STA 291Fall 2009

Lecture 2Dustin Lueker

Parameter◦ Numerical characteristic of the population

Calculated using the whole population Statistic

◦ Numerical characteristic of the sample Calculated using the sample

Basic Terminology

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Each possible sample has the same probability of being selected

The sample size is usually denoted by n

Simple Random Sampling (SRS)

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Population of 4 students: Alf, Buford, Charlie, Dixie

Select a SRS of size n = 2 to ask them about their smoking habits◦ 6 possible samples of size 2

A,B A,C A,D B,C B,D C,D

Example of SRS

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Each of the size possible samples has to have the same probability of being selected◦ How could we do this?

Roll a die Random number generator

How to choose a SRS?

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Convenience sample◦ Selecting subjects that are easily accessible to you

Volunteer sample◦ Selecting the first two subjects who volunteer to take

the survey

What are the problems with these samples?◦ Proper representation of the population◦ Bias

Examples Mall interview Street corner interview

Common Problems when Sampling

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A survey of 300 random individuals was conducted in Louisville that revealed that President Obama had an approval rating of 67%.◦ Is 67% a statistic or parameter?◦ The surveyors stated that only 67% of

Kentuckians approved of President Obama. What is the problem with this statement? Why might the surveyors have chosen Louisville as

their sampling location?

Example

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1936 presidential election of Alfred Landon vs. Franklin Roosevelt◦ Literary Digest sent out over 10 million

questionaires in the mail to predict the election outcome What type of sample is this?

◦ 2 million responses predicted an landslide victory for Alfred Landon

◦ George Gallup used a much small random sample and predicted a clear victory for FDR

FDR won with 62% of the vote

Famous Example

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TV, radio call-in polls◦ “should the UN headquarters continue to be

located in the United States?” ABC poll with 186,000 callers: 67% no Scientific random sample of 500: 28% no

Which sample is more trust worthy? Would any of you call in to give you opinion? Why

or why not?

Other Examples

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Another advantage of random samples◦ Inferential statistical methods can be applied to

state that “the true percentage of all Americans who want the UN headquarters out of the United States is between 24% and 32%”

◦ These methods cannot be applied to volunteer sample

Other Examples

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Whenever you see results from a poll, check whether they come from a random sample

Preferably, it should be stated ◦ Who sponsored and conducted the poll?◦ How were the questions worded?◦ How was the sample selected? ◦ How large was it?

If not, the results may not be trustworthy

Don’t Trust Bad Samples

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Kalton et al. (1978), England Two groups get questions with slightly

different wording◦ Group 1

“Are you in favor of giving special priority to buses in the rush hour or not ?”

◦ Group 2 “Are you in favor of giving special priority to buses in

the rush hour or should cars have just as much priority as buses ?”

Question Wording

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Result: Proportion of people saying that priority should be given to buses.

Question Wording

13

Without reference to cars

With reference to cars

Difference

All respondents 0.69 (n=1076) 0.55 (n=1081) 0.14Women 0.65 (n=585) 0.49 (n=590) 0.16Men 0.74 (n=491) 0.66 (n=488) 0.08Non Car-owners 0.73 (n=565) 0.55 (n=554) 0.18Car owners 0.66 (n=509) 0.54 (n=522) 0.12

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Two questions asked in different order during the cold war◦ (1)“Do you think the U.S. should let Russian newspaper

reporters come here and send back whatever they want?”

◦ (2)“Do you think Russia should let American newspaper reporters come in and send back whatever they want?” When question (1) was asked first, 36% answered “Yes” When question (2) was asked first, 73% answered “Yes” to

question (1)

Question Order

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‘Flavors’ of Statistics Descriptive Statistics

◦ Summarizing the information in a collection of data

Inferential Statistics◦ Using information from a sample to make

conclusions/predictions about the population

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Example 71% of individuals surveyed believed that

the Kentucky Football team will return to a bowl game in 2009◦ Is 71% an example of descriptive or inferential

statistics? From the same sample it is concluded that

at least 85% of Kentucky Football fans approve of Coach Brooks’ job here at UK◦ Is 85% an example of descriptive or inferential

statistics?

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Nominal◦ Gender, nationality, hair color, state of residence

Nominal variables have a scale of unordered categories It does not make sense to say, for example, that green hair

is greater/higher/better than orange hair Ordinal

◦ Disease status, company rating, grade in STA 291 Ordinal variables have a scale of ordered categories,

they are often treated in a quantitative manner (A = 4.0, B = 3.0, etc.) One unit can have more of a certain property than does

another unit

Qualitative Variables

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Quantitative◦ Age, income, height

Quantitative variables are measured numerically, that is, for each subject a number is observed The scale for quantitative variables is called interval

scale

Quantitative Variables

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A survey of Kentucky Football fans obtained the following information◦ Age◦ Whether they preferred the new blue helmet or the old

white helmet◦ The number of games they think the team will win in 2009◦ How they felt the UK vs. U of L game would turn out

U of L in a blowout U of L in a close game UK in a close game UK in a blowout

Are these qualitative or quantitative variables and what is the scale for each?

Example

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A variable is discrete if it can take on a finite number of values◦ Gender◦ Favorite MLB team

Qualitative variables are discrete Continuous variables can take an infinite

continuum of possible real number values◦ Time spent studying for STA 291 per day

27 minutes 27.487 minutes 27.48682 minutes

Can be subdivided into more accurate values Therefore continuous

Discrete and Continuous

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An observational study observes individuals and measures variables of interest but does not attempt to influence the responses◦ Purpose of an observational study is to describe/compare

groups or situations Example: Select a sample of men and women and ask

whether he/she has taken aspirin regularly over the past 2 years, and whether he/she had suffered a heart attack over the same period

Observational Study

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An experiment deliberately imposes some treatment on individuals in order to observe their responses◦ Purpose of an experiment is to study whether the

treatment causes a change in the response Example: Randomly select men and women, divide the

sample into two groups. One group would take aspirin daily, the other would not. After 2 years, determine for each group the proportion of people who had suffered a heart attack.

Experiment

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Observational Studies◦ Passive data collection◦ We observe, record, or measure, but don’t interfere

Experiments◦ Active data production◦ Actively intervene by imposing some treatment in order

to see what happens

◦ Experiments are preferable if they are possible We are able to control more things and be sure our data isn’t

tainted

Which is Preferred?

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