Last Time Hypothesis Testing –1-sided vs. 2-sided Paradox Big Picture Goals –Hypothesis Testing –Margin of Error –Sample Size Calculations Visualization.

Last Time Hypothesis Testing –1-sided vs. 2-sided Paradox Big Picture Goals –Hypothesis Testing –Margin of Error –Sample Size Calculations Visualization –Histograms

Administrative Matters Midterm I, coming Tuesday, Feb. 24 Excel notation to avoid actual calculation –So no computers or calculators Bring sheet of formulas, etc.

Administrative Matters Midterm I, coming Tuesday, Feb. 24 Excel notation to avoid actual calculation –So no computers or calculators Bring sheet of formulas, etc. No blue books needed

Administrative Matters Midterm I, coming Tuesday, Feb. 24 Excel notation to avoid actual calculation –So no computers or calculators Bring sheet of formulas, etc. No blue books needed (will just write on my printed version)

Administrative Matters Midterm I, coming Tuesday, Feb. 24 Material Covered: HW 1 – HW 5

Administrative Matters Midterm I, coming Tuesday, Feb. 24 Material Covered: HW 1 – HW 5 –Note: due Thursday, Feb. 19

Administrative Matters Midterm I, coming Tuesday, Feb. 24 Material Covered: HW 1 – HW 5 –Note: due Thursday, Feb. 19 –Will ask grader to return Mon. Feb. 23

Administrative Matters Midterm I, coming Tuesday, Feb. 24 Material Covered: HW 1 – HW 5 –Note: due Thursday, Feb. 19 –Will ask grader to return Mon. Feb. 23 –Can pickup in my office (Hanes 352)

Administrative Matters Midterm I, coming Tuesday, Feb. 24 Material Covered: HW 1 – HW 5 –Note: due Thursday, Feb. 19 –Will ask grader to return Mon. Feb. 23 –Can pickup in my office (Hanes 352) –So today’s HW not included

Reading In Textbook Approximate Reading for Today’s Material: Pages 261-262, 9-14, 270-276, 30-34 Approximate Reading for Next Class: Pages 279-282, 34-43

Big Picture Hypothesis Testing (Given dist’n, answer “yes-no”) Margin of Error (Find dist’n, use to measure error) Choose Sample Size (for given amount of error) Need better prob. tools

Big Picture Margin of Error Choose Sample Size Need better prob tools Start with visualizing probability distributions (key to “alternate representation”)

Histograms Idea: show rectangles, where area represents

Histograms Idea: show rectangles, where area represents: (a)Distributions: probabilities (b)Lists (of numbers): # of observations

Histograms Idea: show rectangles, where area represents: (a)Distributions: probabilities (b)Lists (of numbers): # of observations Note: will studies these in parallel for a while (several concepts apply to both)

Histograms Idea: show rectangles, where area represents: (a)Distributions: probabilities (b)Lists (of numbers): # of observations Caution: There are variations not based on areas, see bar graphs in text But eye perceives area, so sensible to use it

Histograms Steps for Constructing Histograms: 1.Pick class intervals that contain full dist’n

Histograms Steps for Constructing Histograms: 1.Pick class intervals that contain full dist’n a. Prob. dist’ns: If possible values are: x = 0, 1, …, n, get good picture from choice: [-½, ½), [½, 1.5), [1.5, 2.5), …, [n-½, n+½) where [1.5, 2.5) is “all #s ≥ 1.5 and < 2.5” (called a “half open interval”)

Histograms Steps for Constructing Histograms: 1.Pick class intervals that contain full dist’n a. Prob. dist’ns b. Lists: e.g. 2.3, 4.5, 4.7, 4.8, 5.1 Start with [1,3), [3,7) As above use half open intervals (to break ties)

Histograms Steps for Constructing Histograms: 1.Pick class intervals that contain full dist’n a. Prob. dist’ns b. Lists: e.g. 2.3, 4.5, 4.7, 4.8, 5.1 Start with [1,3), [3,7) Can use anything for class intervals But some choices better than others…

Histograms Steps for Constructing Histograms: 1.Pick class intervals that contain full dist’n 2.Find “probabilities” or “relative frequencies” for each class (a) Probs: use f(x) for [x-½, x+½), etc. (b) Lists: [1,3): rel. freq. = 1/5 = 20% [3,7): rel. freq. = 4/5 = 80%

