# Math W50: How do they know my opinion?
# January, 2000
Monday, January 3
Note: No Class
Tuesday, January 4
Note: No Class
Wednesday, January 5
Note: No Class
Thursday, January 6
Topic: Introduction
Topic: 7 Critical Components @ overheads/seven-critical.shtml
Topic: Measurement
Act: Survey 1 @ data/survey01.shtml
Act: Survey 2 @ data/survey02.txt
Read: Utts 1-3
Vocab:
Vocab: statistics, 7 critical components, %%
individual (unit), variable, value of variable, %%
categorical variable, measurement variable, %%
discrete, continuous, validity, reliabilty, bias, variability
#Note: Class begins 2pm in NH 295
course intro, student intros, why you are in the course
Def of stats
get student ideas (on board)
more than one use of word statistics
descriptive (summary measures)
inferential
Video - FAPP #6 beginning to about 4:00
in diner
some defs (primarily of inferential statistics)
Utts: a collection of procedures and principles for
gaining information in order to make decisions
when faced with uncertainty
Amabile: a way of taming uncertainty, of turning raw data
into arguments that can resolve profound questions
Moore: the science of gaining information from numerical data
Garfunkel: the science of drawing conclusions from data with
the aid of the mathematics of probability
dictionary: the mathematics of the collection, organization,
and interpretation of numerical data, especially the
analysis of a population's characteristics by inference
from sampling
etymology: latin: statisticus - of state affairs
political right down to etymology
key elements
data
uncertainty
information/decision-making ability
science or math?
Video - Against All Odds #1: beginning thru about 13:30, 26:00 to end
[whole thing if time -- but probably not]
Three phases of a statistical study (and outline of course)
1) collect data (statistical design)
2) organize data (data analysis)
3) draw conclusions from data (statistical inference)
Survey 1: Have students add to existing list
collect data from students, have them gather 2 more each
Quiz 0: info about students
-----------
Break
-----------
Fill out survey 1 and survey 2
7 critical components that determine soundness of statistical studies
1) source of funding (why was it done?)
2) researcher contact
3) individuals studied and how selected
4) measurements made (quesitons asked)
5) setting
6) extraneous differences (other explanations for effect)
7) magnitude of claimed effect
example: drug to cure excessive barking in dogs (page 22)
example: US Voters Focus on Selves, Poll says (from Moore S:C&C)
example: most women unhappy in their choice of husbands (page 24)
didn't do this one
How data are organized:
units/individuals/subjects
variables
values of variables
immagine a grid layout
Some terminology
categorical vs. measurement variables
continuous vs. discrete
validity (proxies)
reliability
bias -- systematically off in same direction
variability
pictures of variability/bias possibilities (target)
Some things are not easily or obviously measured:
happiness (happiness newpaper article)
Apply terminology to Survey 1
Summary -- Measuring is a difficult task
example: finding cheapest grocery store
Survey 2: Wording issues
did questions 1 and 2 in class
data: 1) 6-0; 0-6 in expected direction
2) 2-4; 2-4 in expected direction
Wording Pitfalls
Bias (Intentional or unintentional)
Elian Gonzalez
Do you agree that he should be returned to his father in
Cuba? [with US Immigration and Naturalization Service]
Do you agree that he should be allowed to remain with
his relatives in Florida? [agree with boy's attorneys]
Confidentiality and Anonymity (people may lie)
positive AIDS test?
financial and sexual issues
a methods to ask sensitive questions (some answers random)
Desire to Please
"How much do you smoke?" vs. cigarette sales
didn't discuss this particular pair,
put on quiz for tomorrow
Unneccessary complexity, misunderstandings
1992 American Jewish Committee [NY Times July 8, 1994]
Does it seem possible or does it seem impossible to you
that the Nazi extermination of the Jews never happened?
