Math 125
Statistics
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1.
300 is what % of 2,000?
2.
100,000 families -> 0.1 of 1% of these have income greater than
75K. How many families is that?
3.
There are 100 millions eligible voters in the US. The Gallup poll
interviews 5,000 of them. This is equivalent to 1 out of every … ?
4.
In the US (hypothetically), 1 in 500 is in the Army and 3 in 10,000
are officers. What % of Army personnel are officers?
5.
1 out of 1500 people is a Marine and 1/10 of Marines are officers.
What % of population are officers in the Marine?
6.
Without calculator: Sqrt(100,000) is closest to: 30, 100, 300,
1000?
7.
8.
Is Sqrt(0.5) < 0.5 ?
Is Sqrt(2) < 2?
9. Solve for x & y
x + 3y = 1
2x + y = -3
10.
11.
A quart of Vodka is 40% alcohol. A drink is made of OJ and
Vodka (V quarts of Vodka for J quarts of OJ). What is the %
alcohol in the drink? (formula function of V and J)
Tom's mother is 3 times as old as he is. Next year, their ages will
add up to 50. How old is Tom?
12.
Probability:
Throw one die. What is the chance of getting a 1?
Throw two dice. What is the chance of getting two 1's?
13.
Consider two situations:
A) a Coin is tossed 100 times. If it comes up heads 60 times or
more you win.
B) It is tossed 1,000 times. If it comes up heads 600 times or
more, you win. Which is better A or B?
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Statistics: Science of assembling, organizing,
and analyzing data
▪ First gather data (available or to be sampled)
▪ Then analyze the data (graphs, numerical analysis)

Probability theory:
▪ Distributions
▪ Making predictions: Inference
▪ Assessing prediction accuracy for decision making:
Hypothesis testing
Data Analysis
The goal of statistics is to gain information from the data.
First: Data are collected.
Data come from several sources:
1.
Available data:
Census data, Federal agencies, Governmental Statistical Offices.
•
Example: Census Bureau, Triola textbook Datasets, DePaul QRC
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Many more databases are available on the Internet!
2.
New Data:
•
Sampling from population of interest: Observational studies
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Conducting statistical experiments: medical trials, controlled
experiments. When well designed, provide most reliable source of
information!!
Next step after data collection?

Long listings of data are of little value.

Statistical methods come to help us.

Data analysis: methods to display and summarize the data.

In this course we will deal with data on one variable at a time.
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The distribution of the observations is analyzed by
1)
Displaying the data in a graph that shows overall patterns and
unusual observations (histogram, box plot, density curve…)
2)
Computing number statistics that summarize specific aspects of the
data (center and spread).
In election times, politicians want to know the percent of people that will
support them.
During the elections, news and political parties want to know the percent of
votes even before the polls are closed.
Market research firms collects data to study the efficacy of an advertisement,
or the preference of the public towards certain brands or products (ex. Nielsen).
A new Web company wants to know the average age, interests or educational
level of their “visitors”.
WinZip computing, Inc. wants to know the average number of people who
download the freeware version of WinZip and then buy the software.
Common to all these cases:
Investigators want to gather information about a large group of
individuals. This is called the population.
Problem: often not feasible because of
1) time constraints,
2) costs,
3) inconvenience
Solution: Select a part of the population and collect data from this
part. Such a subset is called a sample.
Data collected in the sample are analyzed for drawing conclusions
about the entire population.
In each study, generally there are some numerical facts about the
population that we want to know about.
Examples:
1) at exit polls, two possible quantities are
• the percentage of voters for a certain party;
• the percentage of registered voters that have cast their vote.
2) In Internet surveys:
• average number of visitors to a web site;
• average length of time of each visit;
• average number of sites visited before purchasing a product online;
• other examples?
These quantities are called parameters. Parameters are estimated by
statistics, the numbers calculated from the sample.
NEW YORK, NY – October 11, 2001
A recent study in September 2001 from Nielsen / NetRatings reveals that nearly
56% of the office workers consumed streaming media (live video broadcasts and
instant coverage of the news) in September.
“More than 21 million office workers streamed the web media in September”
“…growing market for Real Networks, Windows Media and Quick Time”.
Did they interview 21 million office workers?
How did they get this result?
Parameter: a numerical measurement describing some
characteristic of a population.
Statistic: a numerical measurement describing some
characteristic of a sample from the population.
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Reliable Data
Since you estimate the parameters from the sample, the sample
must represent the population!
Too often results from surveys are not correct, do you still trust
the exit polls?
This is because the sample is not representative of the
population!!
Selecting a “good” sample
Draw a representative sample by selecting the individuals at random,
like placing the names in a hat and drawing out a handful (the sample)!
A simple random sample of size n is a sample of n individuals from the
chosen population. Each individual in the sample has the same chance to
be selected.
So any sample is chosen using an unbiased impartial method!
If the population is very large, a multistage cluster sampling is often
used. The population is divided into classes or clusters and a simple
random sample is drawn from randomly selected clusters.
Example: household surveys, Nielsen TV preference data,
USA territory is divided into states, towns, wards.
Section 1-2
Types of Data
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Definitions
Quantitative data: numbers representing counts or measurements.
Examples: Weights of athletes, income, gallons of gas pumped, etc.
Qualitative (or categorical) (or attribute) data: can be separated
into different categories that are distinguished by some
nonnumeric characteristic
Example: Sex (male/female), Religious affiliation, “Likes pizza” or not, etc.
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Working with Quantitative Data
Quantitative data can further be
described by distinguishing between:
• discrete and
• continuous.
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Definition

Discrete data
number of possible values is a naturally
‘countable’ number
(i.e. : 0, 1, 2, 3, . . .)
Example: The number of eggs that a hen
lays.
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Definition

Continuous data
infinitely many possible values with continuous
scale coevring a range of values without gaps,
interruptions, or jumps.
Example: The amount of milk that a cow
produces; (example: 2.343115 gallons per day)
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Section 1-3
Critical Thinking
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
Success in the introductory statistics course
typically requires more common sense than
mathematical expertise!

