Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness (centered correctly) –Standard error (measures spread)

Administrative Matters Midterm II, coming Tuesday, April 6

Administrative Matters Midterm II, coming Tuesday, April 6 Numerical answers: –No computers, no calculators

Administrative Matters Midterm II, coming Tuesday, April 6 Numerical answers: –No computers, no calculators –Handwrite Excel formulas (e.g. =9+4^2) –Don’t do arithmetic (e.g. use such formulas)

Administrative Matters Midterm II, coming Tuesday, April 6 Numerical answers: –No computers, no calculators –Handwrite Excel formulas (e.g. =9+4^2) –Don’t do arithmetic (e.g. use such formulas) Bring with you: –One 8.5 x 11 inch sheet of paper

Administrative Matters Midterm II, coming Tuesday, April 6 Numerical answers: –No computers, no calculators –Handwrite Excel formulas (e.g. =9+4^2) –Don’t do arithmetic (e.g. use such formulas) Bring with you: –One 8.5 x 11 inch sheet of paper –With your favorite info (formulas, Excel, etc.)

Administrative Matters Midterm II, coming Tuesday, April 6 Numerical answers: –No computers, no calculators –Handwrite Excel formulas (e.g. =9+4^2) –Don’t do arithmetic (e.g. use such formulas) Bring with you: –One 8.5 x 11 inch sheet of paper –With your favorite info (formulas, Excel, etc.) Course in Concepts, not Memorization

Administrative Matters Midterm II, coming Tuesday, April 6 Material Covered: HW 6 – HW 10

Administrative Matters Midterm II, coming Tuesday, April 6 Material Covered: HW 6 – HW 10 –Note: due Thursday, April 2

Administrative Matters Midterm II, coming Tuesday, April 6 Material Covered: HW 6 – HW 10 –Note: due Thursday, April 2 –Will ask grader to return Mon. April 5 –Can pickup in my office (Hanes 352)

Administrative Matters Midterm II, coming Tuesday, April 6 Material Covered: HW 6 – HW 10 –Note: due Thursday, April 2 –Will ask grader to return Mon. April 5 –Can pickup in my office (Hanes 352) –So today’s HW not included

Administrative Matters Extra Office Hours before Midterm II Monday, Apr. 23 8:00 – 10:00 Monday, Apr :00 – 2:00 Tuesday, Apr. 24 8:00 – 10:00 Tuesday, Apr. 24 1:00 – 2:00 (usual office hours)

Study Suggestions 1.Work an Old Exam a)On Blackboard b)Course Information Section

Study Suggestions 1.Work an Old Exam a)On Blackboard b)Course Information Section c)Afterwards, check against given solutions

Study Suggestions 1.Work an Old Exam a)On Blackboard b)Course Information Section c)Afterwards, check against given solutions 2.Rework HW problems

Study Suggestions 1.Work an Old Exam a)On Blackboard b)Course Information Section c)Afterwards, check against given solutions 2.Rework HW problems a)Print Assignment sheets b)Choose problems in “random” order

Study Suggestions 1.Work an Old Exam a)On Blackboard b)Course Information Section c)Afterwards, check against given solutions 2.Rework HW problems a)Print Assignment sheets b)Choose problems in “random” order c)Rework (don’t just “look over”)

Reading In Textbook Approximate Reading for Today’s Material: Pages , Approximate Reading for Next Class: Pages , ,

Law of Averages Case 2: any random sample CAN SHOW, for n “large” is “roughly” Terminology:  “Law of Averages, Part 2”  “Central Limit Theorem” (widely used name)

Central Limit Theorem Illustration: Rice Univ. Applet Starting Distribut’n user input (very non-Normal) Dist’n of average of n = 25 (seems very mound shaped?)

