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# Inferential Statistics

Module by: David Lane. E-mail the author

Summary: Introduction to inferential statistics.

## Populations and samples

In statistics, we often rely on a sample 1 - that is, a small subset of a larger set of data - to draw inferences about the larger set. The larger set is known as the population. 2

### Example 1

You have been hired by the National Election Commission to examine how the American people feel about the fairness of the voting procedures in the U.S. How will you do it? Who will you ask?

It is not practical to ask every single American how he or she feels about the fairness of the voting procedures. Instead, we query a relatively small number of Americans, and draw inferences about the entire country from their responses. The Americans actually queried constitute our sample of the larger population of all Americans. The mathematical procedures whereby we convert information about the sample into intelligent guesses about the population fall under the rubric of inferential statistics 3.

A sample is typically a small subset of the population. In the case of voting attitudes, we would sample a few thousand Americans, drawn from the hundreds of millions that make up the country. In choosing a sample, it is therefore crucial that it be representative. It must not overrepresent one kind of citizen at the expense of others. For example, something would be wrong with our sample if it happened to be made up entirely of Florida residents. (Recall the controversy surrounding presidential voting in Florida in 2000.) If the sample held only Floridians, it could not be used to infer the attitudes of other Americans. The same problem would arise if the sample were comprised only of Republicans. Inferential statistics are based on the assumption that sampling is random. We trust a random sample to represent different segments of society in close to the appropriate proportions (provided the sample is large enough; see below).

### Example 2

We are interested in examining how many math classes have been taken on the average by current graduating seniors at American colleges and universities during their four years in school. Whereas our population in the last example included all US citizens, now it involves just the graduating seniors throughout the country. This is still a large set since there are thousands of colleges and universities, each enrolling many students. (New York University, for example, enrolls 48,000 students.) It would be costly to examine the transcript of every college senior. We therefore take a sample of college seniors and then make inferences to the entire population based on what we find. To make the sample, we might first choose some public and private colleges and universities across the United States. Then we might sample 50 students from each of these institutions. Suppose that the average number of math classes taken by the people in our sample is 3.2. Then we might speculate that 3.2 approximates the number we would find if we had the resources to examine every senior in the entire population. But we must be careful about the possibility that our sample is non-representative of the population. Perhaps we chose an overabundance of math majors, or chose too many technical institutions that have heavy math requirements. Such bad sampling makes our sample unrepresentative of the population of all seniors. To solidify your understanding of sampling bias 4, consider the following example. Try to identify the population and the sample, and then reflect on whether the sample is likely to yield the information desired.

### Exercise 1

A substitute teacher wants to know how students in the class did on their last test. He asks only the 10 students sitting in the front row to report how they did on their last test and he concludes from them that the class did extremely well. What is the sample? What is the population? Can you identify any problems with choosing the sample in the way that the teacher did?

#### Solution

• The population consists of all students in the class.
• The sample includes the 10 students sitting in the front row.
• The sample is made up of just the 10 students sitting in the front row. The sample is not likely to be representative of the population. Those who sit in the front row tend to be more interested in the class and tend to perform higher on tests. Hence, the sample may perform at a higher level than the population.

### Exercise 2

A coach is interested in how many cartwheels the average college freshmen at his university can do. Eight volunteers from the freshman class step forward. Aftering observing their performance, the coach concludes that college freshmen can do an average of 16 cartwheels in a row without stopping.

#### Solution

• The population is the freshmen at the coach's university.
• The sample is poorly chosen because volunteers are more likely to be able to do cartwheels than the average freshman; people who can't do cartwheels probably did not volunteer!
• In the example, we are also not told of the gender of the volunteers. Were they all women, for example? That might affect the outcome, contributing to the nonrepresentative nature of the sample (if the school is co-ed).

## Simple Random Sampling

Researchers adopt a variety of sampling strategies. The most straightforward is simple random sampling 5. Such sampling requires every member of the population to have an equal chance of being selected into the sample. In addition, the selection of one member must be independent of the selection of every other member. That is, picking one member from the population must not increase or decrease the probability of picking any other member (relative to the others). In this sense, we can say that simple random sampling chooses a sample by pure chance. To check your understanding of simple random sampling, consider the following example. What is the population? What is the sample? Was the sample picked by simple random sampling? Is it biased?

### Exercise 3

A research scientist is interested in studying the experiences of twins raised together versus those raised apart . She obtains a list of twins from the National Twin Registry, and selects two subsets of individuals for her study. First, she chooses all those in the registry whose last name begins with Z. Then she turns to all those whose last name begins with B. Because there are so many names that start with B, however, our researcher decides to incorporate only every other name into her sample. Finally, she mails out a survey and compares characteristics of twins raised apart versus together.

#### Solution

• The population consists of all twins recorded in the National Twin Registry.
• It is important that the researcher only make statistical generalizations to the twins on this list, not to all twins in the nation or world. That is, the National Twin Registry may not be representative of all twins. Even if inferences are limited to the Registry, a number of problems affect the sampling procedure we described. For instance, choosing only twins whose last names begin with Z does not give every individual an equal chance of being selected into the sample. Moreover, such a procedure risks over-representing ethnic groups with many surnames that begin with Z. There are other reasons why choosing just the Z's may bias the sample. Perhaps such people are more patient than average because they often find themselves at the end of the line! The same problem occurs with choosing twins whose last name begins with B. An additional problem for the B's is that the every-other-one procedure disallowed adjacent names on the B part of the list from being both selected. Just this defect alone means the sample was not formed through simple random sampling.