This section is designed to illustrate how
common sense is used when we think critically
about data and statistics.
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Misuse # 1- Bad Sample
Such as:
 Voluntary response sample
(or self-selected sample)
one in which the respondents themselves decide whether to be
included.
In this case, valid conclusions can be made only about the
specific group of people who agree to participate.
Misuse # 2- Sample too Small
Example: Basing a school suspension rate on a sample of only
three students.
Misuse # 3- Inadequate Graphs
You should analyze the
numerical information
not just the shape!
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Misuse # 4- Exaggerated Pictographs
Wrong to multiply
ALL the dimensions!
Misuse # 5- Abusing Percentages
Misleading or unclear percentages are sometimes used. For
example, if you take 100% of a quantity, you take it all. 110% of
an effort does not make sense.
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Other Misuses of Statistics
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Loaded Questions
Order of Questions
Refusals
Correlation & Causality
Self Interest Study
Precise Numbers
Partial Pictures
Deliberate Distortions
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Section 1-4
Design of Experiments
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Key Concept
If sample data are not collected in an appropriate way, the
data may be so completely useless that no amount of
statistical analysis can fix them.
Definitions:
1- Observational studies: observations or measurements are
made, without interference with the subjects (interference can
modify the results).
2- Controlled Experiments: a treatment is applied to the
subjects (experimental units). The effects are then measured
and analyzed.
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Data can be collected in different ways:
 Cross sectional study
data are observed, measured, and collected at one
point in time
 Retrospective (or case control) study
data are collected from the past by going back in time
 Prospective (or longitudinal or cohort) study
data are collected in the future from groups (called
cohorts) sharing common factors
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A possible problem in collecting data:
 Confounding factor: occurs in an experiment when the experimenter
is not able to distinguish between the effects of different factors.
In an observational study, we do not assign subjects to the treatment
and the control groups. We can only observe subjects with a given
condition (treatment group) and subjects without it (control group).
The problem is that often the treatment/condition is badly biased by
confounding factors, that influenced the outcome of the study.
Hidden confounding factors = unobserved variables that affect the
responses being studied.
Example:
Studies on smoking are observational, they compare smokers to non-smokers.
Even though they show evident association between smoking and several
diseases (lung cancer, coronary disease), they do not completely reject the
possibility that a confounding unobserved variable might be the actual cause of
the observed association. For instance a genetic factor that makes people more
likely to smoke and contract the disease.
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In an observational study, a confounding factor can sometimes be controlled.
Block design: Divide the population into groups of similar individuals called
blocks. Randomly assign units to treatments for each block.
Example: Effect of a new treatment against hypertension.
The blocks are: age, sex and smoking habits
General
population
<50 years old
Men
>=50 years old
Women
smokers
smokers
Not smokers
Men
smokers
Not smokers
Women
smokers
Not smokers
Not smokers
More ways to Control the Effects of Variables:
 In addition to Forming Blocks: grouping subjects with similar
characteristics, also:
Blinding: subject does not know he or she is receiving a treatment
or placebo
 Double Blinding: subject and experimenter do not know who is
receiving a treatment or placebo. (ex FDA etc…)
Completely Randomized Experimental Design: subjects are put
into blocks through a process of random selection.
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More definitions (data collection terms):
 Replication: repetition of an experiment when there are enough subjects
to recognize the differences from different treatments.
Sample Size use a sample size that is large enough to see the true nature
of any effects and obtain that sample using an appropriate method, such as
one based on randomness.
Random Sample: members of the population are selected in such a way
that each individual member has an equal chance of being selected.
Simple Random Sample (of size n): subjects selected in such a way that
every possible sample of the same size n has the same chance of being
chosen.
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Methods of Sampling:
Random Sampling
selection so that each
individual member has an
equal chance of being selected.
Stratified Sampling
subdivide the population into at
least two different subgroups that share the same characteristics,
then draw a sample from each subgroup (or stratum)
Systematic Sampling
Select some starting point and then
select every k th element in the population.
Convenience Sampling
use results that are easy to get.
Cluster Sampling
divide the population into sections
(or clusters); randomly select some of those clusters;
choose all members from selected clusters
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Methods of Sampling - Summary
 Random
 Systematic
 Convenience
 Stratified
 Cluster
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How good are the data?

Sampling error: = difference between a sample result and the true
population result;
Cause: chance fluctuations in the sample.

Nonsampling error: or systematic error.
Causes:


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data incorrectly collected (ex: a biased sample)
data incorrectly recorded (ex: defective instrument)
data incorrectly analyzed (ex: wrong calculation, error in formula)
The Statistic will differ from the Parameter according to:
statistic value)= parameter + bias + chance error
▪
▪
Chance error = sampling error
Bias = systematic error (depends on the way the sample was collected)
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