet Toy Example 3: Bimodal distribution • Two lumps of data • Where is center of population??? • Median? Mean and Median Applet Toy Example 3: Bimodal distribution • Two lumps of data • Where is center of population??? • Median? (not compelling) Mean and Median Applet Toy Example 3: Bimodal distribution • Two lumps of data • Where is center of population??? • Median? (not compelling) • Mean? Mean and Median Applet Toy Example 3: Bimodal distribution • Two lumps of data • Where is center of population??? • Median? (not compelling) • Mean? (more sensible) Mean and Median Applet Toy Example 3: Bimodal distribution • Now add one more data point Mean and Median Applet Toy Example 3: Bimodal distribution • Now add one more data point • Small change in mean Mean and Median Applet Toy Example 3: Bimodal distribution • Now add one more data point • Small change in mean • Big change in median Mean and Median Applet Toy Example 3: Bimodal distribution • Now add one more data point • Small change in mean • Big change in median • Note tie color: yellow = red + green Mean and Median Applet Toy Example 3: Bimodal distribution • Now add one more data point Mean and Median Applet Toy Example 3: Bimodal distribution • Now add one more data point • Small change in mean • Big change in median Mean and Median Applet Mean vs. Median: • Which is “better”? Mean and Median Applet Mean vs. Median: • Which is “better”? • Not comparable • Mean better sometimes • Median better other times Mean and Median Applet Mean vs. Median: • Which is “better”? • Not comparable • Mean better sometimes • Median better other times • Depends on Context (specific case) Mean and Median Applet Mean vs. Median: • Which is “better”? • Not comparable • Mean better sometimes • Median better other times • Depends on Context (specific case) • Best you can do: Understand issues, and make choice Mean and Median Applet HW: 1.65 1.67 1.69 And now for something completely different More Lateral Thinking Puzzles: And now for something completely different More Lateral Thinking Puzzles: cycle cycle cycle And now for something completely different More Lateral Thinking Puzzles: cycle cycle cycle tricycle And now for something completely different More Lateral Thinking Puzzles: 0 ________ M. D. Ph. D. And now for something completely different More Lateral Thinking Puzzles: 0 ________ M. D. Ph. D. Two Degrees Below Zero And now for something completely different More Lateral Thinking Puzzles: knee ______________ light And now for something completely different More Lateral Thinking Puzzles: knee ______________ light neon light And now for something completely different More Lateral Thinking Puzzles: ground _______________________ feet feet feet feet feet feet And now for something completely different More Lateral Thinking Puzzles: ground _______________________ feet feet feet feet feet feet six feet underground Big Picture • Margin of Error • Choose Sample Size Need better prob tools Start with visualizing probability distributions Big Picture Start with visualizing probability distributions, Next exploit constant shape property of Binom’l Centerpoint feels p Spread feels n Notions of Spread Textbook: Sections 4.4 and 1.2 Recall parallel development: (a) Probability Distributions (b) Lists of Numbers Study both together Notions of Spread Toy Example: List 1 = -1, 0, 1 List 2 = -10, 0, 10 Notions of Spread Toy Example: • List 1 = -1, 0, 1 List 2 = -10, 0, 10 Both have mean 0 Notions of Spread Toy Example: List 1 = -1, 0, 1 List 2 = -10, 0, 10 • Both have mean 0 • But List 2 is far more spread Notions of Spread Toy Example: List 1 = -1, 0, 1 List 2 = -10, 0, 10 • Both have mean 0 • But List 2 is far more spread • How to measure this? Notions of Spread Approach: Study deviations Notions of Spread Approach: Study deviations = distances to center Notions of Spread Approach: Study deviations = distances to center (a) Probability Distributions Based on Random Variable X Notions of Spread Approach: Study deviations = distances to center (a) Probability Distributions Based on Random Variable X Deviation = X – EX Notions of Spread Approach: Study deviations = distances to center (a) Probability Distributions Based on Random Variable X Deviation = X – EX Recall Center of X Distribution Notions of Spread Approach: Study