Extreme Case of CLT Consequences: roughly Terminology: Called The Normal Approximation to the Binomial

Normal Approx. to Binomial How large n? Bigger is better Could use “n ≥ 30” rule from above Law of Averages But clearly depends on p Textbook Rule: OK when {np ≥ 10 & n(1-p) ≥ 10}

Statistical Inference Idea: Develop formal framework for handling unknowns p & μ e.g. 1:Political Polls e.g. 2a:Population Modeling e.g. 2b:Measurement Error

Statistical Inference A parameter is a numerical feature of population, not sample An estimate of a parameter is some function of data (hopefully close to parameter)

Statistical Inference Standard Error: for an unbiased estimator, standard error is standard deviation Notes:  For SE of, since don’t know p, use sensible estimate  For SE of, use sensible estimate

Statistical Inference Another view: Form conclusions by

Statistical Inference Another view: Form conclusions by quantifying uncertainty

Statistical Inference Another view: Form conclusions by quantifying uncertainty (will study several approaches, first is…)

Confidence Intervals Background:

Confidence Intervals Background: The sample mean,, is an “estimate” of the population mean,

Confidence Intervals Background: The sample mean,, is an “estimate” of the population mean, How accurate?

Confidence Intervals Background: The sample mean,, is an “estimate” of the population mean, How accurate? (there is “variability”, how much?)

Confidence Intervals Idea: Since a point estimate (e.g. or )

Confidence Intervals Idea: Since a point estimate is never exactly right (in particular )

Confidence Intervals Idea: Since a point estimate is never exactly right give a reasonable range of likely values (range also gives feeling for accuracy of estimation)

Confidence Intervals Idea: Since a point estimate is never exactly right give a reasonable range of likely values (range also gives feeling for accuracy of estimation)

Confidence Intervals E.g.

Confidence Intervals E.g. with σ known

Confidence Intervals E.g. with σ known Think: measurement error

Confidence Intervals E.g. with σ known Think: measurement error Each measurement is Normal

Confidence Intervals E.g. with σ known Think: measurement error Each measurement is Normal Known accuracy (maybe)

Confidence Intervals E.g. with σ known Think: population modeling

Confidence Intervals E.g. with σ known Think: population modeling Normal population

Confidence Intervals E.g. with σ known Think: population modeling Normal population Known s.d. (a stretch, really need to improve)

Confidence Intervals E.g. with σ known Recall the Sampling Distribution:

Confidence Intervals E.g. with σ known Recall the Sampling Distribution: (recall have this even when data not normal, by Central Limit Theorem)

Confidence Intervals E.g. with σ known Recall the Sampling Distribution: Use to analyze variation

Confidence Intervals Understand error as: (normal density quantifies randomness in )

Confidence Intervals Understand error as: (distribution centered at μ)

Confidence Intervals Understand error as: (spread: s.d. = )

Confidence Intervals Understand error as: How to explain to untrained consumers?

Confidence Intervals Understand error as: How to explain to untrained consumers? (who don’t know randomness, distributions, normal curves)

Confidence Intervals Approach: present an interval

Confidence Intervals Approach: present an interval With endpoints: Estimate +- margin of error

Confidence Intervals Approach: present an interval With endpoints: Estimate +- margin of error I.e.

Confidence Intervals Approach: present an interval With endpoints: Estimate +- margin of error I.e. reflecting variability

Confidence Intervals Approach: present an interval With endpoints: Estimate +- margin of error I.e. reflecting variability

Confidence Intervals Approach: present an interval With endpoints: Estimate +- margin of error I.e. reflecting variability How to choose ?

Confidence Intervals Choice of Confidence Interval Radius

Confidence Intervals Choice of Confidence Interval Radius, i.e. margin of error,

Confidence Intervals Choice of Confidence Interval Radius, i.e. margin of error, : Notes: No Absolute Range (i.e. including “everything”) is available

Confidence Intervals Choice of Confidence Interval Radius, i.e. margin of error, : Notes: No Absolute Range (i.e. including “everything”) is available From infinite tail of normal dist’n

Confidence Intervals Choice of Confidence Interval Radius, i.e. margin of error, : Notes: No Absolute Range (i.e. including “everything”) is available From infinite tail of normal dist’n So need to specify desired accuracy

Confidence Intervals Choice of margin of error,

Confidence Intervals Choice of margin of error, : Approach: Choose a Confidence Level

Confidence Intervals Choice of margin of error, : Approach: Choose a Confidence Level Often 0.95

Confidence Intervals Choice of margin of error, : Approach: Choose a Confidence Level Often 0.95 (e.g. FDA likes this number for approving new drugs, and it is a common standard for publication in many fields)

