Communicating Quantitative Information

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Communicating Quantitative Information

• Inflation
• Election district
• Polling, predictions, confidence intervals,
  margin of error
• Homework Identify topic for Project 1. Postings.
  Prepare for Midterm

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Inflation

• is when goods and services cost more over time
• money is worth less
• Government agencies do the analysis on a
  'shopping cart' of goods and services and
  calculates (and publishes) a number
• If annual inflation is 2 .02 , it means that
  something that cost 100 last year would cost
  102 this year (on average)
• old_cost (1 inflation_rate) is the
  new_cost

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Hint

• Need to change the percentage into a fraction
• 2 becomes .02
• Need to add 1
• Multiply old by 1.02
• Hint if inflation is positive (if goods and
  services are increasing in price), then new must
  be more than oldneed to multiply by something
  that increases..

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Exercises

• If inflation is 4, what would new prices be for
  something
• 50
• 10
• If inflation is 12, what would new prices be for
  something
• 50
• 10

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History

• Mostly, there is inflation, though deflation is
  possible (and generally not good for economy)
• Central banks ('the fed') try to regulate
  inflation by changes in the interest rates
• Calculation is complex
• Consider computers
• digital cameras

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What is meant by Grade Inflation?

• ?

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Dental expenses

• Yes, expenses have gone up, but have they gone up
  faster than inflation, that is, faster than
  everything
• Look at the graph
• Gray line versus blue line
• NOTE both are increases

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Pie chart versus Bar graph

• Pie is to show parts of a whole
• For example, different categories of spending
• Bar graphs can show categories, also.
• Better than pie charts if categories are not
  everything
• Bar graphs good for showing different time
  periods
• Horizontal (x-axis) typically holds the time
• Clustered bar good for comparisons
• Stacked bar good for parts of a whole

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On graphs

• Graphs and diagrams are for showing context.
  Telling a story (the relevant story)
• Complexity is okay
• Want to encourage AND reward study
• Remember definitions, denominator, distribution,
  difference (context), dimension
• Dimension may be axis in graph
• gapminder uses color, size of 'dot', and timing
• Napoleon matching to/from Moscow color,
  thickness of line, geography, temperature

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On re-districting

• One technique is to concentrate known voters of
  one type to remove from other districts
• Are voters so predictable?
• Do the qualities of the individual
  representatives count?

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New topic(s)

• Measurement
• Polling and sampling

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Measurements

• Measuring something can require defining a system
  / process
• Competitive figure skating
• operational definition
• likely voter
• someone who voted in x of last general elections
  and/or y of primaries
• And knows the voting place
• Fixed place and time
• For surveys answered a specific question in the
  context of other questions,

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Source

• The Cartoon guide to Statistics by Larry Gonick
  and Woollcott SmithHarperResource

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Caution

• Procedures (formulas) presented without proof,
  though, hopefully, motivated
• Go over process different ways
• Next class models of population, subpopulations
  in sample

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Task

• Want to know the percentage (proportion) of some
  large group
• adults in USA
• television viewers
• web users
• For a particular thing
• think the president is doing a good job
• watched specific program
• viewed specific commercial
• visited specific website

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Strategy Sampling

• Ask a small group
• phone
• solicitation at a mall
• other?
• Monitor actions of a small group, group defined
  for this purpose
• Monitor actions of a panel chosen ahead of time

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Quality of sample

• Recall discussion on students who 'took the bait'
  to take special survey
• More on quality of sample later
• More on adjusting data from panel for statement
  about total population later

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Two approaches

• Estimating with confidence intervalc in general
  population based on proportionphatin sample
• Hypothesis testingH0 (null hypothesis) p p0
  versusHa p gt p0

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Estimation process

• Construct a sample of size n and determine phat
• Ask who they are voting for (for now, let this
  be binomial choice)
• Use this as estimate for actual proportion p.
• but the estimate has a margin of error. This
  means The actual value is within a range
  centered at phat UNLESS the sample was really
  strange.
• The confidence value specifies what the chances
  are of the sample being that strange.

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Statement

• I'm 95 sure that the actual proportion is in the
  following range.
• phat m lt p lt phat m
• Notice if you want to claim more confidence, you
  need to make the margin bigger.

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Image from Cartoon book

• You are standing behind a target.
• An arrow is shot at the target, at a specific
  point in the target. The arrow comes through to
  your side.
• You draw a circle (more complex than/- error)
  and sayChances arethe target point is inthis
  circle unless shooterwas 'way off' . Shooter
  would only be way off X percent of the
  time.(Typically X is 5 or 1.)