deviations = distances to center (a) Probability Distributions Deviation = X – EX Notions of Spread Approach: Study deviations = distances to center (a) Probability Distributions Deviation = X – EX (b) Lists of Numbers Notions of Spread Approach: Study deviations = distances to center (a) Probability Distributions Deviation = X – EX (b) Lists of Numbers Based on x1 , x2 ,, xn . Notions of Spread Approach: Study deviations = distances to center (a) Probability Distributions Deviation = X – EX (b) Lists of Numbers Based on x1 , x2 ,, xn . Deviations are: x1 x , x2 x ,, x xn . Notions of Spread Approach: Study deviations = distances to center (a) Probability Distributions Deviation = X – EX (b) Lists of Numbers Recall: Average, i.e. center of list Based on x1 , x2 ,, xn . Deviations are: x1 x , x2 x ,, x xn . Notions of Spread How to summarize deviations: Mean Absolute Deviation Notions of Spread How to summarize deviations: Mean Absolute Deviation (a) Probability Distributions E |X – EX| Notions of Spread How to summarize deviations: Mean Absolute Deviation (a) Probability Distributions E |X – EX| (# spaces to center) Notions of Spread How to summarize deviations: Mean Absolute Deviation (a) Probability Distributions E |X – EX| (make positive, thus distance to center) Notions of Spread How to summarize deviations: Mean Absolute Deviation (a) Probability Distributions E |X – EX| (Summarize as average distance) Notions of Spread How to summarize deviations: Mean Absolute Deviation (a) Probability Distributions E |X – EX| (b) Lists of Numbers n 1 n x x i 1 i . Notions of Spread How to summarize deviations: Mean Absolute Deviation (a) Probability Distributions E |X – EX| (b) Lists of Numbers n 1 n x x i 1 i (# spaces to center) Notions of Spread How to summarize deviations: Mean Absolute Deviation (a) Probability Distributions E |X – EX| (b) Lists of Numbers n 1 n x x i 1 i (make positive, thus distance to center) Notions of Spread How to summarize deviations: Mean Absolute Deviation (a) Probability Distributions E |X – EX| (b) Lists of Numbers n 1 n x x i 1 i (Summarize as average distance) Notions of Spread How to summarize deviations: Mean Absolute Deviation Problems: Notions of Spread How to summarize deviations: Mean Absolute Deviation Problems: • Hard to find distribution (i.e. measure error) Notions of Spread How to summarize deviations: Mean Absolute Deviation Problems: • Hard to find distribution (i.e. measure error) • No calculus for minimizing (later in course) Notions of Spread How to summarize deviations: Mean Absolute Deviation Problems: • Hard to find distribution (i.e. measure error) • No calculus for minimizing (later in course) • No “shortcut formula” (later) Notions of Spread How to summarize deviations: Mean Absolute Deviation Problems: • Hard to find distribution (i.e. measure error) • No calculus for minimizing (later in course) • No “shortcut formula” (later) Hence will not study further here Notions of Spread How to summarize deviations: Mean Absolute Deviation Problems: • Hard to find distribution (i.e. measure error) • No calculus for minimizing (later in course) • No “shortcut formula” (later) Hence will not study further here (but is important in more advanced courses) Notions of Spread More common summary of deviations: Standard Deviation Notions of Spread More common summary of deviations: Standard Deviation (a) Probability Distributions E X EX 2 Notions of Spread More common summary of deviations: Standard Deviation (a) Probability Distributions E X EX 2 Greek “sigma” (lower case) Notions of Spread More common summary of deviations: Standard Deviation (a) Probability Distributions E X EX 2 Essentially root of average square deviations Notions of Spread More common summary of deviations: Standard Deviation (a) Probability Distributions E X EX 2 Square root makes units same as X (e.g. ft, not ft2 = sq. ft) Notions of Spread More common summary of deviations: Standard Deviation (a) Probability Distributions E X EX 2 (b) Lists of Numbers s n 1 n 1 x x i 1 2 i Notions of Spread More common summary of deviations: Standard Deviation (a) Probability Distributions E X EX 2 (b) Lists of Numbers s n 1 n 1 x x i 1 2 i Again root of average square deviations Notions of Spread More common summary of deviations: Standard Deviation (a) Probability