1. Chapter 3: Producing Data
Inferential Statistics
Sampling
Designing Experiments

2. Inferential Statistics
We start with a question about a group or groups.
The group(s) we are interested in is(are) called the population(s).
Examples
What is the average number of car accidents for a person over 65 in the United States?
For the entire world, is the IQ of women the same as the IQ of men?
How many times a day should I feed my goldfish?
Which is more effective at lowering the heartrate of mice, no drug (control), drug A, drug B, or drug C?

3. Inferential Statistics
Example 1: What is the average number of car accidents for a person over 65 in the United States?
How many populations are of interest?
One
What is the population of interest?
All people in the U.S. over age 65.

4. Inferential Statistics
Example 2: For the entire world, is the IQ of women the same as the IQ of men?
How many populations are of interest?
Two
What are the populations of interest?
All women and all men

5. Inferential Statistics
Example 3: How many times a day should I feed my goldfish?
How many populations are of interest?
One
What is the population of interest?
All pet goldfish

6. Inferential Statistics
Example 4: Which is more effective at lowering the heartrate of mice, no drug (control), drug A, drug B, or drug C?
How many populations are of interest?
Four
What are the populations of interest?
All mice taking no drug, all mice taking drug A, all mice taking drug B, all mice taking drug C

7. Inferential Statistics
Suppose we have no previous information about these questions. How could we answer them?
Census
Advantages
We get everyone, we know the truth
Disadvantages
Expensive, Difficult to obtain, may be impossible.
Sample
Less expensive. Feasible.
Uncertainty about the truth. Instead of surety we may have error.

8. Inferential Statistics
Suppose we have no previous information about these questions. How could we answer them?
If we take a census, we have everyone and we have no need for inference. We know.
If we take a sample, we make inference from the sample to the whole population.
For these four questions, it is not likely we can get a census. We will need to use a sample.
Obviously, for each population we are interested in, we must get a separate sample.

9. Inferential Statistics
General Idea of Inferential Statistics
We take a sample from the whole population.
We summarize the sample using important statistics.
We use those summaries to make inference about the whole population.
We realize there may be some error involved in making inference.

10. Inferential Statistics
Example (1988, the Steering Committee of the Physicians' Health Study Research Group)
Question: Can Aspirin reduce the risk of heart attack in humans?
Sample: Sample of 22,071 male physicians between the ages of 40 and 84, randomly assigned to one of two groups. One group took an ordinary aspirin tablet every other day (headache or not). The other group took a placebo every other day. This group is the control group.
Summary statistic: The rate of heart attacks in the group taking aspirin was only 55% of the rate of heart attacks in the placebo group.
Inference to population: Taking aspirin causes lower rate of heart attacks in humans.

11. Sampling a Single Population
Basics for sampling
Sampling should not be biased: no favoring of any individual in the population.
Example of a biased sample
Select goldfish from a particular store
The selection of an individual in the population should not affect the selection of the next individual – independence.
Example of non-independent sample
Choosing cards from a deck without replacement

12. Sampling a Single Population
Basics for sampling
Sampling should be large enough to adequately cover the population.
Example of a small sample
Suppose only 20 physicians were used in the aspirin study.
Sampling should have the smallest variability possible.
We know there is some error want to minimize it.

13. Sampling a Single Population
Sampling Techniques
Simple Random Sample (SRS): every member of the population has an equal chance of being selected.
Population
Sample

14. Sampling a Single Population
Sampling Techniques
Simple Random Sample (SRS): every member of the population has an equal chance of being selected
Assign every individual a number and randomly select 30 numbers using a random number table (or computer generated random numbers).
Example: Obtain a list of all SSN for individuals in the U.S. who are over 65. Using a random number table, select 50 of them.
Table B at the back of the book is random digits.

15. Sampling a Single Population
Sampling Techniques
Stratified Random Sample: Divide the population into several strata. Then take a SRS from each stratum.
Strata 1
Population
Strata 2
Sample
Strata 3

16. Sampling a Single Population
Sampling Techniques
Stratified Random Sample
Advantage: Each stratum is guaranteed to be randomly sampled
Example: Obtain a list of all SSN for individuals in the U.S. who are over 65. Divide up the SSNs into region of the country (time zones). Then randomly sample 30 from each time zone.

