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Any characteristic of elements of the population is called a variable .
Quantitative variables can be expressed as numbers.
Qualitative variables cannot be expressed as numbers-usually expressed as categories.
Populations and Samples
A census measures the variable for every element of the population.
A census is time-consuming and expensive, unless the population is very small.
Instead of dealing with the entire population, a subset, called a sample , is usually selected for study.
Suppose you want to determine voter opinion on a ballot measure. You survey potential voters among pedestrians on Main Street during lunch.
What is the population?
What is the sample?
What is the variable being measured?
Solution: The population consists of all the people who intend to vote on the ballot measure.
Solution: The sample consists of all the people you interviewed on Main Street who intend to vote on the ballot measure.
Solution: The variable being measured is the voter’s intent to vote “yes” or “no” on the ballot measure.
Qualitative data with a natural ordering is called ordinal .
For example, a ranking of a pizza on a scale of “Excellent” to “Poor” is ordinal.
Qualitative data without a natural ordering is called nominal .
For example, eye color is nominal.
Suppose you survey potential voters among the people on Main Street during lunch to determine their political affiliation and age, as well as their opinion on the ballot measure.
Classify the variables as quantitative or qualitative.
Political affiliation is a qualitative variable (categories)
Age is a quantitative variable (numbers)
Opinion on the ballot measure is a qualitative variable (categories)
Common Sources of Bias
Faulty sampling : The sample is not representative.
Faulty questions : The questions are worded to influence the answers.
Faulty interviewing : Interviewers fail to survey the entire sample, misread questions, and/or misinterpret answers.
Common Sources of Bias, cont’d
Lack of understanding or knowledge : The person being interviewed does not understand the question or needs more information.
False answers : The person being interviewed intentionally gives incorrect information.
Suppose you wish to determine voter opinion regarding eliminating the capital gains tax. You survey potential voters on a street corner near Wall Street in New York City.
Identify a source of bias in this poll.
Solution: One source of bias in choosing the sample is that people who work on Wall Street would benefit from the elimination of the tax and are more likely to favor the elimination than the average voter may be.
This is faulty sampling.
Suppose a car manufacturer wants to test the reliability of 1000 alternators. They will test the first 30 from the lot for defects.
Identify any potential sources of bias.
Solution: One source of bias could be that the first 30 alternators are chosen for the sample. It may be that defects are either much more likely at the beginning of a production run or much less likely at the beginning. In either case, the sample would not be representative.
This is potentially faulty sampling.
Simple Random Samples
Given a population and a desired sample size, a simple random sample is any sample chosen in such a way that all samples of the same size are equally likely to be chosen.
Simple Random Samples, cont’d
One way to choose a simple random sample is to use a random number generator or table.
A random number generator is a computer or calculator program designed to produce numbers with no apparent pattern.
A random number table is a table produced with a random number generator.
An example of the first few rows of a random number table is shown on the next slide.
Random Number Table
Choose a simple random sample of size 5 from 12 semifinalists: Astoria, Beatrix, Charles, Delila, Elsie, Frank, Gaston, Heidi, Ian, Jose, Kirsten, and Lex.
Example 5, cont’d
Solution: Assign numerical labels to the population elements, in any order, as shown below:
Choose a random spot in the table to begin.
One option is to start at the top of the third column and to read down, looking at the last 2 digits in each number. This choice is arbitrary. There are many ways to use this table.
Numbers that correspond to population labels are recorded, ignoring duplicates, until 5 such numbers have been found.
The numbers located are 01, 06, 10, 11, and 07.
The simple random sample consists of Beatrix, Gaston, Heidi, Kirsten, and Lex.
Choose a simple random sample of size 8 from the states of the United States of America.
We randomly choose to start at the top row, left column of the number table and read the last 2 digits of each entry across the row.