Transcript MAT 1000 - Wayne State University

MAT 1000
Mathematics in Today's World
Last Time
1. Two types of observational study
2. Three methods for choosing a sample
Last Time
Population: the collection of all individuals being
studied
Census: an observational study that observes the
entire population
Sample survey: an observational study that only
observes some of the population
Sample: the group of individuals chosen in a sample
survey
Last Time
Methods of choosing a sample
Voluntary response sample: the individuals in the
sample volunteer to participate
Convenience sample: the individuals in the sample
are chosen because they are the easiest to reach
Simple random sample: every group in the population
has the same chance of being the sample
Today
1. What does a sample tell us about the
population?
2. Practical problems in sample surveys.
What do samples tell us?
Example
Suppose we are studying the ages of Wayne State
students.
Structure of the data:
Individuals: WSU students
Variable: Age
Population: all WSU students
What do samples tell us?
Example
We could try and determine the age of every
individual in the population, but this is time
consuming and expensive.
Better to use a sample.
What do samples tell us?
Example
We hope the mix of ages of students in our sample
gives a good reflection of the mix of ages of all WSU
students.
With a good sampling method (like SRS), we can be
reasonably sure this is the case.
What happens after we collect data on our sample?
What do samples tell us?
Example
That data will be a big list of numbers—one age for
each student in the sample.
A big list of numbers is not very informative. Better to
try and summarize.
One way to summarize a list of numbers is to take the
average (mean).
If our sample is good, the average age of the students
in our sample ought to be close to the average age of
all WSU students.
What do samples tell us?
The goal in a sample survey is to use information
about a sample to “infer” information about the
entire population.
The word “infer” means “derive” or “estimate.”
Here “information” refers to some kind of numbers
(“average age” in the example).
So we can restate the goal of a sample survey
slightly…
What do samples tell us?
The goal in a sample survey is to use numbers that
describe a sample to estimate the value of a number
that describes the entire population.
Numbers that describe a sample are called statistics.
Numbers that describes a population are called
parameters.
In the example, the average age of all WSU students is
the parameter, and the statistic is the average age of
the students in our sample.
What do samples tell us?
One common type of sample survey is the opinion
poll.
Example
On November 4th there will be an election for the
Governor of Michigan. A recent poll of 750 Michigan
voters found that 45% would vote for Rick Snyder (R)
and 42% for Mark Schauer (D).
What do samples tell us?
Population: all Michigan voters
Variable: which candidate a person will vote for
Unlike “age,” the variable here is not numeric. So we
can’t summarize the survey results with an average.
Instead we use a proportion (percent).
Statistic: proportion of the sample who support Rick
Snyder
Parameter: proportion of the population who support
Rick Snyder (unknown)
What do samples tell us?
Goal: make a good estimate of the parameter
The sampling method we use matters.
Convenience sample: ask WSU professors who they
plan to vote for
Voluntary response sample: Rush Limbaugh asks
listeners to call in
Will these samples give a good representation of the
population of all Michigan voters?
What do samples tell us?
We say that convenience sampling and voluntary
response sampling are biased sampling methods.
The samples we get using these methods are probably
not representative of the population.
Bias: in repeated samples, the sample statistic
consistently misses the population parameter in the
same direction
To avoid bias, use a simple random sample.
What do samples tell us?
If we use a simple random sample of 750 Michigan
voters, every group of 750 people has the same
chance of being chosen.
Could we just by chance get a simple random sample
of 750 WSU professors?
Sure.
We can never know for sure that our sample
represents the whole population. If we use an SRS, the
best we can say is that we are confident that our
sample represents the whole population.
What do samples tell us?
A good way to understand this: think of taking a SRS
many many times.
Each time we get a different sample.
In each sample the proportion of people who plan on
voting for Rick Snyder will be slightly different.
So we get different statistics.
But, with an SRS all of these different statistics should
be close to each other, and to the parameter.
What do samples tell us?
For any sampling method we use, there will be some
variability in the statistics we get.
Variability: different samples from the same
population may yield different values of the sample
statistic
Variability and bias are two different ways a sample
can fail to represent the population.
Bias and Variability
Consider shooting arrows (statistics) at a target (the parameter):
Bias means the archer systematically misses in the same direction.
Variability means that the arrows are scattered.
What do samples tell us?
To reduce bias, use random sampling.
To reduce variability, use larger samples.
What do samples tell us?
In a sample survey the discrepancy between the
statistic and the parameter is called “error” (doesn’t
mean mistake).
What are some sources of error?
• Bias
• Variability
These are called “sampling errors.”
What do samples tell us?
There are other types of errors in a sample survey
called “nonsampling errors.”
Processing errors: data entry, calculations
Response error: individuals answer incorrectly (lie or
misremember)
Even the wording of questions can be a source of
error!
Concerns when Asking Survey
Questions
• Deliberate bias
• Unintentional bias
• Desire to please
• Asking the uninformed
• Unnecessary complexity
• Ordering of questions
• Confidentiality and anonymity
Deliberate Bias
• “If you found a wallet with $20 in it, would you
return the money?”
• “If you found a wallet with $20 in it, would you
do the right thing and return the money?”
Unintentional Bias
• “I have taught several students over the
past few years.”
o How many students do you think I have taught?
o How many years am I referring to?
• “Over the past few days, how many
servings of fruit have you eaten?”
o How many days are you considering?
o What constitutes a serving?
Desire to Please
• “Is your instructor doing a good job
presenting the course material in a clear and
interesting way?”
 Yes
 No
Asking the Uninformed:
Case Study
Washington Post National Weekly Edition (April 10-16, 1995, p. 36)
• A 1978 poll done in Cincinnati
asked people whether they
“favored or opposed repealing the
1975 Public Affairs Act.”
o There was no such act!
o About one third of those asked expressed an
opinion about it.
Unnecessary Complexity
• “Do you sometimes find that you have
arguments with your family members and coworkers?”
o Arguments with family members
o Arguments with co-workers
o “sometimes find” (vague or unclear)
Ordering of Questions
• “How often do you normally go out on a
•
date? about ___ times a month.”
“How happy are you with life in general.”
o Strong association between these questions.
o If the ordering is reversed, then there would be no
strong association between these questions
Confidentiality and Anonymity
• Confidential answer
o respondent is known, but the information is a secret
o facilitates follow-up studies
• Anonymous answer
o the respondent is not known, or cannot be linked to
his/her response
o usually yields more truthful answers