excursions in modern mathematics sixth edition · excursions in modern mathematics sixth edition...

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Excursions in Modern Mathematics

Sixth Edition

Peter Tannenbaum

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Chapter 13Collecting Statistical Data

Censuses,

Surveys, and

Clinical Studies

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Collecting Statistical DataOutline/learning Objectives

To identify whether a given survey or poll is

biased.

To list and discuss the quality of several

sampling methods.

To identify components of a well-constructed

clinical study.

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Collecting Statistical DataOutline/learning Objectives

To define key terminology in the data collection

process.

To estimate the size of a population using the

capture-recapture method.

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Collecting Statistical Data

13.1 The Population

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Population

Every statistical statement refers, directly or indirectly, to some group of individuals or objects.

N-value

Given a specific population, an obviously relevant question is, “How many individuals or objects are there in that population?”

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Census

The process of collecting data by going

through every member of the population.

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Over the last 45 years, the United States Fish and Wildlife

Service has been able to keep a remarkably accurate tally of the

number of bald eagle breeding pairs in the lower 48 states.

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A tremendous amount of effort has gone into collecting and

verifying these N-values, which, for a wildlife population, are

of remarkable accuracy. The above figure summarizes the

population numbers over the period 1963-2000.

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Census

Mandated by the U.S. constitution

Must be carried out every 10 years

Adjusted by the 14th Amendment and courts

Plagued by undercounts

1999 Supreme Court ruling

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13.2 Sampling

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Survey

The practical alternative to a census is to collect data only from some members of the population and use that data to draw conclusions and make inferences about the entire population.

Poll

When the data collection is done by asking questions.

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Sample

The subgroup chosen to provide the data.

Sampling

The act of selecting a sample.

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Target population

The most important step in a survey is to

distinguish the population for which the survey

applies.

Sampling frame

The actual subset of the population from which

the sample will be drawn.

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Are Polls Always Right?

1936 US presidential election pitted

Alfred Landon against incumbent

Franklin D. Roosevelt.

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Public Opinion Polls

Selection bias

When the choice of the sample has a built-in tendency

to exclude a particular group or characteristics within

the population.

Response rate

The percentage of respondents out of the total sample.

Nonresponse bias

When the response rate to a survey is low.

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Convenience Sampling

In convenience sampling the selection of which

individuals are in the sample is dictated by what

is easiest for the data collector.

A classic example is when interviewers set up

at a fixed location such as a mall or outside a

supermarket and ask passersby to be a part of

a public opinion poll.

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Quota Sampling

Quota sampling is a systematic effort to force the sample to be representative of a given population through the use of quotas– the sample should have so many women, so many men, so many blacks, so many whites, so many people living in urban areas, so many people living in rural areas, and so on.

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13.3 Random

Sampling

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Random sampling

Sampling methods that use randomness as

part of their design.

Random sample

Any sample obtained through random

sampling.

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Simple Random Sampling

It is based on the same principle a lottery is.

Any set of numbers of a given size has an

equal chance of being chosen as any other set

of numbers of that size.

Implementing on a national level is very costly

and would take a great deal of time.

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Stratified Sampling

The alternative to simple random sampling.

Break the sampling frame into categories,

called strata, and then randomly choose a

sample from these strata.

How a Gallop poll is based.

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13.4 Sampling:

Terminology and

Key Concepts

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Statistic

To describe any kind of numerical information drawn from a sample.

Parameter

An estimate for some unknown measure of the population.

Sampling error

To describe the difference between a parameter and a statistic used to estimate that parameter.

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Chance error

The result of the basic fact that a sample,

being just a sample, can only give us

approximate information about the population.

Sampling variability

Different samples are likely to produce different

statistics for the same population, even when

the samples are chosen in exactly the same

way.

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Sample bias

The result of choosing a bad sample and is a

much more serious problem than chance error.

Sample proportion

The size of the sample, denoted by n (to

contrast with N, the size of the population).

The ratio n/N is the sample proportion.

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Example 1

As part of a sixth-grade statistics project, the

teacher brings to class a candy jar full of

gumballs of two different colors: red and green.

The assignment is to estimate the proportion of

red gumballs in the jar. To do this, the jar is

shaken well, and one of the students draws 25

gumballs from the jar. Of these, 8 are red and

17 are green.

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Example 1

1. (a)

(b)

(c)

(d)

2. (a)

(b)

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Example 1

3. (a)

(b)

(c)

(d)

4. (a)

(b)

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In 1988, “Dear Abby” concluded that the

amount of cheating among married couples

is much less than people believe.

Status Women Men

Faithful 127,318 44,807

Unfaithful 22,468 15,747

Total 149,786 60,550

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13.5 The Capture-

Recapture Method

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The Capture-Recapture Method

Step 1. Capture (sample): Capture (choose) a sample of size n1, tag (mark, identify) the animals (objects, people), and release them back into the general population.

Step 2. Recapture (resample): After a certain period of time, capture a new sample of size n2, and take an exact head count of the tagged individuals. Let’s call this number k.

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Small Fish in a Big Pond

A large pond is stocked with catfish. As part of a research project we need to estimate the number of catfish in the pond.

Step 1. For our first sample we capture a predetermined number n1 of catfish, say n1 = 200. The fish are tagged and released unharmed back in the pond.

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Step 2. After giving enough time for the released fish to

mingle and disperse throughout the pond, we capture a

second sample of n2 catfish. While n2 does not have to

equal n1, it is a good idea for the two samples to be of

approximately the same order of magnitude. Let’s say

that n2 = 250.

Of the 250 catfish in the second sample, 35 have tags

(were part of the original sample).

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The ratio of tagged fish in the second sample is

the same as the ratio of tagged fish in the pond.

35/250 200/N

which in turn gives

N 200 X 250/35 1428.57

A sensible conclusion is that there are

approximately N = 1400 catfish in the pond.

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13.6 Clinical

Studies

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Clinical Studies Terminology

Clinical study (trial). Studies concerned with

determining whether a single variable or

treatment can cause a certain effect.

Confounding variables. All other possible

contributing causes that could produce the

same effect in a clinical study.

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Controlled study. The subjects are divided

into two different groups.

– Treatment group. Subjects receiving the actual

treatment.

– Control group. Subjects that are not receiving any

treatment.

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Randomized controlled study. The subjects

are assigned to the treatment group or the

control group randomly.

Placebo effect. A critical confounding

variable from the generally accepted principle

that just the idea that one is getting a

treatment, can produce positive results.

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Placebo. A make-believe form of treatment–

a harmless pill, an injection of saline solution,

or any other fake type of treatment intended to

look like the real treatment.

Controlled placebo study. A controlled

study in which the subjects in the control group

are given a placebo.

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Blind. A study in which neither the members of the treatment group nor the members of the control group know to which of the two groups they belong.

Double-blind study. A controlled placebo study in which neither the subjects nor the scientist conducting the experiment know which subjects are in the treatment group and which are in the control group.

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Collecting Statistical DataConclusion

Census

Sample/ Survey/ Sample Bias

Simple Random/Stratified Sampling

Confounding Variables

Controlled Study

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