Confidence Intervals Choice of margin of error, : Approach: Choose a Confidence Level Often 0.95 (e.g. FDA likes this number for approving new drugs, and it is a common standard for publication in many fields) And take margin of error to include that part of sampling distribution

Confidence Intervals E.g. For confidence level 0.95, want 0.95 = Area

Confidence Intervals E.g. For confidence level 0.95, want distribution 0.95 = Area

Confidence Intervals E.g. For confidence level 0.95, want distribution 0.95 = Area = margin of error

Confidence Intervals Computation: Recall NORMINV

Confidence Intervals Computation: Recall NORMINV takes areas (probs)

Confidence Intervals Computation: Recall NORMINV takes areas (probs), and returns cutoffs

Confidence Intervals Computation: Recall NORMINV takes areas (probs), and returns cutoffs Issue: NORMINV works with lower areas

Confidence Intervals Computation: Recall NORMINV takes areas (probs), and returns cutoffs Issue: NORMINV works with lower areas Note: lower tail included

Confidence Intervals So adapt needed probs to lower areas….

Confidence Intervals So adapt needed probs to lower areas…. When inner area = 0.95,

Confidence Intervals So adapt needed probs to lower areas…. When inner area = 0.95, Right tail = 0.025

Confidence Intervals So adapt needed probs to lower areas…. When inner area = 0.95, Right tail = Shaded Area = 0.975

Confidence Intervals So adapt needed probs to lower areas…. When inner area = 0.95, Right tail = Shaded Area = So need to compute as:

Confidence Intervals Need to compute:

Confidence Intervals Need to compute: Major problem: is unknown

Confidence Intervals Need to compute: Major problem: is unknown But should answer depend on ?

Confidence Intervals Need to compute: Major problem: is unknown But should answer depend on ? “Accuracy” is only about spread

Confidence Intervals Need to compute: Major problem: is unknown But should answer depend on ? “Accuracy” is only about spread Not centerpoint

Confidence Intervals Need to compute: Major problem: is unknown But should answer depend on ? “Accuracy” is only about spread Not centerpoint Need another view of the problem

Confidence Intervals Approach to unknown

Confidence Intervals Approach to unknown : Recenter, i.e. look at dist’n

Confidence Intervals Approach to unknown : Recenter, i.e. look at dist’n

Confidence Intervals Approach to unknown : Recenter, i.e. look at dist’n Key concept: Centered at 0

Confidence Intervals Approach to unknown : Recenter, i.e. look at dist’n Key concept: Centered at 0 Now can calculate as:

Confidence Intervals Computation of:

Confidence Intervals Computation of: Smaller Problem: Don’t know

Confidence Intervals Computation of: Smaller Problem: Don’t know Approach 1: Estimate with (natural approach: use estimate)

Confidence Intervals Computation of: Smaller Problem: Don’t know Approach 1: Estimate with Leads to complications

Confidence Intervals Computation of: Smaller Problem: Don’t know Approach 1: Estimate with Leads to complications Will study later

Confidence Intervals Computation of: Smaller Problem: Don’t know Approach 1: Estimate with Leads to complications Will study later Approach 2: Sometimes know

Research Corner How many bumps in stamps data? Kernel Density Estimates Depends on Window ~1?

Research Corner How many bumps in stamps data? Kernel Density Estimates Depends on Window ~2?

Research Corner How many bumps in stamps data? Kernel Density Estimates Depends on Window ~7?

Research Corner How many bumps in stamps data? Kernel Density Estimates Depends on Window ~10?

Research Corner How many bumps in stamps data? Kernel Density Estimates Depends on Window Early Approach: Use data to choose window width

Research Corner How many bumps in stamps data? Kernel Density Estimates Depends on Window Challenge: Not enough info in data for good choice

Research Corner How many bumps in stamps data? Kernel Density Estimates Depends on Window Alternate Approach: Scale Space

Research Corner Scale Space: Main Idea: Don’t try to choose window width

Research Corner Scale Space: Main Idea: Don’t try to choose window width Instead use all of them

Research Corner Scale Space: Main Idea: Don’t try to choose window width Instead use all of them Terminology from Computer Vision

Research Corner Scale Space: Main Idea: Don’t try to choose window width Instead use all of them Terminology from Computer Vision (goal: teach computers to “see”)