## Sample size matters

Recall that the definition of a random sample is a sample in which every member of the population has an equal chance of being selected. This means that the sampling procedure rather than the results of the procedure define what it means for a sample to be random. Random samples, especially if the sample size is small, are not necessarily representative of the entire population. For example, if a random sample of 20 subjects were taken from a population with an equal number of males and females, there would be a nontrivial probability (0.06) that 70% or more of the sample would be female. (To see how we obtain this probability, click here.) Such a sample would not be representative, although it would be drawn randomly. Only a large sample size makes it likely that our sample is representative of the population. For this reason, inferential statistics needs to take into account the sample size when it attempts to generalize results from samples to populations. In later chapters, you'll see what kinds of mathematical techniques ensure this sensitivity to sample size.

## More sophisticated sampling

Sometimes it is not feasible to build a sample using simple random sampling. To see the problem, consider the fact that both Dallas and Houston are competing to be hosts of the 2012 Olympics. Imagine that you are hired to assess whether most Texans prefer Houston to Dallas as the host, or the reverse. Given the impracticality of obtaining the opinion of every single Texan, you must construct a sample of the Texas population. But now notice how difficult it would be to proceed by simple random sampling. For example, how will you contact those individuals who don't vote and don't have a phone? Even among people you find in the telephone book, how can you identify those who have just relocated to California (and had no reason to inform you of their move)? What do you do about the fact that since the beginning of the study, an additional 4,212 people took up residence in the state of Texas? As you can see, it is sometimes very difficult to develop a truly random procedure. For this reason, other kinds of sampling techniques have been devised. We now discuss two of them.

## Random Assignment

In experimental research, populations are often hypothetical. For example, in an exeriment comparing the effectiveness of a new anti-depressant drug with a placebo 6, there is no actual populaton of individuals taking the drug. In this case, a specified population of people with some degree of depression is defined and a random sample is taken from this populaton. The sample is then randomly divided into two groups; one group is assigned to the treatment condition (drug) and the other group is assigned to the control condition (placebo). This random division of the sample into two groups is called random assignment. Random assignment is critical for the validity of an experiment. For example, consider the bias that could be introduced if the first 20 subjects to show up at the experiment were assigned to the experimental group and the second 20 subjects were assigned to the control group. It is possible that subjects who show up late tend to be more depressed than those who show up early thus making the experimental group less depressed than the control group even before the treatment was administered.

In exerimental research of this kind, failure to assign subjects randomly to groups is generally more serious than having a non-random sample. Failure to randomize (the former error) invalidates the experimental findings. Non-random samples (the latter error) simply restricts the generalizeability of the results.

## Stratified Sampling

Since simple random sampling often does not ensure a representative sample, a sampling method called stratified random sampling 7 is sometimes used to make the sample more representative of the population. This method can be used if the population has a number of distinct "strata" or groups. In stratified sampling, you first identify members of your sample who belong to each group. Then you randomly sample from each of those subgroups in such a way that the sizes of the subgroups in the sample are proportional to their sizes in the population.

Let's take an example: Suppose you were interested in views of capital punishment at an urban university. You have the time and resources to interview 200 students. The student body is diverse with respect to age; many older people work during the day and enroll in night courses (average age is 39), while younger students generally enroll in day classes (average age of 19). It is possible that night students have different views about capital punishment than day students. If 70% of the students were day students, it makes sense to ensure that 70% of the sample consisted of day students. Thus, your sample of 200 students would consist of 140 day students and 60 night students. The proportion of day students in the sample and in the population (the entire university) would be the same. Inferences to the entire population of students at the university would therefore be more secure.

## Footnotes

1. A sample is a subset of a population, often taken for the purpose of statistical inference. Generally, one tries uses a random sample. See also: bias, stratified random sample.
2. A population is the complete set of observations a researcher is interested in. Contrast this with a sample which is a subset of a population. A population can be defined in a manner convenient for a researcher. For example, one could define a population as all girls in fourth grade in Houston, Texas. Or, a different population is the set of all girls in fourth grade in the United States. Inferential statistics are computed from sample data in order to make inferences about the population.
3. The branch of statistics concerned with drawing conclusions about a population from a smaller sample. This is generally done through random sampling, followed by inferences made about central tendency, or any of a number of other aspects of a distribution.
4. (i) A sampling method is biased if each individual does not have an equal chance of being selected. A sample of internet users found reading an online statistics book would be a biased sample of all internet users. It would give a distorted view of what the average internet user is like. (ii) An estimator is biased if it systematically overestimates of underestmates the parameter it is estimating. The average squared deviation of sample scores from their mean is a biased estimate of the variance since it tends to underestimate the population variance.
5. The process of selecting a subset of a population for the purposes of statistical inference. Random sampling means that every member of the population is equally likely to be chosen. When this rule is violated, the sample is said to be biased. See also: stratified random sampling.
6. A device used in clinical trials, the placebo is visually indistinguishable from the study medication, but in reality has no medical effect (often, a sugar pill). A group of subjects chosen randomly takes the placebo, the others take one or another type of medication. This is done to prevent confounding the medical and psychological effects of the drug. Even a sugar pill can lead some patients to report improvement and side effects.
7. In stratified random sampling, the population is divided into a number of subgroups (or strata). Random samples are then taken from each subgroup with sample sizes proportionatal to the size of the subgroup in the population. For instance, if a population contained equal numbers of men and women, and the variable of interest is suspected to vary by gender, one might conduct stratified random sampling to insure a representative sample.

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