(22% possible)
Does it seem possible to you that the Nazi extermination
of the Jews never happened, or do you feel certain that it
happened? (1% possible)
-------------- only got this far --------------------
"do you own stock?" cartoon [do on monday instead]
Asking the uninformed
will see an example in a video
1975 Public Affairs Act [didn't exist] (page 35)
in 1978 1/3 expressed opinion
in 1995 nearly 1/2 expressed opinion
with leading political bias, 53% expressed opinion
and tended to go with political leanings
Ordering of questions, additional information
peer pressure example (34)
Open/closed questions
museum example
Levi jeans example (page)
Defining your terms
adolescent sex: increasing or decreasing? (page 37)
unemployment
changes have been made to survey questsions
- define week as sunday to saturday so people don't
underreport weekend work
- redesign questions so definitions of 'work', 'looking
for work', 'on layoff' are uniform
- emphasize difference between 'on layoff' and 'fired'
- also split long questions into series of short
questions
"The Truth but not the Whole Truth"
Friday, January 7
Topic: Sampling
Topic: Observation and Experimentation
Act: Random Samples of %%
Circles @ http://www.calvin.edu/~rpruim/cgi-bin/random-digits.cgi
Act: What comes to mind @ data/words.txt
Read: Utts 4
Read: Utts 5
Due: HW #1 @ hw01.shtml
Vocab:
Vocab: observational study, experiment, %%
unit, population, sample, sampling frame, sample survey, %%
census, margin of error, "1 Over Root n Rule", %%
simple random sampling, stratified random sampling, %%
cluster sampling, systematic sampling, %%
random digit dialing, %%
multi-stage sampling, convenience sample, response rate, %%
treatment, explanatory variable, %%
response variable (outcome variable), control, %%
interaction, %%
confounding, placebo, placebo effect, Hawthorne effect, %%
experimenter effect, double-blind, single-blind %%
Basic categories of studies
sample survey -- ask a bunch of people a question
experiment -- looking for relationships, cause/effect
key: treatment
observational study -- looking for relationship, but no treatment
meta-analysis
case study (popular in media -- clip from ABC news?)
First 3 all have something in common: you don't measure every unit
To conduct a study properly
1) get a representative sample
2) get a large enough sample
3) decide between observational study and experiment
Sampling
terminology: unit, population, sample, sampling frame,
sample survey, census,
margin of error
sampling vs. census: samples are often possible, faster, accurate
SRS: sampling circles using random digits
Video -- Against All Odds #14: skip 35:20-41:10
[14 without bead sampling, 21 minutes with]
Sampling Methods -- make sure they understand
SRS
stratified random sampling
cluster sampling
systematic sampling
random-digit dialing
multistage sampling
Sampling difficulties
wrong sampling frame
not reaching selected individuals
low response rate
volunteer sample
haphazard or convenience sample
Literary Digest poll (1936)
Alf Landon predicted to get 3-2 victory
volunteer response to sample from poor frame
George Gallup
Quiz 1
-------------------------------------
Break: do the three words experiment
-------------------------------------
Experiment terminology and set-up
treatment, explanatory variable, response variable, control
can't get cause/effect from observational study alone
individual divided into groups
each group gets different treatment
measurements taken and comparison made between groups
looking for cause/effect: treatment -> response
Video -- FAPP segment on Physicians Health Study
Problems with experiments
Placebo effect
Gastric freezing to relieve ulcer pain
(34% in gf group, 38% in placebo group)
Lack of control, confounding variables -- randomize
1940 propaganda experiment [Germany occupied France]
Interaction -- measure and report possible variables
can turn possible confounding variables into possible
interaction variables by measuring
nicotine patch and smokers at home [didn't mention]
Hawthorne effect -- not always possible to avoid this problem
new curricula
Experimentor Bias -- blindness
Ecological validity/generalizability
what is the population, was setting a factor?
over weekend:
how many words come to mind from 4 others
==
Monday, January 10
Topic: Statistical Summaries
Topic: Distributions
Read: Utts 7
Read: Utts 8
Act: How Many Raisins?
Due: HW #2 @ hw01.shtml
Vocab:
Vocab: mean, median, mode, outlier, range, stemplot, histogram, %%
shape, symmetric, bell-shaped, unimodal, bimodal, skewed, %%
five-number summary, quartile, boxplot, interquartile range, %%
variance, standard deviation, frequency curve, normal curve, %%
proportion, percentile, standardized score, z-score, %%
standard normal distribution, "68-95-99.7 Rule"
Where are we now?
many problems with statistical studies are not mathematical
7 critical components
Pam Plantinga left job because she was told what to find
you can't do good statistics unless you start with good data
individual issues
four issues: validity, reliability, bias, variablility
measurement
must decide what and how to measure -- not always easy
ask about any problems with survey 1
wording
dealing with people is especially hard
do you own stock cartoon
proxies (validity/reliability trade-off)
sampling/assignment issues
good samples are representative and large enough
1/root n rule
experiment vs. observation, the role of treatment
randomness used to reduce bias
moving into a phase of "what do we do with all this data?"
but first ...
... Ethics of experiments
informed consent
use of doctors in physicians health study
kids in art experiment
human subjects & review boards
Stanley Milgram (Yale): shock and memory
done 1960's, probably not doable today
Penny's data collection in grad school
risk: cost/benefit analysis
reasonable hope, reasonable doubt criteria for clinical trials
(did friday)
Some specific examples and issues
Nazi data
give them article (2 versions) and then discuss
from Bouma's class
Yes No
good use 6 2 (1 of 2 struggled)
criticism 5 3
twins studies -- ideal matched pairs?