Research Corner Scale Space: Main Idea: Don’t try to choose window width Instead use all of them Terminology from Computer Vision: –Oversmoothing: coarse scale view (zoomed out – macroscopic perception)

Research Corner Scale Space: Main Idea: Don’t try to choose window width Instead use all of them Terminology from Computer Vision: –Oversmoothing: coarse scale view –Undersmoothing: fine scale view (zoomed in – microscopic perception)

Research Corner Scale Space: View 1: Rainbow colored movie

Research Corner Scale Space: View 2: Rainbow colored overlay

Research Corner Scale Space: View 3: Rainbow colored surface

Research Corner Scale Space: Main Idea: Don’t try to choose window width Instead use all of them Challenge: how to do statistical inference?

Research Corner Scale Space: Main Idea: Don’t try to choose window width Instead use all of them Challenge: how to do statistical inference? Which bumps are really there?

Research Corner Scale Space: Main Idea: Don’t try to choose window width Instead use all of them Challenge: how to do statistical inference? Which bumps are really there? (i.e. statistically significant)

Research Corner Scale Space: Challenge: how to do statistical inference? Which bumps are really there? (i.e. statistically significant)

Research Corner Scale Space: Challenge: how to do statistical inference? Which bumps are really there? (i.e. statistically significant) Address this next time

Confidence Intervals E.g. Crop researchers plant 15 plots with a new variety of corn.

Confidence Intervals E.g. Crop researchers plant 15 plots with a new variety of corn. The yields, in bushels per acre are:

Confidence Intervals E.g. Crop researchers plant 15 plots with a new variety of corn. The yields, in bushels per acre are: Assume that = 10 bushels / acre

Confidence Intervals E.g. Find: a)The 90% Confidence Interval for the mean value, for this type of corn. b)The 95% Confidence Interval. c)The 99% Confidence Interval. d)How do the CIs change as the confidence level increases? Solution, part 1 of Class Example 11:

Confidence Intervals E.g. Find: a)90% Confidence Interval for Next study relevant parts of E.g. 11:

Confidence Intervals E.g. Find: a)90% Confidence Interval for Use Excel

Confidence Intervals E.g. Find: a)90% Confidence Interval for Use Excel Data in C8:C22

Confidence Intervals E.g. Find: a)90% Confidence Interval for Steps: - Sample Size, n

Confidence Intervals E.g. Find: a)90% Confidence Interval for Steps: - Sample Size, n - Average,

Confidence Intervals E.g. Find: a)90% Confidence Interval for Steps: - Sample Size, n - Average, - S. D., σ

Confidence Intervals E.g. Find: a)90% Confidence Interval for Steps: - Sample Size, n - Average, - S. D., σ - Margin, m

Confidence Intervals E.g. Find: a)90% Confidence Interval for Steps: - Sample Size, n - Average, - S. D., σ - Margin, m - CI endpoint, left

Confidence Intervals E.g. Find: a)90% Confidence Interval for Steps: - Sample Size, n - Average, - S. D., σ - Margin, m - CI endpoint, left - CI endpoint, right

Confidence Intervals E.g. Find: a)90% CI for : [119.6, 128.0]

Confidence Intervals An EXCEL shortcut: CONFIDENCE

Confidence Intervals An EXCEL shortcut: CONFIDENCE

Confidence Intervals An EXCEL shortcut: CONFIDENCE Note: same margin of error as before

Confidence Intervals An EXCEL shortcut: CONFIDENCE

Confidence Intervals An EXCEL shortcut: CONFIDENCE Inputs: Sample Size

Confidence Intervals An EXCEL shortcut: CONFIDENCE Inputs: Sample Size S. D.

Confidence Intervals An EXCEL shortcut: CONFIDENCE Inputs: Sample Size S. D. Alpha

Confidence Intervals An EXCEL shortcut: CONFIDENCE Careful: parameter α

Confidence Intervals An EXCEL shortcut: CONFIDENCE Careful: parameter α is: 2 tailed outer area

Confidence Intervals An EXCEL shortcut: CONFIDENCE Careful: parameter α is: 2 tailed outer area So for level = 0.90, α = 0.10

Confidence Intervals E.g. Find: a)90% CI for μ: [119.6, 128.0]

Confidence Intervals E.g. Find: a)90% CI for μ: [119.6, 128.0] b)95% CI for μ: [118.7, 128.9]

Confidence Intervals E.g. Find: a)90% CI for μ: [119.6, 128.0] b)95% CI for μ: [118.7, 128.9] c)99% CI for μ: [117.1, 130.5]

Confidence Intervals E.g. Find: a)90% CI for μ: [119.6, 128.0] b)95% CI for μ: [118.7, 128.9] c)99% CI for μ: [117.1, 130.5] d)How do the CIs change as the confidence level increases?