Distributions E X EX 2 (b) Lists of Numbers s n 1 n 1 x x i 1 2 i Again square root makes units same as X Notions of Spread More common summary of deviations: Standard Deviation (a) Probability Distributions E X EX 2 (b) Lists of Numbers n s Reason for 1 n 1 instead of 1 n 1 1 n x x i 1 2 i discussed later Notions of Spread E.g. 1: for list 10, 11, 12 Notions of Spread E.g. 1: for list 10, 11, 12, x 11 Notions of Spread E.g. 1: for list 10, 11, 12, deviations are: -1, 0, 1 x 11 Notions of Spread E.g. 1: x 11 for list 10, 11, 12, deviations are: -1, 0, 1 s 1 2 1 2 0 1 1 2 2 Notions of Spread E.g. 1: deviations are: -1, 0, 1 • x 11 for list 10, 11, 12, Same as for list -1, 0, 1 s 1 2 1 2 0 1 1 2 2 Notions of Spread E.g. 1: x 11 for list 10, 11, 12, deviations are: -1, 0, 1 s • Same as for list -1, 0, 1 • s does not feel location 1 2 1 2 0 1 1 2 2 Notions of Spread E.g. 1: x 11 for list 10, 11, 12, deviations are: -1, 0, 1 s • Same as for list -1, 0, 1 • s does not feel location • Can shift all numbers in list 1 2 1 2 0 1 1 2 2 Notions of Spread E.g. 1: x 11 for list 10, 11, 12, deviations are: -1, 0, 1 s 1 2 1 2 • Same as for list -1, 0, 1 • s does not feel location • Can shift all numbers in list, and 0 1 1 not change spread 2 2 Notions of Spread E.g. 1: for list 10, 11, 12, E.g. 2: for list 0, 11, 22 Notions of Spread E.g. 1: for list 10, 11, 12, E.g. 2: for list 0, 11, 22 x 11 Notions of Spread E.g. 1: for list 10, 11, 12, E.g. 2: for list 0, 11, 22 Dev’ns are: -11, 0, 11 x 11 Notions of Spread E.g. 1: for list 10, 11, 12, E.g. 2: for list x 11 0, 11, 22 Dev’ns are: -11, 0, 11 s 1 2 11 2 0 11 11 2 2 Notions of Spread E.g. 1: for list 10, 11, 12, E.g. 2: for list Dev’ns are: -11, 0, 11 • x 11 0, 11, 22 s 1 2 11 2 s for E.g. 2 is 11 times larger 0 11 11 2 2 Notions of Spread E.g. 1: for list 10, 11, 12, E.g. 2: for list x 11 0, 11, 22 Dev’ns are: -11, 0, 11 s 1 2 11 2 • s for E.g. 2 is 11 times larger • Since #s are 11 times more spread (on number line) 0 11 11 2 2 Notions of Spread E.g. 3: x f(x) 0 0.1 1 0.2 4 0.7 Notions of Spread E.g. 3: x f(x) 0 0.1 1 0.2 Expected Value (center) 4 0.7 Notions of Spread E.g. 3: x f(x) 0 0.1 1 0.2 4 0.7 Expected Value (center) EX = (0.1)0 + (0.2)1 + (0.7)4 = 3 Notions of Spread E.g. 3: x f(x) x-EX 0 0.1 -3 1 0.2 -2 4 0.7 1 EX = (0.1)0 + (0.2)1 + (0.7)4 = 3 Notions of Spread E.g. 3: x f(x) x-EX (x-EX)2 0 0.1 -3 9 1 0.2 -2 4 4 0.7 1 1 Notions of Spread E.g. 3: x f(x) x-EX (x-EX)2 0 0.1 -3 9 1 0.2 -2 4 4 0.7 1 1 So E(X – EX)2 = (0.1)9 + (0.2)4 + (0.7)1 Notions of Spread E.g. 3: x f(x) x-EX (x-EX)2 0 0.1 -3 9 1 0.2 -2 4 4 0.7 1 1 So E(X – EX)2 = (0.1)9 + (0.2)4 + (0.7)1 = 2.4 Notions of Spread E.g. 3: x f(x) x-EX (x-EX)2 0 0.1 -3 9 1 0.2 -2 4 4 0.7 1 1 So E(X – EX)2 = (0.1)9 + (0.2)4 + (0.7)1 = 2.4 So σ(X) = 2.4 = sqrt(2.4) Notions of Spread Quantity closely related to standard deviation Notions of Spread Quantity closely related to standard deviation is its square Notions of Spread Quantity closely related to standard deviation is its square, called the variance (a) Probs: var(X) Notions of Spread Quantity closely related to standard deviation is its square, called the variance (a) Probs: var(X) = [σ(X)]2 Notions of Spread Quantity closely related to standard deviation is its square, called the variance (a) Probs: var(X) = [σ(X)]2 = σ2 Notions of Spread Quantity closely related to standard deviation is its square, called the variance (a) Probs: var(X) = [σ(X)]2 = σ2 = E(X - EX)2 Notions of Spread Quantity closely related to standard deviation is its square, called the variance (a) Probs: var(X) = [σ(X)]2 = σ2 = E(X - EX)2 (b) Lists: s 2 n 1 n 1 x x i 1 2 i Notions of Spread Excel Computation for Lists: Notions of Spread Excel Computation for Lists: • Can do manually (using SUM & SQRT) Notions of Spread Excel Computation for Lists: • Can do manually (using SUM & SQRT) • But faster to use: – STDEV – VAR Notions of Spread Excel Computation for Lists: • Can do manually (using SUM & SQRT) • But faster to use: • – STDEV – VAR Application same as for other functions Notions of Spread HW: C19: Calculate the standard deviation for the following lists, and compare qualitatively in terms of spread: (a) 1, 3, 3, 1 (1.15) (b) -6, -4, -4, -6 (1.15) (c) 1, 5, 5, 1 (2.31) (d) 1, 1, 1, 1 (0) Notions of Spread HW: 1.79a 4.77 (2.689, 1.710)