Chapter 7: Data for Decisions For All Practical Purposes...

Chapter 7: Data for Decisions

Lesson Plan

Sampling

Bad Sampling Methods

Simple Random Samples

Cautions About Sample Surveys

Experiments

Experiments Versus Observational

Studies

Inference: From Sample to Population

Confidence Intervals

Mathematical Literacy in Today’s World, 9th ed.

For All Practical Purposes

© 2013 W. H. Freeman and Company


Sampling

Statistics The science of collecting, organizing, and interpreting data.

How is the data produced? Sampling and experiments.

Sampling Gather information about a large group of individuals.

Time, cost, and inconvenience forbid contacting every individual.

Instead, gather information about only part of the group in order to draw conclusions about the whole.

Population – The entire

group of individuals about

whom we want information.

Sample – Part of the population from

which we actually collect information

used to draw conclusions about the

whole.




If personal choice is involved in selecting the sample, the following

could happen:

Results could become biased.

The sample may not be a true representation of the population.

Bias – The design

of a statistical

study that

systematically

favors a certain

outcome.

1. Convenience Samples

Interviewer chooses the sample from individuals close at hand (easiest to reach).

Example: Mall surveys

2. Voluntary Response Sample

People who choose themselves by responding to a general appeal.

People with strong opinions are most likely to respond; can cause bias.

Examples: Online polls, call-ins, write-ins.


Simple Random Samples

Simple Random Sample (SRS)

An SRS of size n consists of n individuals from the population

chosen in such a way that every set of n individuals has an equal

chance to be the sample actually selected.

Choosing a sample by chance avoids bias by giving all individuals

an equal chance to be chosen (a good sampling method).

Examples of SRS

Draw names from a hat: Place all the names of the people in the

population into a hat and draw out a handful (the sample).

Slow and inconvenient

Use the table of random digits: A more efficient way of randomly

selecting the sample without bias.

For smaller samples, tables of random digits are used.

For larger samples, computers do the random digit sampling.


Simple Random Sample

Two Steps in Choosing a Simple Random Sample 1. Give each member of the population a numerical label of the

same length.

Example: 100 items can be labeled with two digits 01, 02, …, 99, 00

2. To choose the random sample, select a line in the digit table.

A table of random digits – A list

of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8,

9 with these two properties:

1. Each entry in the table is equally

likely to be any of the 10 digits

from 0 through 9 .

2. The entries are independent of

one another. That is, knowledge

of one part of the table gives no

information about the other part.

For a sample size of n, start reading off numbers of length of the labels until n individuals are selected from the population.

When selecting the n individuals for the sample from the random digits table:

1. Do not use any group of digits not used as a label.

2. Do not use any repeats.


Simple Random Sample

Using the Random Digit Table

Example:

A group of 70 people were labeled 01, 02, 03, …, 69, 70.

In the random digits table, line 104 was selected and three lucky

winners were selected.

Reading off two-digit labels from line 104…

52 was selected first; 71 was skipped over (because it is not in the

range of labels); 13 was chosen; then 88, 89, and 93 were skipped

over (out of range); and 07 was chosen. Winners: 52 ,13, and 07


Cautions About Sample Surveys

Sample surveys of large populations require the following:

A good sampling design (can be done with SRS)

An accurate and complete list of the population

Participation of all individuals selected for the sample

A question posed that is neutral and clear

Bias can occur due to the following:

Problems with obtaining an accurate and complete population list

Undercoverage – Occurs when some groups in the population are left

out of the process of choosing the sample.

Example: Homeless, prison inmates, students in dormitories, etc.

Problems with getting 100% participation of sampled people

Nonresponse – Occurs when an individual chosen for the sample

cannot be contacted or refuses to participate.

Problems with posing a misleading or confusing question


Experiments

Observation versus Experiments Observational Study – Example: sample survey

Observes individuals and measures variable of interest but does not attempt to influence the response.

Purpose is to describe some group or situation.

Experiment

Deliberately imposes some treatment on individuals in order to observe their responses.

Purpose is to study whether the treatment causes a change in the response.


Experiments

Examining Cause and Effect Between Variables Experiments are the preferred method for examining the effect of

one variable on another.

By imposing specific treatment of interest and controlling other influences, we can pin down cause and effect.

Uncontrolled Experiment

When it is not possible to control outside factors that can

influence the outcome.

Confounding – The variables, whether part of a study or not, are

said to be confounded when their effects on the outcome cannot

be distinguished from each other.

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Experiments

Example: GMAT Prep Class

A college only offers a GMAT

exam preparation course online,

whereas in the past it was only

offered live. The students who

take the online course score an

average of 10% higher on the

GMAT exam than those who took

the live course in the past. Can we

conclude that the online course is

more effective?

No, there may be confounding

variables.

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Experiments

Randomized Comparative Experiment (helps confounding)

The outside effects and confounding variables act on all groups.

An experiment to compare two or more treatments in which people,

animals, or things are assigned to treatments by chance.

Randomized – The subjects are assigned to treatments by chance.

Comparative – Compares two or more treatments.


Experiments versus Observational Studies

Placebo Effect

The effect of a dummy treatment (such as an inert pill in a

medical experiment) on the response of the subjects.

The tendency to respond favorably to any treatment.

Double-Blind Experiments

An experiment in which neither the experimental subjects nor the

persons who interact with them know which treatment each

subject received. This helps to eliminate possible influences or

biases between the subjects and workers — everyone is kept

“blind.”

Observational Study

Does not try to manipulate the environment (such as assigning

treatments to people); it simply observes the measurements of

variables of interest that result from people’s free choices.

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Statistical Inference

When the sample was chosen at random from a population, we

can infer conclusions about the wider population from these data.

Statistical inference works only if the data comes from random

samples or a randomized comparative experiment.

Parameter is a number that describes the population.

A parameter is a fixed number (in practice we do not know its value).

A statistic is a number that describes a sample.

The value of a statistic is known when we have taken a sample, but it

can change from sample to sample.



Example:

A random sample of 2500 people was chosen from the

population and asked a question: “Do you like getting new

clothes but find shopping for clothes frustrating and time

consuming?” 1650 people agreed.

Infer that 66% of the population agrees.

ˆ 1650/ 2500 0.66 66%p

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Sampling Distribution

The distribution of values taken by the statistic in all possible

samples of the same size from the same population.

For a fixed number of trials, a distribution with larger sample sizes

will have less variation and the values will lie closer to the mean.



Sample Proportion Choose a SRS of size n from a large population that contains

population proportion p of successes. Then the sample proportion of successes is:

Then…

Shape: For large sample sizes , the sampling distribution of is approximately normal.

Center: The mean of the sampling distribution of is p.

Spread: The standard deviation of the sampling distribution is:

For the shopping example…

With a mean p = 0.6 and n = 2500, stand. dev. is

30n p̂

p̂

ˆcount of successes in the sample

pn

1p p

n

0.6 1 0.6.0098

2500


Confidence Intervals

The 68-95-99.7 Rule 68% of the observations fall within ± 1 standard deviation of the mean.

95% of the observations fall within ± 2 standard deviations of the mean.

99.7% of the observations fall within ± 3 standard deviations of the mean.

95% Confidence Interval An interval obtained from the sample data by a method that in 95% of all

samples will produce an interval containing the true population parameter

A 95% confidence interval for p is approximately:

ˆ ˆ1

ˆ 2p p

pn

Chapter 7: Data for Decisions For All Practical Purposes...

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