Confidence Intervals E.g. Find: a)90% CI for μ: [119.6, 128.0] b)95% CI for μ: [118.7, 128.9] c)99% CI for μ: [117.1, 130.5] d)How do the CIs change as the confidence level increases? –Intervals get longer

Confidence Intervals E.g. Find: a)90% CI for μ: [119.6, 128.0] b)95% CI for μ: [118.7, 128.9] c)99% CI for μ: [117.1, 130.5] d)How do the CIs change as the confidence level increases? –Intervals get longer –Reflects higher demand for accuracy

Confidence Intervals HW: 6.11 (use Excel to draw curve & shade by hand) 6.13, 6.14 (7.30,7.70, wider) 6.16 (n = 2673, so CLT gives Normal)

Choice of Sample Size Additional use of margin of error idea

Choice of Sample Size Additional use of margin of error idea Background: distributions Small n Large n

Choice of Sample Size Could choose n to make = desired value

Choice of Sample Size Could choose n to make = desired value But S. D. is not very interpretable

Choice of Sample Size Could choose n to make = desired value But S. D. is not very interpretable, so make “margin of error”, m = desired value

Choice of Sample Size Could choose n to make = desired value But S. D. is not very interpretable, so make “margin of error”, m = desired value Then get: “ is within m units of, 95% of the time”

Choice of Sample Size Given m, how do we find n?

Choice of Sample Size Given m, how do we find n? Solve for n (the equation):

Choice of Sample Size Given m, how do we find n? Solve for n (the equation): (where is n in this?)

Choice of Sample Size Given m, how do we find n? Solve for n (the equation): (use of “standardization”)

Choice of Sample Size Given m, how do we find n? Solve for n (the equation):

Choice of Sample Size Given m, how do we find n? Solve for n (the equation): [so use NORMINV & Stand. Normal, N(0,1)]

Choice of Sample Size Graphically, find m so that: Area = 0.95

Choice of Sample Size Graphically, find m so that: Area = 0.95 Area = 0.975

Choice of Sample Size Thus solve:

Choice of Sample Size Thus solve:

Choice of Sample Size Thus solve:

Choice of Sample Size (put this on list of formulas)

Choice of Sample Size Numerical fine points:

Choice of Sample Size Numerical fine points: Change this for coverage prob. ≠ 0.95

Choice of Sample Size Numerical fine points: Change this for coverage prob. ≠ 0.95 Round decimals upwards, To be “sure of desired coverage”

Choice of Sample Size EXCEL Implementation: Class Example 11, Part 2:

Choice of Sample Size Class Example 11, Part 2: Recall: Corn Yield Data

Choice of Sample Size Class Example 11, Part 2: Recall: Corn Yield Data Gave

Choice of Sample Size Class Example 11, Part 2: Recall: Corn Yield Data Gave Assumed σ = 10

Choice of Sample Size Class Example 11, Part 2: Recall: Corn Yield Data Resulted in margin of error, m

Choice of Sample Size Class Example 11, Part 2: How large should n be to give smaller (90%) margin of error, say m = 2?

Choice of Sample Size Class Example 11, Part 2: How large should n be to give smaller (90%) margin of error, say m = 2? Compute from:

Choice of Sample Size Class Example 11, Part 2: How large should n be to give smaller (90%) margin of error, say m = 2? Compute from: (recall 90% central area, so use 95% cutoff)

Choice of Sample Size Class Example 11, Part 2: How large should n be to give smaller (90%) margin of error, say m = 2? Compute from:

Choice of Sample Size Class Example 11, Part 2: How large should n be to give smaller (90%) margin of error, say m = 2? Compute from:

Choice of Sample Size Class Example 11, Part 2: How large should n be to give smaller (90%) margin of error, say m = 2? Compute from:

