Stat 155, Section 2, Last Time

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Stat 155, Section 2, Last Time

• Binomial Distribution
• Normal Approximation
• Continuity Correction
• Proportions (different scale from counts)
• Distribution of Sample Means
• Law of Averages, Part 1
• Normal Data ? Normal Mean
• Law of Averages, Part 2
• Everything (averaged) ? Normal

2
Reading In Textbook

• Approximate Reading for Todays Material
• Pages 382-396, 400-416
• Approximate Reading for Next Class
• Pages 425-428, 431-439

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Chapter 6 Statistical Inference

• Main Idea
• Form conclusions by
• quantifying uncertainty
• (will study several approaches,
• first is)

4
Section 6.1 Confidence Intervals

• Background
• The sample mean, , is an estimate of
  the population mean,
• How accurate?
• (there is variability, how much?)

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Confidence Intervals

• Recall the Sampling Distribution
• (maybe an approximation)

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Confidence Intervals

• Thus understand error as
• How to explain to untrained consumers?
• (who dont know randomness,
• distributions, normal curves)

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Confidence Intervals

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

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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 distn
• So need to specify desired accuracy

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Confidence Intervals

• HW 6.1

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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

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Confidence Intervals

• E.g. For confidence level 0.95, want
• 
  distribution
• 0.95 Area
• margin of
  error

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Confidence Intervals

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

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Confidence Intervals

• So adapt needed probs to lower areas.
• When inner area 0.95,
• Right tail 0.025
• Shaded Area 0.975
• So need to compute

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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

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Confidence Intervals

• Approach to unknown
• Recenter, i.e. look at distn
• Key concept
• Centered at 0
• Now can calculate as

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Confidence Intervals

• Computation of
• Smaller Problem Dont know
• Approach 1 Estimate with
• Leads to complications
• Will study later
• Approach 2 Sometimes know

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Confidence Intervals
138
139.1
113
132.5
140.7
109.7
118.9
134.8
109.6
127.3
115.6
130.4
130.2
111.7
105.5

• 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

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Confidence Intervals

• E.g. Find
• The 90 Confidence Interval for the mean value
  , for this type of corn.
• The 95 Confidence Interval.
• The 99 Confidence Interval.
• How do the CIs change as the confidence level
  increases?
• Solution, part 1 of
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg22.xls

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Confidence Intervals

• An EXCEL shortcut
• CONFIDENCE
• Careful parameter is
• 2 tailed outer area
• So for level 0.90, 0.10

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Confidence Intervals

• HW 6.5, 6.9, 6.13, 6.15, 6.19

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Choice of Sample Size

• Additional use of margin of error idea
• Background distributions
• Small n
  Large n

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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

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Choice of Sample Size

• Given m, how do we find n?
• Solve for n (the equation)

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Choice of Sample Size

• Graphically, find m so that
• Area 0.95 Area
  0.975

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Choice of Sample Size

• Thus solve

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Choice of Sample Size

• Numerical fine points
• Change this for coverage prob. ? 0.95
• Round decimals upwards,
• To be sure of desired coverage

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Choice of Sample Size

• EXCEL Implementation
• Class Example 22, Part 2
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg22.xls
• HW 6.22 (1945), 6.23

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Interpretation of Conf. Intervals

• 2 Equivalent Views
• Distribution
  Distribution
• 95
• pic 1
  pic 2

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Interpretation of Conf. Intervals

• Mathematically
• pic 1
  pic 2
• no pic

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Interpretation of Conf. Intervals

• Frequentist View If repeat the experiment
  many times,
• About 95 of the time, CI will contain
• (and 5 of the time it wont)

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Confidence Intervals

• Nice Illustration
• Publishers Website
• Statistical Applets
• Confidence Intervals
• Shows proper interpretation
• If repeat drawing the sample
• Interval will cover truth 95 of time

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Interpretation of Conf. Intervals

• Revisit Class Example 17
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg17.xls
• Recall Class HW
• Estimate of Male Students at UNC
• C.I. View Class Example 23
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg23.xls
• Illustrates idea
• CI should cover 95 of time

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Interpretation of Conf. Intervals

• Class Example 23
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg23.xls
• Q1 SD too small ? Too many cover
• Q2 SD too big ? Too few cover
• Q3 Big Bias ? Too few cover
• Q4 Good sampling ? About right
• Q5 Simulated Bi ? Shows natural varn

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Interpretation of Conf. Intervals

• HW 6.27, 6.29, 6.31

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And now for somethingcompletely different.

