Probability and Statistics Todays Goals

1
Probability and StatisticsTodays Goals

• Understand what constitutes an acceptable sample
  for a scientific study
• Understand the pitfalls in observational studies
• HW 10 (due Wed. April 22) Ch 5 27,68, 69 Plus
  One web problem.
• Article will be due April 24

2
What is statistics?

• Statistics are numbers that summarize the results
  of a study.
• Statistical inference formalizes the process of
  learning through observation.
• Statistics is the field that studies how to
  efficiently collect informative data, explore and
  interpret these data and draw conclusions based
  on them.

3
Statistical Inference

• Designing Experiments
• What data should be collected?
• Sampling questions
• Analyzing data from experiments
• Sample statistics
• Pictorial representations
• Confidence intervals
• Hypothesis testing.
• Regressions for correlation analysis

4
Example 1

• An engineer starting a company has developed a
  new coating for pipes to resist corrosion. He
  believes that it will perform better than
  standard coatings and wants to convince clients
  that the new coating is better.
• He obtains a measure of corrosion by applying new
  coating to one section of the pipe and standard
  coating to a second section of the pipe.
• First pipe (new coating) corrosion free 256 days,
  
• second pipe (standard coating) corrosion free 243
  days.
• Can we conclude that the new coating works better
  than the old coating?
• What would you do differently?

5
What can we do to improve things?

• Collect multiple observations.
• So suppose we collect 100 observations of the
  coating effectiveness. (Controlling for the
  exposure for coated and non-coated)
• We then compute the mean number of corrosion free
  days for both treatments, the new coating and the
  standard coating.
• Suppose now that the mean number of corrosion
  free days for the new coating is 250 and the mean
  number of corrosion free days for the standard
  coating is 242.
• Can we conclude that the coating with the better
  mean effectiveness will definitely outperform the
  coating with the worse mean effectiveness?

6
Example 2

• A parts' supplier tells us that only 1 out of
  every 100 batteries shipped to us will have a
  lifetime of less than 100 hours.
• We are shipped a lot of 100,000 batteries.
• We could look at all of the batteries, but to do
  so would cost us way too much money.
• What might we decide to do?

7
Example 2

• Suppose we decided to randomly sample 200
  batteries.
• And suppose 10 of the 200 batteries have life
  less than 100 hours.
• What can we say to the manufacturer that might
  convince him or her that the shipment we received
  does not meet the specifications?
• In this case, we want to say something like the
  following
• If only one out of every 100 batteries has a
  lifetime of less than 100 hours, then the
  probability that in a random sample of 200
  batteries 10 would have lifetimes less than 100
  hours is very small, say less than 1 in 1000.
• How could we have gotten such a bad sample?

8
Notation

• Population collection of all potential
  observations.
• Sample the subset of the population that is
  actually observed. A collection of observations.
• Observing all units in the population (called
  census) is usually expensive or infeasible.
• Statistical inference extrapolates from a sample
  to the population being sampled.

9
Examples

• Population Diameters of all shafts in a lot.
• Sample Diameters of the shafts that are
  actually measured.
• Population Employment status of all eligible
  adults in the US.
• Sample Employment status of subjects who are
  interviewed.
• Population Lifetimes of the items made by a
  certain manufacturing process.
• Sample Lifetimes of the subset of items tested

10
Inferential Statistics

• Make inferences about a given population based on
  a sample drawn from it. The inferences support
  decisions.
• E.g. Deciding whether or not to accept the lot
  based on the diameters of the sample taken.
• Issues
• The sample must be representative of the
  population.
• Random sample
• The inferences/decisions based on a sample may be
  in error ?Sampling Error
• Quantified by the tools of probability.

11
Exercise 1

• A consumer magazine article asks,
• HOW SAFE IS THE AIR IN AIRPLANES? and then
  says that its study of air quality is based on
  measurements taken on 158 different flights of
  U.S. based airlines.
• Identify the population and the sample.

12
Exercise 2

• For each of the following situations, identify
  the population and the sample and comment on
  whether you think that the sample is or isn't
  representative.
• A member of Congress wants to know what his
  constituents think of proposed legislation on
  health insurance. His staff reports that 228
  letters have been received on the subject, of
  which 193 oppose the legislation.
• A machinery manufacturer purchases voltage
  regulators from a supplier.There are reports that
  variation in the output voltage of the regulators
  is affecting the performance of the finished
  products. To assess the quality of the supplier's
  production, the manufacturer subjects a sample of
  5 regulators from the last shipment to careful
  laboratory analysis.