17. Sampling a Single Population
Sampling Techniques
Cluster Sample: Divide the population into several strata or clusters. Then take a SRS of clusters using all the observations in each.
Strata 1
Strata 2
Strata 3
Strata 4
Strata 5
Strata 6
Strata 7
Strata 8
Strata 9
Strata 1
Strata 4
Strata 9
Population
Sample

18. Sampling a Single Population
Sampling Techniques
Cluster Sample
Advantage: May be the only feasible method, given resoures.
Example: Obtain a list of all SSNs for individuals in the U.S. who are over 65. Sort the SSNs by the last 4 digits making each set of 100 a cluster. Use a random number table to pick the clusters. You may get the 4100’s, 5600’s and 8200’s for example.

19. Sampling a Single Population
Sampling Techniques
Multi-Stage Sample: Divide the population into several strata. Then take a SRS from a random subset of all the strata.
Strata 1
Strata 2
Strata 3
Strata 4
Strata 5
Strata 6
Strata 7
Strata 8
Strata 9
Strata 1
Strata 4
Strata 9
Population
Sample

20. Sampling a Single Population
Sampling Techniques
Multi-Stage Sample
Advantage: May be the only feasible method, given resources.
Example: Obtain a list of all SSN for individuals in the U.S. who are over 65. Divide up the SSNs into 50 states. Randomly select 10 states. Then randomly sample 40 from each of the selected states.

21. Sampling a Single Population
Sampling Problems
Voluntary response
Internet surveys
Call-in surveys
Convenience sampling
Sampling friends
Sampling at the mall
Dishonesty
Asking personal questions
Not enough time to respond honestly

22. Sampling a Single Population
Undercoverage – Some groups in the population are left out when the sample is taken
Nonresponse – An individual chosen for the sample can’t be contacted or does not cooperate
Response Bias – Results that are influenced by the behavior of the respondent or interviewer
For example, the wording of questions can influence the answers
Respondent may not want to give truthful answers to sensitive questions

23. Sampling More than One Population
We sample from more than one population when we are interested in more than one variable.
As previously discussed, one variable is chosen to be the response variable and the other is selected as the explanatory variable.
Examples
Comparing decibel levels of 4 different brands of speakers
Determining time to failure of 3 different types of lightbulbs
Comparing GRE scores for students from 5 different majors

24. Sampling More than One Population
Example 1: Comparing decibel levels of 4 different brands of speakers
What is the explanatory variable?
Brand
What is the response variable?
Decibel Level
Number of Populations?
Four
Number of Samples needed?

25. Sampling More than One Population
Example 2: Determining time to failure of 3 different types of lightbulbs
What is the explanatory variable?
Type
What is the response variable?
Time to Failure
Number of Populations?
Three
Number of Samples needed?

26. Sampling More than One Population
Example 3: Comparing GRE scores for students from 5 different majors
What is the explanatory variable?
Major
What is the response variable?
GRE score
Number of Populations?
Five
Number of Samples needed?

27. Sampling More than One Population
Important Considerations
Each sample should represent the population it corresponds to well.
Samples from more than one population should be as close to each other in every respect as possible except for the explanatory variable. Otherwise we may have confounding variables.
Two variables are confounded if we cannot determine which one caused the differences in the response.

28. Sampling More than One Population
Important Considerations
Examples of Confounding
Suppose we compared the decibel levels of the four different speaker brands, each with a different measuring instrument
We wouldn’t know if the differences were due to the different brands or different instruments.
Brand and Instrument are then confounded.
Suppose we compared the time to failure of the three different types of lightbulbs, each in a different light socket.
We wouldn’t know if the differences were due to the different types of lightbulbs or different light sockets.
Type and Socket confounded.

29. Sampling More than One Population
Important Considerations
Examples of Confounding
Suppose we obtained GRE scores for each major, each from a different university.
We wouldn’t know if the differences were due to the different majors or different universities.
Major and University are then confounded.
Confounding can be avoided by using good sampling techniques, which will be explained shortly

30. Sampling More than One Population
Important Considerations
It is also possible that more than one (possibly several) explanatory variable can influence a given response variable.
Example
Perhaps both the type of lightbulb and the type of light socket influence the time to failure of a lightbulb.
It is likely that different types of lightbulbs work better for different sockets.
This concept is known as interaction.
Interaction: The responses for the levels of one variable differ over the levels of another variable.