• A fun dance video
• http//ebaumsworld.com/2006/07/robotdance.html
• Suggested by David Moltz

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Sec. 6.2 Tests of Significance

• Hypothesis Tests
• Big Picture View
• Another way of handling random error
• I.e. a different view point
• Idea Answer yes or no questions, under
  uncertainty
• (e.g. from sampling or measurement error)

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Hypothesis Tests

• Some Examples
• Will Candidate A win the election?
• Does smoking cause cancer?
• Is Brand X better than Brand Y?
• Is a drug effective?
• Is a proposed new business strategy effective?
• (marketing research focuses on this)

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Hypothesis Tests

• E.g. A fast food chain currently brings in
  profits of 20,000 per store, per day. A new
  menu is proposed. Would it be more profitable?
• Test Have 10 stores (randomly selected!) try
  the new menu, let average of their daily
  profits.

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Fast Food Business Example

• Simplest View for
• new menu looks better.
• Otherwise looks worse.
• Problem New menu might be no better (or even
  worse), but could have
• by bad luck of sampling
• (only sample of size 10)

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Fast Food Business Example

• Problem How to handle quantify gray area in
  these decisions.
• Note Can never make a definite conclusion e.g.
  as in Mathematics,
• Statistics is more about real life
• (E.g. even if or
  , that might be bad luck of sampling, although
  very unlikely)

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Hypothesis Testing

• Note Can never make a definite conclusion,
• Instead measure strength of evidence.
• Approach I (note different from text)
• Choose among 3 Hypotheses
• H Strong evidence new menu is better
• H0 Evidence is inconclusive
• H- Strong evidence new menu is worse

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Caution!!!

• Not following text right now
• This part of course can be slippery
• I am breaking this down to basics
• Easier to understand
• (If you pay careful attention)
• Will tie things together later
• And return to textbook approach later

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Hypothesis Testing

• Terminology
• H0 is called null hypothesis
• Setup H, H0, H- are in terms of parameters,
  i.e. population quantities
• (recall population vs. sample)

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Fast Food Business Example

• E.g. Let true (over all stores) daily
  profit from new menu.
• H (new is
  better)
• H0 (about the
  same)
• H- (new is worse)

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Fast Food Business Example

• Base decision on best guess
• Will quantify strength of the evidence using
  probability distribution of
• E.g. ? Choose H
• ? Choose
  H0
• ? Choose
  H-

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Fast Food Business Example

• How to draw line?
• (There are many ways,
• here is traditional approach)
• Insist that H (or H-) show strong evidence
• I.e. They get burden of proof
• (Note one way of solving
• gray area problem)

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Fast Food Business Example

• Assess strength of evidence by asking
• How strange is observed value ,
• assuming H0 is true?
• In particular, use tails of H0 distribution as
  measure of strength of evidence

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Fast Food Business Example

• Use tails of H0 distribution as measure of
  strength of evidence
• 
  distribution
• 
  under H0
• observed
  value of
• Use this probability to measure
• strength of evidence

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Hypothesis Testing

• Define the p-value, for either H or H-, as
• Pwhat was seen, or more conclusive H0
• Note 1 small p-value ? strong evidence
  against H0, i.e. for H (or H-)
• Note 2 p-value is also called observed
  significance level.

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Fast Food Business Example

• Suppose observe ,
• based on
• Note , but is this
  conclusive?
• or could this be due to natural sampling
  variation?
• (i.e. do we risk losing money from new menu?)

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Fast Food Business Example

• Assess evidence for H by
• H p-value Area

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Fast Food Business Example

• Computation in EXCEL
• Class Example 22, Part 1
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg24.xls
• P-value 0.094.
• 1 in 10, could be random variation,
• not very strong evidence