Choice of Sample Size Class Example 11, Part 2: How large should n be to give smaller (90%) margin of error, say m = 2? Compute from:

Choice of Sample Size Class Example 11, Part 2: How large should n be to give smaller (90%) margin of error, say m = 2? Compute from: Round up, to be safe in statement

Choice of Sample Size Class Example 11, Part 2: Excel Function to round up: CIELING

Choice of Sample Size Class Example 11, Part 2: How large should n be to give smaller (90%) margin of error, say m = 2? n = 68

Choice of Sample Size Now ask for higher confidence level: How large should n be to give smaller (99%) margin of error, say m = 2?

Choice of Sample Size Now ask for higher confidence level: How large should n be to give smaller (99%) margin of error, say m = 2? Similar computations: n = 166

Choice of Sample Size Now ask for smaller margin: How large should n be to give smaller (99%) margin of error, say m = 0.2?

Choice of Sample Size Now ask for smaller margin: How large should n be to give smaller (99%) margin of error, say m = 0.2? Similar computations: n = 16588

Choice of Sample Size Now ask for smaller margin: How large should n be to give smaller (99%) margin of error, say m = 0.2? Similar computations: n = Note: serious round up

Choice of Sample Size Now ask for smaller margin: How large should n be to give smaller (99%) margin of error, say m = 0.2? Similar computations: n = (10 times the accuracy requires 100 times as much data)

Choice of Sample Size Now ask for smaller margin: How large should n be to give smaller (99%) margin of error, say m = 0.2? Similar computations: n = (10 times the accuracy requires 100 times as much data) (Law of Averages: Square Root)

Choice of Sample Size HW: 6.29, 6.30 (52), 6.31

And now for something completely different…. An interesting advertisement:

C.I.s for proportions Recall: Counts:

C.I.s for proportions Recall: Counts: Sample Proportions:

C.I.s for proportions Calculate prob’s with BINOMDIST

C.I.s for proportions Calculate prob’s with BINOMDIST (but C.I.s need inverse of probs)

C.I.s for proportions Calculate prob’s with BINOMDIST, but note no BINOMINV

C.I.s for proportions Calculate prob’s with BINOMDIST, but note no BINOMINV, so instead use Normal Approximation Recall:

Normal Approx. to Binomial Example: from StatsPortal For Bi(n,p): Control n Control p See Prob. Histo. Compare to fit (by mean & sd) Normal dist’n

C.I.s for proportions Recall Normal Approximation to Binomial

C.I.s for proportions Recall Normal Approximation to Binomial: For

C.I.s for proportions Recall Normal Approximation to Binomial: For is approximately

C.I.s for proportions Recall Normal Approximation to Binomial: For is approximately

C.I.s for proportions Recall Normal Approximation to Binomial: For is approximately So use NORMINV

C.I.s for proportions Recall Normal Approximation to Binomial: For is approximately So use NORMINV (and often NORMDIST)

C.I.s for proportions Main problem: don’t know p

C.I.s for proportions Main problem: don’t know p Solution: Depends on context: CIs or hypothesis tests

C.I.s for proportions Main problem: don’t know p Solution: Depends on context: CIs or hypothesis tests Different from Normal

C.I.s for proportions Main problem: don’t know p Solution: Depends on context: CIs or hypothesis tests Different from Normal, since now mean and sd are linked

C.I.s for proportions Main problem: don’t know p Solution: Depends on context: CIs or hypothesis tests Different from Normal, since now mean and sd are linked, with both depending on p

C.I.s for proportions Main problem: don’t know p Solution: Depends on context: CIs or hypothesis tests Different from Normal, since now mean and sd are linked, with both depending on p, instead of separate μ & σ.