31. Sampling More than One Population
Randomized Experiment
The key to a randomized experiment: the treatment (explanatory variable) is randomly assigned to the experimental units or subjects.
Random Assignment
Control
Population
Statistics
Simple Random Sample
Compare
Treatment

32. Sampling More than One Population
Randomized Experiment
Example: Suppose that before we want to test the effect of aspirin on the physicians, we wish to do a study on the effect of aspirin on mice, comparing heart rates.
We obtain a random sample of 100 mice.
We randomly assign 50 mice to receive a placebo.
We randomly assign 50 mice to receive aspirin.
After 20 days of administering the placebo and aspirin, we measure the heart rates and obtain summary statistics for comparison.

33. Sampling More than One Population
Randomized Experiment
The single greatest advantage of a randomized experiment is that we can infer causation.
Through randomization to groups, we have controlled all other factors and eliminated the possibility of a confounding variable.
Unfortunately or perhaps fortunately, we cannot always use a randomized experiment
Often impossible or unethical, particularly with humans.

34. Sampling More than One Population
Observational Study
We are forced to select samples from different pre-existing populations
Pre-existing Population 1
Pre-existing Population 2
Statistics
Simple Random Sample
Compare

35. Sampling More than One Population
Observational Study
Advantage: The data is much easier to obtain.
Disadvantages
We cannot say the explanatory variable caused the response
There may be lurking or confounding variables
Observational studies should be more to describe the past, not predict the future.
Case-Control Study: A study in which cases having a particular condition are compared to controls who do not. The purpose is to find out whether or not one or more explanatory variables are related to a certain disease.
Although you can’t usually determine cause and effect, these studies are more efficient and they can reduce the potential confounding variables.

36. Sampling More than One Population
Observational Study
Example 1: Suppose we are interested in comparing GRE scores for students in five different majors
We cannot do a randomized experiment because we cannot randomly assign individuals to a specific major. The individuals decide that for themselves.
Thus, we observe students from 5 different pre-existing populations: the five majors.
We obtain a random sample of size 15 from each of the five majors.
We calculate statistics and compare the 5 groups.
Can we say being in a specific major causes someone to get a higher GRE score?
What are some possible confounding variables?
How might we reduce the effect of these confounding variables?

37. Sampling More than One Population
Observational Study
Example 2: Suppose we are interested finding out which age group talks the most on the telephone: 0-10 years, years, years, or years
We cannot do a randomized experiment because we cannot randomly assign individuals to an age group.
Thus, we observe (through polling or wire tapping) individuals from 4 different pre-existing populations: the four age groups.
We obtain a random sample of size 25 from each of the four age groups.
We calculate statistics and compare the 4 groups.
Can we say being in a specific age group causes someone to talk more on the telephone?
What are some possible confounding variables?
How might we control these confounding variables?

38. Inference Overview
Recall that inference is using statistics from a sample to talk about a population.
We need some background in how we talk about populations and how we talk about samples.

39. Inference Overview Describing a Population
It is common practice to use Greek letters when talking about a population.
We call the mean of a population m.
We call the standard deviation of a population s and the variance s2.
When we are talking about percentages, we call the population proportion p. (or pi).
It is important to know that for a given population there is only one true mean and one true standard deviation and variance or one true proportion.
There is a special name for these values: parameters.

40. Inference Overview Describing a Sample
It is common practice to use Roman letters when talking about a sample.
We call the mean of a sample .
We call the standard deviation of a sample s and the variance s2.
When we are talking about percentages, we call the sample proportion p.
There are many different possible samples that could be taken from a given population. For each sample there may be a different mean, standard deviation, variance, or proportion.
There is a special name for these values: statistics.

41. Inference Overview
We use sample statistics to make inference about population parameters
Population
Sample
Mean:
Standard Deviation:
Proportion: m s s p p

42. Sampling Variability
There are many different samples that you can take from the population.
Statistics can be computed on each sample.
Since different members of the population are in each sample, the value of a statistic varies from sample to sample.

43. Sampling Distribution
The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.
We can then examine the shape, center, and spread of the sampling distribution.

44. Bias and Variability
Bias concerns the center of the sampling distribution. A statistic used to a parameter is unbiased if the mean of the sampling distribution is equal to the true value of the parameter being estimated.
To reduce bias, use random sampling. The values of a statistic computed from an SRS neither consistently overestimates nor consistently underestimates the value of the population parameter.
Variability is described by the spread of the sampling distribution.
To reduce the variability of a statistic from an SRS, use a larger sample. You can make the variability as small as you want by taking a large enough sample.

45. Bias and Variability