Communicating Quantitative Information

InflationElection district

Polling, predictions, confidence intervals, margin of error

Homework: Identify topic for Project 1. Postings. Prepare for Midterm

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

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 old—need to multiply by something that increases…..

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

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

What is meant by Grade Inflation?

• ?

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

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

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), dimensionDimension: may be axis in graph

gapminder uses color, size of 'dot', and timingNapoleon matching to/from Moscow: color, thickness

of line, geography, temperature

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?

New topic(s)

• Measurement

• Polling and sampling

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, …

Source

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

Caution

• Procedures (formulas) presented without proof, though, hopefully, motivated

• Go over process different ways

• Next class: models of population, subpopulations in sample

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

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

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

Two approaches

• Estimating with confidence intervalc in general population based on proportionphat

in sample

• Hypothesis testing:H0 (null hypothesis) p = p0 versusHa p > p0

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.

Statement

• I'm 95% sure that the actual proportion is in the following range….

• phat – m <= p <= phat + m

• Notice: if you want to claim more confidence, you need to make the margin bigger.

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 are:the 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%.)

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

Actual (mathematical) process• Can use these techniques

when n*p>=5 and n*(1-p)>=5• 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 <= (p-phat)/sd(p) <=1.96)

n

pp )1(*

Sample size must be this big

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

n

pp )1(*

Notes

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

n

pp )1(*

Formula

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

Level of confidence

1-leg or 2-leg Standard deviations (z-score)

80% .10 or .20 1.28

90% .05 or .10 1.64

95% .025 or .05 1.96

99% .005 or .01 2.58

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%

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.

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

99% confidence means

• [Give fraction positive]

• [Give fraction negative]

Why

• Confidence intervals given mainly for 95% and 99%??

• History, tradition, doing others required more computing….

Let's ask a question

• How many of you watched the last Super Bowl? World Cup? – Sample is whole class

• How many registered to vote?– Sample size is number in class 18 and older

• ????

Excel: columns A & B

students

watchers

psample =B2/B1

sd =SQRT(B3*(1-B3)/B1)

Ztransform for 95% =1.96

margin =B5*B4

lower =MAX(0,B3-B6)

upper =MIN(B3+B6,1)

Variation of book problem

• Say sample was 300 (not 1000).• sd(phat) = square_root ((.55)*(.45)/300)

= .0287Bigger 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 .606

You/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!

Divisor smaller

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!

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 >= phat – m and <= phat + m

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.

Hypothesis testing

• Null hypothesisp = p0

• Alternate hypothesisp > 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!)

Hypothesis testing, continued

• Test statistic is

Z = (.55-50)/sqrt(.5*.5)/sqrt(1000)

= 3.16

Use Excel =1-normsdist(3.16)

P(z>=3.16) = .0008

Reject Null hypothesis. Chances are .0008 that it is true (that p = p0)

n

pp

pp

)1(

ˆ

00

0

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!!!

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/

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 18• Presentation and project 1 paper due Nov. 4

• (Guide to midterm is on-line. Reviewing will assume you have studied the guide.)