PHS used only middle-aged men, what about women? minorities
1 in 5 men has heart attack before age 65
1 in 17 women has heart attack before age 65
(did friday)
AIDS and slow process of clinical trials
measuring easier but less reliable things
pressure to release drugs before effectiveness demonstrated
(mentioned friday, didn't mention more here)
domestic violence: warn and release or arrest
can a randomized experiment be done? [no informed consent]
[didn't do here]
Raisins guesses
Looking at the data: stemplots, histograms
choosing bin-size
subdividing stems
Measures of center -- what is a typical value? mean, median, mode
what is unusual? outliers
Measures of spread
range, standard deviation
Five-number summary & boxplots
Some shape descriptions
symmetric, skewed, bell-shaped, unimodal, bimodal
Intro to frequency curves and cummulative probability (proportions)
generalization of 5-number summary
deciles, percentiles from standard tests
-----------------------------------------------
Quiz 2
get data from survey 1, survey 2, raisin counts
break
-----------------------------------------------
Normal distributions
symmetric, bell-shaped, determined by mean and standard dev.
Empirical Rule: 68-95-99.7
standardized scores (z-scores)
charts and computers to get other values (chart on page 137)
examples of approx. normal distributions
height (male 18-74) N(5'9",3") = N(69,3)
height (female 18-74) N(5'3.5",2.5")= N(63.5,2.5)
height (male 18-24) N(5'10",3") = N(70,2.8)
height (female 18-24) N(5'4.3",2.6")= N(64.3,2.6)
height (male 11) N(146cm,8cm) = N(57.5,3.15)
weight (male 18-24) N(162,29.1)
weight (female 18-24) N(134,27)
1994 SAT mean Verbal = 423, 7% above 600, 42% below 400
approx s.d. 120
now renormalized to N(500,100)
what percent score 800?
1987 CA women's salaries mean=11,600; s.d.=10,500
Tuesday, January 11
Topic: More Pictures of Data
Topic: Relationships between Categorical Variables
Topic: Chi-Squared
Read: Utts 9
Due: HW #3 @ hw02.shtml
Vocab: pie chart, bar chart, pictogram, line graph %%, scatter plot
comments on quiz 2
abstraction in mathematics
precise use of language in mathematics
Look at survey 1 data
correct typos, errors, etc.
Common problems with plots, graphs, and pictures
1) missing labels
2) scale doesn't start at 0
3) changes in labeling along an axis
4) misleading units
5) poor information
Picture Checklist
overall impression
1) is message clear?
2) is purpose clear?
10) is there any clutter?
source
3) is source given?
4) is source reliable?
labeling
5) is labeling clear?
6) do axes start at 0?
7) is scale constant?
8) are there any breaks along axis? are they easy to spot?
9) was inflation adjustment made?
Banner chart and follow-up letters
Utts, Figure 9.9 (page 149) and fixed version
Read: Utts 12
Act: Golf Balls in the Yard @ data/golfballs.shtml
Vocab: contingency table, cell, row, column, conditional percentage, %%
rate, test statistic, chi-sqaured statistic, p-value, %%
statistical significance, proportion, odds, relative risk, %%
odds ratio, Simpson's paradox %%
Physician's Health Study data
attack no att. total rate/1000
Aspirin 104 10,933 11,037 9.4
Placebo 189 10,845 11,034 17.1
Total 293 21,778 22,071
Question of the day: why is this data so compelling?
significance: how unusual is this? (chi-squared)
magnitude: how big is this?
Look at data from Survey 2 (in class) and simulate with cards
results: 4-2 split not so unusual, 6-0 split more compelling
Golf ball distribution and test statistics
4-sided dice, computer simulation
Chi-squared statistic (on golf ball data again)
what should we expect if there is no association?
how can we adjust our measurement to account for sample size?
obs exp diff n dif chi sq. P-value cum prob
137 121.5 15.5 1.97737 8.46914 0.0372487 0.962751
138 121.5 16.5 2.24074
107 121.5 -14.5 1.73045
104 121.5 -17.5 2.52058
Return to Physician's Health Survey
significance: do chi squared
P-value: interpretted the same for all statistical tests!