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Mathematical basis

• Samples are themselves normally distributed
• if sample and p satisfy certain conditions.
• Most samples produce phat values that are close
  to the p value of the whole population.
• Only a small number of samples produce values
  that are way off.
• Think of outliers of normal distribution

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Actual (mathematical) process
Sample size must be this big

• Can use these techniques when npgt5 and
  n(1-p)gt5
• The phat values are distributed close to normal
  distribution with standard deviation sd(p)
• Can estimate this using phat in place of p in
  formula!
• Choose the level of confidence you want (again,
  typically 5 or 1). For 5 (95 confident),
  look up (or learn by heart the value 1.96 this
  is the amount of standard deviations such that
  95 of values fall in this area. So .95 is
  P(-1.96 lt (p-phat)/sd(p) lt1.96)

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Notes

• p is less than 1 so (1-p) is positive.
• Margin of error decreases as p varies from .5 in
  either direction. (Check using excel).
• if sample produces a very high (close to 1) or
  very low value (close to 0), p (1-p) gets
  smaller
• (.9)(.1) .09 (.8)(.2) .16, (.6)(.4) .24
  (.5).5).25

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Notes

• Need to quadruple the n to halve the margin of
  error.

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Formula

• Use a value called the z transform
• 95 confidence, the value is 1.96

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Mechanics

• Process is
• Gather data (get phat and n)
• choose confidence level
• Using table, calculate margin of error.
• Book example 55 (.55 of sample of 1000) said
  they backed the politician)
• sd(phat) square_root ((.55)(.45)/1000)
  .0157
• Multiply by z-score (e.g., 1.96 for a 95
  confidence) to get margin of error
• So p is within the range .550 (1.96)(.0157)
  and .550 (1.96)(.0157)
• .519 to .581 or 51.9 to 58.1

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Example, continued

• 51.9 to 58.1
• may round to 52 to 58
• or
• may say 55 plus or minus 3 percent.
• What is typically left out is that there is a
  1/20 chance that the actual value is NOT in this
  range.

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95 confident means

• 95/100 probability that this is true
• 5/100 chance that this is not true
• 5/100 is the same as 1/20 so,
• There is only a 1/20 chance that this is not
  true.
• Only 1/20 truly random samples would give an
  answer that deviated more from the real
• ASSUMING NO INTRINSIC QUALITY PROBLEMS
• ASSUMING IT IS RANDOMLY CHOSEN

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99 confidence means

• Give fraction positive
• Give fraction negative

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Why

• Confidence intervals given mainly for 95 and
  99??
• History, tradition, doing others required more
  computing.

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Let's ask a question

• How many of you watched the last Super Bowl?
• Sample is whole class
• How many registered to vote?
• Sample size is number in class 18 and older
• ????

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Excel columns A B
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Variation of book problem
Divisor smaller

• Say sample was 300 (not 1000).
• sd(phat) square_root ((.55)(.45)/300)
  .0287
• Bigger number. The circle around the arrow is
  larger. The margin is larger because it was
  based on a smaller sample. Multiplying by 1.96
  get .056, subtracting and adding from the .55 get
• .494 to .606You/we are 95 sure that true
  value is in this range.
• Oops may be better, but may be worse. The fact
  that the lower end is below .5 is significant for
  an election!

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Exercise

• Determine / choose / read
• size of sample n
• proportion in sample (phat)
• claimed confidence level (and consult table).
• Hint go back to Mechanics slide and Table slide
  and plug in the numbers!

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Exercise

• size of sample is n
• proportion in sample is phat
• confidence level produces factor called the
  z-score
• Can be anything but common values are 80, 90,
  95, 99)
• Use table. For example, 95 value is 1.96 99
  is 2.58
• Calculate margin of error m
• m zscore sqrt((phat)(1-phat)/n)
• Actual value is gt phat m and lt phat m

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Hypothesis testing

• Pre-election polling
• Repeat example
• Source (again) The Cartoon Guide to Statistics by
  Gonick and Smith
• See also for Jury selection, product inspection,
  etc.

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Hypothesis testing

• Null hypothesis
• p p0
• Alternate hypothesisp gt p0
• Do a test and decide if there is evidence to
  reject the Null hypothesis. (Need more evidence
  to reject than to keep).
• Similar analysis (not giving proof!)

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Hypothesis testing, continued

• Test statistic is
• Z (.55-50)/sqrt(.5.5)/sqrt(1000)
• 3.16
• Use Excel 1-normsdist(3.16)
• P(zgt3.16) .0008
• Reject Null hypothesis. Chances are .0008 that it
  is true (that p p0)

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Project I

• Paper or presentation on news story involving
  mathematics and/or quantitative reasoning
• Involving the audience is good
• Everybody be ready with paper or ready to
  present. Some presentations may go to next class.
• Use multiple sources
• Explain the mathematics!!!

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Ways to get topic

• Topic, assignment in other course that involves
  quantitative information
• Double dipping
• Alternative compare how two different
  newspapers/writers/media treat the same topic.
  There must be real differences.
• Variant (special case) election polling. Talk
  about similarities and differences, perhaps
  definition of 'toss-up', how they describe
  sources,?
• Paulos TV series http//abcnews.go.com/Technology
  /WhosCounting/

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Homework

• Topic for project 1 due by October 20
• You can re-use any topic you or anyone else
  posted
• You can re-use spreadsheet or diagram topics
• You can use topics I suggested
• You can use topics from another class
• YOU MUST post your proposal even if it is a topic
  I suggested.
• Midterm is October 30
• Presentation and project 1 paper due Nov. 6
• (Guide to midterm is on-line. Reviewing will
  assume you have studied the guide.)