Histograms Steps for Constructing Histograms: 1.Pick class intervals that contain full dist’n 2.Find “probabilities” or “relative frequencies” for each class 3.Above each interval, draw rectangle where area represents class frequency

Histograms 3.Above each interval, draw rectangle where area represents class frequency (a) Probs: If width = 1, then area = width x height = height So get area = f(x), by taking height = f(x)

Histograms 3.Above each interval, draw rectangle where area represents class frequency (a) Probs: If width = 1, then area = width x height = height So get area = f(x), by taking height = f(x) E.g. Binomial Distribution

Binomial Prob. Histograms From Class Example 5 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg5.xls Construct Prob. Histo: Create column of x values Compute f(x) values Make bar plot

Binomial Prob. Histograms Make bar plot –“Insert” tab –Choose “Column” –Right Click – Select Data (Horizontal – x’s, “Add series”, Probs) –Resize, and move by dragging –Delete legend –Click and change title –Right Click on Bars, Format Data Series: Border Color, Solid Line, Black Series Options, Gap Width = 0

Binomial Prob. Histograms From Class Example 5 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg5.xls Construct Prob. Histo: Create column of x values Compute f(x) values Make bar plot Make several, for interesting comparison

Binomial Prob. Histograms From Class Example 5a

Binomial Prob. Histograms From Class Example 5a Compare Different p

Binomial Prob. Histograms From Class Example 5a Compare Different p: Surprisingly similar “mound” shape

Binomial Prob. Histograms From Class Example 5a Compare Different p: Surprisingly similar “mound” shape (will exploit this fact)

Binomial Prob. Histograms From Class Example 5a Compare Different p: Centerpoint moves as p grows

Binomial Prob. Histograms From Class Example 5a Compare Different p: Centerpoint moves as p grows (will quantify, and use this, too)

Binomial Prob. Histograms Important point: Binomial shows common shape across p

Binomial Prob. Histograms Important point: Binomial shows common shape across p Mound Shape (like dumping dirt out of a truck)

Binomial Prob. Histograms Important point: Binomial shows common shape across p Mound Shape (like dumping dirt out of a truck) What about n?

Binomial Prob. Histograms From Class Example 5b Compare Different n

Binomial Prob. Histograms From Class Example 5b Compare Different n: Again very similar mound shape

Binomial Prob. Histograms From Class Example 5b Compare Different n: Again very similar mound shape (will exploit this fact)

Binomial Prob. Histograms From Class Example 5b Compare Different n: Center does not appear to move

Binomial Prob. Histograms From Class Example 5b Compare Different n: Center does not appear to move, but check axes!

Binomial Prob. Histograms From Class Example 5b Compare Different n: Center does not appear to move, but check axes! (will quantify, and use this, too)

Binomial Prob. Histograms From Class Example 5b Compare Different n: But width of bump does seem to change

Binomial Prob. Histograms From Class Example 5b Compare Different n: But width of bump does seem to change (will quantify, and use this, too)

Binomial Prob. Histograms Important point: Binomial shows common shape across p & n Mound Shape (like dumping dirt out of a truck)

Binomial Prob. Histograms Important point: Binomial shows common shape across p & n Mound Shape (like dumping dirt out of a truck) Question for later: How can we put this work?

And now for something (sort of) different Recall survey from first class meeting

And now for something (sort of) different Recall survey from first class meeting Display Results?

And now for something (sort of) different Recall survey from first class meeting Display Results? Use “bar graph”

And now for something (sort of) different Bar Graph from Survey, on major

And now for something (sort of) different Bar Graph from Survey, on major Business biggest (true for many years)

And now for something (sort of) different Bar Graph from Survey, on major Business biggest Biology 2 nd (fairly new)

And now for something (sort of) different Bar Graph from Survey, on major Business biggest Biology 2 nd Variety of others Welcome!

And now for something (sort of) different Bar Graph from Survey, on major Labels, not Class Intervals

And now for something (sort of) different Bar Graph from Survey, on major Thin bars Now OK

And now for something (sort of) different Bar Graph from Survey, on major Study Counts, not rel. freq.

And now for something (sort of) different Bar Graph from Survey, on major Study Counts, not rel. freq. (not areas)

And now for something (sort of) different Bar Graph from Survey, on year

And now for something (sort of) different Bar Graph from Survey, on year Distribution makes sense?