chi-sqared table (degrees of freedom)
magnitude: relative risk
percentage having trait = (# with trait / total #) * (100%)
proportion having trait = (# with trait / total #)
i.e. probablility written as decimal
risk of having trait = # with trait / total #
odds of having trait = # with / # without to 1
= # with to # without
odds against trait = # without / # with to 1
= # without to # with
relative risk: = one risk / other risk
increased risk: = change / original (* 100%)
Misrepresenting risk [covered only implicitly in discussion of PHS]
1) no baseline risk given
2) no time period given
3) unclear population (may not apply to you)
Preview Simpson's Paradox -- Berkeley Admissions example
Wednesday, January 12
Topic: Probability and Randomness
Read: Utts 15
Read: Utts 16
Vocab: probability, relative frequency, personal probability, coherent,
Vocab: mutually exclusive events, independents events,
Vocab: cummulative probability, expected value, four probability rules
Due: HW #4 @ hw02.shtml
Review Chi-Squared
Berkeley admissions (1=men 2=women) (Utts page 221, exercise 14)
video clip -- FAPP #10 2:04:45 -- 2:08:40 [didn't show]
Expected counts are printed below observed counts
accept reject Total
1 450 550 1000
416.67 583.33
2 175 325 500
208.33 291.67
Total 625 875 1500
Chi-Sq = 2.667 + 1.905 +
5.333 + 3.810 = 13.714
DF = 1, P-Value = 0.000
------------------------------------------------
Simpson's Parodox
If we divide by programs applied to, we see a different story
------------------------------------------------
Expected counts are printed below observed counts
acceptA rejectA Total
1 400 250 650
403.45 246.55
2 50 25 75
46.55 28.45
Total 450 275 725
Chi-Sq = 0.029 + 0.048 +
0.255 + 0.418 = 0.751
DF = 1, P-Value = 0.386
------------------------------------------------
Expected counts are printed below observed counts
acceptB rejectB Total
1 50 300 350
79.03 270.97
2 125 300 425
95.97 329.03
Total 175 600 775
Chi-Sq = 10.665 + 3.111 +
8.783 + 2.562 = 25.120
DF = 1, P-Value = 0.000
------------------------------------------------
hospital example (Utts chapter 12, pages 213-215)
give combined results first, then separate
survive die s rate d rate
standard 505 595 .46 .54
new 195 905 .18 .82
total 700 1500
standard 5 95 .05 .95
new 100 900 .10 .90
total 105 995
discrimination example (Utts, chapter 12, pages 215-217) ??
death penalty
326 cases, white defendant: 19/160 get death pen. (.119)
black defendant: 17/166 get death pen. (.102)
when separated by victim's race, see different story
[overhead from Moore 207]
point: statistically significant means that the effect is not
likely to be due to chance alone, but there may be
some other factor than the obvious one that is
reason
Probability
random: long-term predictability vs short-term unpredictability
law of large numbers / "law" of small numbers
scale: 0 to 1 (0% to 100%)
personal vs. mathematical (relative frequency)
4 Rules and applications
axiomatic method
four rules "overhead"
examples
probability of losing luggage is 1/176 (Krantz)
P(heart attack kills) = .33, P(cancer kills) = .2
[assuming death]
estimated probability of grades
probability of two girls (P(boy) about .512)
probability of winning 2 of 3, 3 of 5 given an estimate
for each game
video -- Life By the Numbers (#4 Prob)
02:00 (or 08:20) - 27:00: intro to prob., Graunt, casinos
27:00 - 42:25: polling, polio, prob assesses results
Thursday, January 13
Topic: More Probability
Topic: Sampling Distributions
Act: Sampling Milk Lids
Vocab: expected value, false positive, false negative, %%
gamblers fallacy, statistic, parameter, sampling distribution %%
HW/quiz questions/comments
P-value
observational study & cause/effect
equal portions on normal curve
pictures on HW
Long-term vs. short-term (Free-Throw simulation in Excel)
gambler's fallacy, "law" of small numbers"
Expected Value of Lottery Ticket
insurance
False Positives/False Negatives
data from Utts 303
Quiz 5
Sampling
hands on (milk jug lids)
web simulation
video -- Life by the Numbers (#4)
Read: Utts 17
Read: Utts 18 (Categorical Parts)
Due: HW #5 @ hw03.shtml
Friday, January 14
Topic: Confidence Intervals
Topic: Hypothesis Testing
Act: Colors of Reeses' Pieces
Read: Utts 19
Read: Utts 20 (optional)
Read: Utts 21
Due: HW #6 @ hw04.shtml
Vocab: confidence level, confidence interval, margin of error,%%
hypothesis testing, test statistic,%%
null hypothesis, alternative hypothesis, p-value
Questions on HW/Quiz, etc
bin size for histograms (Old Faithful eruption times)
Nothing new today: just going to put all the pieces together
Two inference tasks
estimating a paramter
1) get sample
2) compute statistic from sample
3) determine the quality of that statistic as an
estimate for paramter
a) confidence level
b) confidence interval, margin of error
testing a hypothesis
Video -- Against All Odds #23 (01:40 - 20:15)
Woburn Leukemia, BLS stats, example computations
Confidence Intervals for proportions
conditions under which the math applies
1) parameter must have a fixed (unknown value) for population
2) simple random sample or repeatable experiment
3) sample includes 5 of each outcome
4) population at least 10 times size of sample
the math:
distribution of sample proportions (statistic) will be
approximately normal N(p,root(p(1-p)/n)
example: suppose fair coin (50% heads)
flip it 100 times
flip it 400 times
flip it 1600 times
for each: ______% of time with _______
______% of time between _____ and _____
example: public opinion poll (suppose 30% rate)
sample size 1500
unknown p, what do we do?