13
Exercise 2

• For each of the following situations, identify
  the population and the sample and comment on
  whether you think that the sample is or isn't
  representative.
• A member of Congress wants to know what his
  constituents think of proposed legislation on
  health insurance. His staff reports that 228
  letters have been received on the subject, of
  which 193 oppose the legislation.
• True if the sample is representative
• False if not

14
Exercise 2

• For each of the following situations, identify
  the population and the sample and comment on
  whether you think that the sample is or isn't
  representative.
• A machinery manufacturer purchases voltage
  regulators from a supplier.There are reports that
  variation in the output voltage of the regulators
  is affecting the performance of the finished
  products. To assess the quality of the supplier's
  production, the manufacturer subjects a sample of
  5 regulators from the last shipment to careful
  laboratory analysis.
• True if sample is representative
• False if not

15
Sampling

• In order for the inferences to be reliable, we
  must have a representative sample
• A haphazard or opportunistic sample is prone to
  bias
• Sampling designs
• Simple Random Sample (SRS)
• Independence the selection of one unit has no
  influence on the selection of other units
• Lack of bias each unit has the same chance of
  being chosen
• All possible samples are equally
  likely.

16
What might cause samples to not be random?
17
What might cause samples to not be random?

• A professor uses students.
• A survey is done by phone.
• who might get missed?
• A survey is done on the internet.
• who might get missed?
• A survey is left in a doctors office.
• who might choose to complete or not complete?

18
Sampling Designs

• Stratified random sample
• Divide the population into relatively homogeneous
  subpopulations, called strata
• Take a SRS from each stratum
• e.g. estimation of out-of-spec raw material can
  be done by taking separate SRS from the materials
  supplied by different vendors
• More accurate estimate plus separate estimates
  for each vendor

19
Example

• You want to determine the relationship between
  the weather and the number of children driven to
  school in cars.
• Is it better to observe everyday for a week or
• five Thursdays in a row?
• Why?
• What if you had the resources to observe on 50
  days?

20
Designing Experimentscontrols

• A key question in all scientific explorations is
  compared to what?
• Having a control is essential for interpreting
  the results of a study.
• A controlled study is one of the following
• A study containing a control group
• A study in which the investigator assigns
  treatment and non-treatment (control).

21
Designing Experimentscontrols

• A key question in all scientific explorations is
  compared to what?
• Having a control is essential for interpreting
  the results of a study.
• A controlled study is one of the following
• A study containing a control group
• A study in which the investigator assigns
  treatment and non-treatment (control).
• When there is no control, evidence is often
  called anecdotal, meaning that, while it may be
  true, it doesnt prove anything.

22
Exercise 3

• Which of the following are based on a sample, and
  not just anecdotal?
• Seatbelts are a bad safety measure 10 drivers
  saved their lives by being thrown free of their
  car wrecks last year in Massachusetts.
• Out of 102 customers at a quick-stop market, 31
  made their purchases with a credit card.
• Out of 100 cars manufactured in the night shift,
  95 were fully defect-free.
• Hoan says that adding a few drops of almond
  flavoring makes her cookies taste better.

23
Anecdotal vs sample

• Seatbelts are a bad safety measure 10 drivers
  saved their lives by being thrown free of their
  car wrecks last year in Massachusetts.
• What kind of data would we want to look at to see
  if seatbelts are a good idea or not?

24
Designing ExperimentsRandomized versus
Observational

• In a randomized study the researcher assigns
  treatment randomly.
• In an observational study, the researcher
  observes differences between two groups
  treatment and control but does not assign
  treatment.

25
Designing ExperimentsRandomized versus
Observational

• Observational studies can be very hard to
  interpret, since the subjects have often
  self-selected into the treatment and control
  groups.

26
Observational Studies Example 1

• Pet a day keeps the doctor away
• Those who owned pets had contact with a doctor
  8.42 times in a year.
• Those who did not own pets had contact 9.49
  times.
• What interpretation is being made?
• Could there be another interpretation?

27
Observational Studies Example 1

• Pet a day keeps the doctor away
• Those who owned pets had contact with a doctor
  8.42 times in a year.
• Those who did not own pets had contact 9.49
  times.
• Does Pet cause good health or
• Does good health cause pet or
• Does something else cause both?

28
?
Good Health
Pet Ownership
29
?
Good Health
Pet Ownership
30
A desire to care for others
Good Health
?
Pet Ownership
31
Do storks bring babies?
?
Human baby births
Stork Nests
32
Do babies causing stork nests?
?
Human baby births
Stork Nests
33
Are babies born (and stork nests built) in
particular months?
month of year
?
Human baby births
Stork Nests
34
Observational Studies Example 2

• Likely to have a nervous breakdown?

35
Observational Studies Example 2

• Likely to have a nervous breakdown?

Does marriage make women crazy? Do crazy women
make a point of getting married? Is there a 3rd
issue, say rich woman are more likely to marry
and more likely to be crazy.
36
Observational Studies Example 3

• A common finding is that African Americans do not
  go to college at the same rates as the general
  population.
• It has been hypothesized that this could be
  explained by income.
• But, it is consistently found that African
  Americans do not go to college at the same rates,
  even controlling for income.
• Discussion over?