And now for something (sort of) different Bar Graph from Survey, on year Different color stresses different data

And now for something (sort of) different Bar Graph from Survey, on year Shorter & fewer labels appear as horizontal

Histograms Steps for Constructing Histograms: 1.Pick class intervals that contain full dist’n 2.Find “probabilities” or “relative frequencies” for each class 3.Above each interval, draw rectangle where area represents class frequency

Histograms HW: 5.21b (make & print an Excel plot)

Histograms 3.Above each interval, draw rectangle where area represents class frequency (a) Probs

Histograms 3.Above each interval, draw rectangle where area represents class frequency (a) Probs (b) Lists

Histograms 3.Above each interval, draw rectangle where area represents class frequency (a) Probs (b) Lists: e.g. 2.3, 4.5, 4.7, 4.8, 5.1 same e.g. as above

Histograms 3.Above each interval, draw rectangle where area represents class frequency (a) Probs (b) Lists: e.g. 2.3, 4.5, 4.7, 4.8, 5.1

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1 1 2 3 4 5 6 7

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) 1 2 3 4 5 6 7

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) From above discussion 1 2 3 4 5 6 7

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) From above discussion (will see: not very good) 1 2 3 4 5 6 7

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) 1 2 3 4 5 6 7

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) 20Total Frequency = 100% 15 10 5 1 2 3 4 5 6 7

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) 20Total Frequency = 100% 15So each is 20% 10 5 1 2 3 4 5 6 7

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) 20Total Frequency = 100% 1520% = Area 10 5 1 2 3 4 5 6 7

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) 20Total Frequency = 100% 1520% = Area = 2 * height 10 5 1 2 3 4 5 6 7

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) 20Total Frequency = 100% 1520% = Area = 2 * ht = 2 * (10% / unit) 10 5 1 2 3 4 5 6 7

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) 20Total Frequency = 100% 1520% = Area = 2 * ht = 2 * (10% / unit) 10 5 1 2 3 4 5 6 7 % per unit

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) 20Total Frequency = 100% 1520% = Area = 4 * ht 10 5 1 2 3 4 5 6 7 % per unit

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) 20Total Frequency = 100% 1520% = Area = 4 * ht = 4 * (5% / unit) 10 5 1 2 3 4 5 6 7 % per unit

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) 20 20% = Area = 4 * ht = 4 * (5% / unit) 15 10 5 1 2 3 4 5 6 7 % per unit

Histograms Rectangles - area represents class frequency 2.3, 4.5, 4.7, 4.8, 5.1, Class Intervals [1,3), [3,7) 20 15 10 5 1 2 3 4 5 6 7 % per unit

Histograms Note: This histogram hides structure in data: 2.3, 4.5, 4.7, 4.8, 5.1 20 15 10 5 1 2 3 4 5 6 7 % per unit

Histograms Quite sparse region 2.3, 4.5, 4.7, 4.8, 5.1 20 15 10 5 1 2 3 4 5 6 7 % per unit

Histograms Quite dense region 2.3, 4.5, 4.7, 4.8, 5.1 20 15 10 5 1 2 3 4 5 6 7 % per unit

Histograms Endpoints way off 2.3, 4.5, 4.7, 4.8, 5.1 20 15 10 5 1 2 3 4 5 6 7 % per unit

Histograms General Major Challenge: Choice of Class Intervals 20 15 10 5 1 2 3 4 5 6 7 % per unit

Histograms Try for “better” choice : 2.3, 4.5, 4.7, 4.8, 5.1 1 2 3 4 5 6 7

Histograms Try for “better” choice : 2.3, 4.5, 4.7, 4.8, 5.1 [2,4) [4,5) [5,6) 1 2 3 4 5 6 7

Histograms Now build histogram as above (areas): 2.3, 4.5, 4.7, 4.8, 5.1 60 30 1 2 3 4 5 6 7 % per unit

Histograms Note: much better visual impression 2.3, 4.5, 4.7, 4.8, 5.1 60 30 1 2 3 4 5 6 7 % per unit

Histograms Note: much better visual impression Histogram better reflects “structure in data” 60 30 1 2 3 4 5 6 7 % per unit