C.I.s for proportions Case 1: Margin of Error and CIs: 95%

C.I.s for proportions Case 1: Margin of Error and CIs: 95% 0.975

C.I.s for proportions Case 1: Margin of Error and CIs: 95% So:

C.I.s for proportions Case 1: Margin of Error and CIs:

C.I.s for proportions Case 1: Margin of Error and CIs: Continuing problem: Unknown

C.I.s for proportions Case 1: Margin of Error and CIs: Continuing problem: Unknown Solution 1: “Best Guess”

C.I.s for proportions Case 1: Margin of Error and CIs: Continuing problem: Unknown Solution 1: “Best Guess” Replace by

C.I.s for proportions Solution 2: “Conservative”

C.I.s for proportions Solution 2: “Conservative” Idea: make sd (and thus m) as large as possible

C.I.s for proportions Solution 2: “Conservative” Idea: make sd (and thus m) as large as possible (makes no sense for Normal)

C.I.s for proportions Solution 2: “Conservative” Idea: make sd (and thus m) as large as possible (makes no sense for Normal)

C.I.s for proportions Solution 2: “Conservative” Idea: make sd (and thus m) as large as possible (makes no sense for Normal)

C.I.s for proportions Solution 2: “Conservative” Idea: make sd (and thus m) as large as possible (makes no sense for Normal) zeros at 0 & 1

C.I.s for proportions Solution 2: “Conservative” Idea: make sd (and thus m) as large as possible (makes no sense for Normal) zeros at 0 & 1 max at

C.I.s for proportions Solution 1: “Conservative” Can check by calculus so

C.I.s for proportions Solution 1: “Conservative” Can check by calculus so Thus

C.I.s for proportions Solution 1: “Conservative” Can check by calculus so Thus

C.I.s for proportions Example: Old Text Problem 8.8

C.I.s for proportions Example: Old Text Problem 8.8 Power companies spend time and money trimming trees to keep branches from falling on lines.

C.I.s for proportions Example: Old Text Problem 8.8 Power companies spend time and money trimming trees to keep branches from falling on lines. Chemical treatment can stunt tree growth, but too much may kill the tree.

C.I.s for proportions Example: Old Text Problem 8.8 Power companies spend time and money trimming trees to keep branches from falling on lines. Chemical treatment can stunt tree growth, but too much may kill the tree. In an experiment on 216 trees, 41 died.

C.I.s for proportions Example: Old Text Problem 8.8 Power companies spend time and money trimming trees to keep branches from falling on lines. Chemical treatment can stunt tree growth, but too much may kill the tree. In an experiment on 216 trees, 41 died. Give a 99% CI for the proportion expected to die from this treatment.

C.I.s for proportions Example: Old Text Problem 8.8 Solution: Class example 12, part 1

C.I.s for proportions Class e.g. 12, part 1 Sample Size, n

C.I.s for proportions Class e.g. 12, part 1 Sample Size, n Data Count, X

C.I.s for proportions Class e.g. 12, part 1 Sample Size, n Data Count, X Sample Prop., Check Normal Approximation

C.I.s for proportions Class e.g. 12, part 1 Sample Size, n Data Count, X Sample Prop., Check Normal Approximation

C.I.s for proportions Class e.g. 12, part 1 Sample Size, n Data Count, X Sample Prop., Best Guess Margin of Error

C.I.s for proportions Class e.g. 12, part 1 Sample Size, n Data Count, X Sample Prop., Best Guess Margin of Error

C.I.s for proportions Class e.g. 12, part 1 Sample Size, n Data Count, X Sample Prop., Best Guess Margin of Error (Recall 99% level & 2 tails…)

C.I.s for proportions Class e.g. 12, part 1 Sample Size, n Data Count, X Sample Prop., Best Guess Margin of Error Conservative Margin of Error

C.I.s for proportions Class e.g. 12, part 1 Best Guess CI: [0.121, 0.259]

C.I.s for proportions Class e.g. 12, part 1 Best Guess CI: [0.121, 0.259] Conservative CI: [0.102, 0.277]

C.I.s for proportions Example: Old Text Problem 8.8 Solution: Class example 12, part 1 Note: Conservative is bigger

C.I.s for proportions Example: Old Text Problem 8.8 Solution: Class example 12, part 1 Note: Conservative is bigger Since

C.I.s for proportions Example: Old Text Problem 8.8 Solution: Class example 12, part 1 Note: Conservative is bigger Since Big gap

C.I.s for proportions Example: Old Text Problem 8.8 Solution: Class example 12, part 1 Note: Conservative is bigger Since Big gap So may pay substantial price for being “safe”

C.I.s for proportions HW: 8.7 Do both best-guess and conservative CIs: 8.11, 8.13a, 8.19