note that (p)(1-p) < .25, so use .25
note that p-hat is usually very close to p, especially if
the sample is large
example: sample 1600 people and 500 people say yes
example: Reeses' pieces
Testing a hypothesis
1) determine null and alternative hypotheses
2) collect data
3) compute test statistic
test stat is a measure of how true the
null hypothesis seems to be
4) determine likelihood of such an extreme test
statistic if null hypothesis is true (p-value)
5) make a decision
Testing Hypotheses for Proportions
test statistic is z-score
example: predicting election outcomes
==
Monday, January 17
Topic: More Inference for Proportions
#Topic: Confidence Intervals
#Topic: Hypothesis Testing for proportions
#Topic: Chi-Squared & Hypothesis Testing
Read: Utts 21
Read: Utts 22
Due: HW #7 @ hw04.shtml
Vocab: one-sided hypothesis test, two-sided hypothesis test,%%
type 1 error, type 2 error, (power of a test)
Answer questions and do examples of inference procedures
200 british couples, 10 with wife taller than husband
61/165 correct in 1 out of 4 esp test
55 quitters out of 120 volunteers assigned to use nicotine patch
24 of 120 placebo users quit
quitting measured after 8 weeks
Hugo -- 11 times in 30 rolls; how unusual is that?
105 times in 300 rolls?
have students roll 30 dice several times and count number
of 6's rolled (work in pairs)
hypothesis testing
ganzfeld experiments
122 successes out of 355 trials
drinking and sex
77 of 404 men, 16 of 138 wemen
report of smoking results above
High smoking cessation rates were observed in the active
nicotine pathc group at 8 weeks (46.7% vs 20%) (P<.001)
and at 1 year (27.5% vs 14.2%) (P = 0.11).
Tuesday, January 18
Topic: More about Inference
Topic: Significance and Importance
Read: Utts 23
Due: HW #8 @ hw05.shtml
Wednesday, January 19
Topic: Risk Assessment
Read: Utts 12.3-12.4
Read: Utts 16
Greatest Risks (problem 11.4 on page 191 of Utts)
have students rank risks of 10 to 30 different items
discuss how to measure relative risk
Act: Video: Are We Scaring Ourselves to Death? @ videos/stossel.shtml
Thursday, January 20
Topic: Test
Friday, January 21
Topic: Work on Projects
Due: Report on video (via email) @ videos/stossel.shtml
==
Monday, January 24
Topic: Consumer Price Index @ overheads/prices.shtml
Due: HW #9 @ hw05.shtml
Vocab: inflation, Consumer Price Index, price index, base year,%%
Index of Leading Economic Indicators
Review of video -- comments and questions
progress report on projects
Consumer Price Index
Break
hand out exams
inference review
example -- Hugo 5/12; how unusual is that?
example -- 100 coin tosses does coin look fair if ...
45 heads
40 heads
35 heads
comments on test 1
Tuesday, January 25
Topic: Time Series
Topic: Wrap-Up
Due: HW #10 @ hw06.shtml
Vocab: time series, long-term trend, seasonal variation,%%
seasonal adjustment, cycle
A look at Calvin Tuition Data
Time series
plot Calvin Tuition Data
plot CPI (with Calvin Price Index?)
births
Dow Jones
postage?
Things to watch for
cherry picking data
choice of units ($ vs inflation adjustments, etc)
vertical axis doesn't start at zero (magnifies steepness)
Wednesday, January 26
Topic: Test
Due: HW #11 @ hw06.shtml
Thursday, January 27
Note: no class
Friday, January 28
Note: no class
== end of calendar