Histograms General Comments: Total area under histogram is 100%

Histograms General Comments: Total area under histogram is 100% So label vertical axis as “% per unit”

Histograms General Comments: Total area under histogram is 100% So label vertical axis as “% per unit” Synonym for “Class Interval” is “bin”

Histograms General Comments: Total area under histogram is 100% So label vertical axis as “% per unit” Synonym for “Class Interval” is “bin” (think of relative frequency as counting observations that “fall into bins”)

Histograms General Comments: Total area under histogram is 100% So label vertical axis as “% per unit” Synonym for “Class Interval” is “bin” (think of relative frequency as counting observations that “fall into bins”) Choice of bins is critical

Histograms General Comments: Total area under histogram is 100% So label vertical axis as “% per unit” Synonym for “Class Interval” is “bin” (think of relative frequency as counting observations that “fall into bins”) Choice of bins is critical Common Simplification: Equally spaced

Histograms General Comments: Choice of bins is critical Common Simplification: Equally spaced But still have choice of binwidth (also very challenging)

Histograms HW: C15 For the data: 0.8, 2.1, 2.6, 0.9, 2.2, 0.8, 2.2, 0.9 a)Make histograms using the bins: i.[0,1), [1,2), [2,3) ii.[0.5,1.5), [1.5,2.5), [2.5,3.5) iii.[0,1), 1,3) (Interesting to look at differences)

Histograms HW: C15 For the data: 0.8, 2.1, 2.6, 0.9, 2.2, 0.8, 2.2, 0.9 a)Make histograms using the bins: i.[0,1), [1,2), [2,3) ii.[0.5,1.5), [1.5,2.5), [2.5,3.5) iii.[0,1), 1,3) b) Why are bins [0,2), [1,3) inappropriate here? c) Why are bins [1,2), [2,5) inappropriate here?

Histogram Real Data Example Buffalo Snow Fall Data Annual totals (in inches)

Histogram Real Data Example Buffalo Snow Fall Data Annual totals (in inches) For Buffalo, N.Y.

Histogram Real Data Example Buffalo Snow Fall Data Annual totals (in inches) For Buffalo, N.Y. 63 years, ranging from ~30 to ~120

Histogram Real Data Example Buffalo Snow Fall Data Annual totals (in inches) For Buffalo, N.Y. 63 years, ranging from ~30 to ~120 A lot of snow, due to “lake effect”

Histogram Real Data Example Buffalo Snow Fall Data Annual totals (in inches) For Buffalo, N.Y. 63 years, ranging from ~30 to ~120 A lot of snow, due to “lake effect” Any patterns in data?

Histogram Real Data Example Buffalo Snow Fall Data Data Available in Class Example 6 Left hand column of spreadsheet: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg6.xls

Histogram Real Data Example Buffalo Snow Fall Data Data Available in Class Example 6 Left hand column of spreadsheet: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg6.xls Now do histogram analysis Using Excel

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Data Tab

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Data Tab Push Data Analysis Button

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Data Tab Push Data Analysis Button Pulls up:

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Data Tab Push Data Analysis Button Pulls up: Choose:

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Pulls Up:

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Pulls Up: Link input data

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Pulls Up: Link input data Empty for default

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Pulls Up: Link input data Empty for default Choose here

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Pulls Up: Link input data Empty for default Choose here And location

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Pulls Up: Link input data Empty for default Choose here And location Get Histo Plot

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Manually Chart Result???

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Manually Chart Result??? Twiddle Output (similar to above): Delete Series Legend

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Manually Chart Result??? Twiddle Output (similar to above): Delete Series Legend Format Data Series – Gap Width  0

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Manually Chart Result??? Twiddle Output (similar to above): Delete Series Legend Format Data Series – Gap Width  0 Format Data Series – Border Color  Black

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Manually Chart Result??? Twiddle Output (similar to above): Delete Series Legend Format Data Series – Gap Width  0 Format Data Series – Border Color  Black Chart Tools – Design – Choose Titled

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Manually Chart Result??? Twiddle Output (similar to above): Delete Series Legend Format Data Series – Gap Width  0 Format Data Series – Border Color  Black Chart Tools – Design – Choose Titled Type in Title

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Result:

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Result: Unround numbers for bin edges

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Result: Unround numbers for bin edges Hard to interpret

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Data centered around 90

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Data centered around 90 Most data between 50 and 130

Histogram Real Data Example Buffalo Snow Fall Data – Excel Default Histo Data centered around 90 Most data between 50 and 130 Assymetric Distribution

Histogram Real Data Example Buffalo Snow Fall Data – Smaller binwidth

Histogram Real Data Example Buffalo Snow Fall Data – Smaller binwidth Chosen by me Binwidth = 5, << ~13 from EXCEL default

Histogram Real Data Example Buffalo Snow Fall Data – Smaller binwidth Chosen by me Binwidth = 5, << ~13 from EXCEL default Nicer edge numbers

Histogram Real Data Example Buffalo Snow Fall Data – Smaller binwidth Chosen by me Binwidth = 5, << ~13 from EXCEL default Nicer edge numbers Data centered around 84 (now more precise)

Histogram Real Data Example Buffalo Snow Fall Data – Smaller binwidth Chosen by me Binwidth = 5, << ~13 from EXCEL default Nicer edge numbers Data centered around 84 (now more precise) Bar graph rougher (fewer points in each bin)

Histogram Real Data Example Buffalo Snow Fall Data – Smaller binwidth Chosen by me Binwidth = 5, << ~13 from EXCEL default Nicer edge numbers Data centered around 84 (now more precise) Bar graph rougher (fewer points in each bin) Suggests 3 main groups

Histogram Real Data Example Buffalo Snow Fall Data – Smaller binwidth Chosen by me Binwidth = 5, << ~13 from EXCEL default Nicer edge numbers Data centered around 84 (now more precise) Bar graph rougher (fewer points in each bin) Suggests 3 main groups (called “modes” or “clusters”)

Histogram Real Data Example Buffalo Snow Fall Data – Smaller binwidth Chosen by me Binwidth = 5, << ~13 from EXCEL default Nicer edge numbers Data centered around 84 (now more precise) Bar graph rougher (fewer points in each bin) Suggests 3 main groups (called “modes” or “clusters”) (can’t see this above: bin width is important)

Histogram Real Data Example Buffalo Snow Fall Data – Larger binwidth

Histogram Real Data Example Buffalo Snow Fall Data – Larger binwidth Chosen by me Binwidth = 30, >> ~13 from EXCEL default

Histogram Real Data Example Buffalo Snow Fall Data – Larger binwidth Chosen by me Binwidth = 30, >> ~13 from EXCEL default Bar graph is “smooth” (since many points in each bin)

Histogram Real Data Example Buffalo Snow Fall Data – Larger binwidth Chosen by me Binwidth = 30, >> ~13 from EXCEL default Bar graph is “smooth” (since many points in each bin) Only one mode (cluster)???

Histogram Real Data Example Buffalo Snow Fall Data – Larger binwidth Chosen by me Binwidth = 30, >> ~13 from EXCEL default Bar graph is “smooth” (since many points in each bin) Only one mode (cluster)??? Quite symmetric?

Histogram Real Data Example Buffalo Snow Fall Data – Larger binwidth Chosen by me Binwidth = 30, >> ~13 from EXCEL default Bar graph is “smooth” (since many points in each bin) Only one mode (cluster)??? Quite symmetric? (different from above: bin width is important)

Histogram Real Data Example HW: 1.28 [data in ta01_005.xls] ((c) loses bump near 50) 1.36 [data in ex01_036.xls] ((a) 4 (b) 2 (c) 1) 1.37 1.39

Research Corner Histo Bin Width (serious issue)

Research Corner Histo Bin Width (serious issue) Interesting Data Set: Hidalgo Stamps

Research Corner Histo Bin Width (serious issue) Interesting Data Set: Hidalgo Stamps Famous among postage stamp collectorsFamous among postage stamp collectors

Research Corner Histo Bin Width (serious issue) Interesting Data Set: Hidalgo Stamps Famous among postage stamp collectorsFamous among postage stamp collectors Printed in Mexico, 1800’s, over ~70 yearsPrinted in Mexico, 1800’s, over ~70 years

Research Corner Histo Bin Width (serious issue) Interesting Data Set: Hidalgo Stamps Famous among postage stamp collectorsFamous among postage stamp collectors Printed in Mexico, 1800’s, over ~70 yearsPrinted in Mexico, 1800’s, over ~70 years Very different paper thicknesses…Very different paper thicknesses…

Research Corner Histo Bin Width (serious issue) Interesting Data Set: Hidalgo Stamps Famous among postage stamp collectorsFamous among postage stamp collectors Printed in Mexico, 1800’s, over ~70 yearsPrinted in Mexico, 1800’s, over ~70 years Very different paper thicknesses…Very different paper thicknesses… How many paper sources?How many paper sources?

Research Corner Histo Bin Width (serious issue) Interesting Data Set: Hidalgo Stamps Famous among postage stamp collectorsFamous among postage stamp collectors Printed in Mexico, 1800’s, over ~70 yearsPrinted in Mexico, 1800’s, over ~70 years Very different paper thicknesses…Very different paper thicknesses… How many paper sources?How many paper sources? Unknown, since records are lostUnknown, since records are lost

Research Corner Histo Bin Width (serious issue) Interesting Data Set: Hidalgo Stamps Famous among postage stamp collectorsFamous among postage stamp collectors Printed in Mexico, 1800’s, over ~70 yearsPrinted in Mexico, 1800’s, over ~70 years Very different paper thicknesses…Very different paper thicknesses… How many paper sources?How many paper sources? Unknown, since records are lostUnknown, since records are lost Study histogram of stamp thicknessesStudy histogram of stamp thicknesses

Research Corner Movie over binwidth

Research Corner Movie over binwidth Shows very wide range

Research Corner Movie over binwidth Shows very wide range (much different (much different visual impressions) visual impressions)

Research Corner Movie over binwidth Shows very wide range (much different (much different visual impressions) visual impressions) How many bumps?

Research Corner Movie over binwidth Shows very wide range (much different (much different visual impressions) visual impressions) How many bumps? Answer published in literature: 2, 3, 5, 7, 10

Research Corner Movie over binwidth Shows very wide range (much different (much different visual impressions) visual impressions) How many bumps? Answer published in literature: 2, 3, 5, 7, 10 Very challenging question

Research Corner How many bumps? Believe in 2?

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

Notions of Center Textbook: Sections 4.4 and 1.2 Recall parallel development: (a)Probability Distributions (b)Lists of Numbers

Notions of Center Textbook: Sections 4.4 and 1.2 Recall parallel development: (a)Probability Distributions (b)Lists of Numbers Study 1 st, since easier

Notions of Center (b)Lists of Numbers “Average” or “Mean”

Notions of Center (b)Lists of Numbers “Average” or “Mean” of x 1, x 2, …, x n Mean = =

Notions of Center (b)Lists of Numbers “Average” or “Mean” of x 1, x 2, …, x n Mean = = common notation

Notions of Center (b)Lists of Numbers “Average” or “Mean” of x 1, x 2, …, x n Mean = = (as before) Greek sigma for sum means “sum over I = 1,…,n”

Notions of Center HW: C16: for the data of 1.57, find the mean using the Excel function AVERAGE (10.03)

Notions of Center Generalization of Mean: “Weighted Average”

Notions of Center Generalization of Mean: “Weighted Average” Idea: allow non-equal weights on s:

Notions of Center Generalization of Mean: “Weighted Average” Idea: allow non-equal weights on s: Where,

Notions of Center Generalization of Mean: “Weighted Average” E.g.: ordinary mean has each

Notions of Center Generalization of Mean: “Weighted Average” E.g.: ordinary mean has each (constant weights)

Notions of Center Generalization of Mean: “Weighted Average” Intuition: Corresponds to finding balance point of weights on number line

Notions of Center HW: C17: Calculate (and think about as “balance point”) weighted average of 1, 2, 3, 10 for the weights: a.¼, ¼, ¼, 1/4, (ordinary avg.)(4) b.0.1, 0.1, 0.1, 0.7 (more on 10)(7.6) c.0.3, 0.3, 0.3, 0.1 (less on 10)(2.8) d.1/3, 1/3, 1/3, 0 (none on 10)(2) e.0, 1, 0, 0 (all on 2)(2)

Last Time Hypothesis Testing –1-sided vs. 2-sided Paradox Big Picture Goals –Hypothesis Testing –Margin of Error –Sample Size Calculations Visualization.

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Last Time Hypothesis Testing –1-sided vs. 2-sided Paradox Big Picture Goals –Hypothesis Testing –Margin of Error –Sample Size Calculations Visualization.

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