Math 105, Topics in Mathematics

1. If we are studying the income distribution of US population, then the population is the US population.
   
   
2. If we are studying the income distribution of the immigrant American population, then the population is the immigrant American population.
   
   
3. If we are studying the growth of the fish population in Clinton Lake, then the population is the fish population in Clinton Lake.
   
   
4. If we are studying African elephants, then the population is the population of African elephants.

Often, we focus on a particular characteristic (like height, weight, annual income) of such populations in these examples, and consider the population as a collection of numbers. For example, if we are studying income distribution of the US population, we look at list of annual incomes of the whole U S population as the population.

Also note, that this list of numbers or the population is unknown for a statistician; because if it was known then there will be nothing for the statistician to study there.

The N-VALUE: The total number of members in the population under study is called the N-value of the population. In fact, it is more commonly known as the population size.

The N-value is often unknown and must be estimated because either an accurate head count of all the members in the population is impossible or too expensive. In case this N-value will be unknown, and you may need to estimate the N-value by statistical methods. The following is a method of estimating N-values.

1.2 The Capture-recapture Method: Estimating N-value

Suppose we want to estimate the number of fish in Clinton Lake. Let N be the number of fish in the lake. Using the capture-recapture method, we do the following.

Step 1. (The capture) Capture a sample of m fish, tag them, and release them back into the water.

Step 2. (The recapture) After everything has settled down, capture a new sample of n fish. Count the number of tagged fish. Suppose that k of them are tagged. It is reasonable to assume that, approximatley, m/N=k/n.

We have an estimate N of N given by

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1.3 Statistical Studies: Census, Surveys, Public Opinion Polls, Clinical Studies

Census

Article 1 and Article 2 of the Constitution of the United States mandates that a national census be conducted every ten years. By census we mean an official enumeration of the population. Not only in the United States, but all over the world, a census is conducted every ten years. Following are some comments about census:

1. Originally the intent of the census was to count heads for taxes and representation. That is why it may also become a political issue as it did during the year 2000 census.
2. Census is one major source of data about the population, and the United Nations assumes a role in the worldwide census.
3. Census has often failed to count all members of the population. It is believed that a complete count is not really possible.
4. In the 2000 census, the U.S. population was counted by using statistical techniques. The Congress and the administration fought over this law, and the law was challenged in the courts.

Surveys

A more realistic and economical alternative to census is to collect data only from a small subgroup and then use this data to make inferences about the whole population. This approach is called a survey, and the subgroup of the population from which the data is collected is called a sample.

The basic idea behind a survey is that if we can find a "representative" sample of the whole population (that means it is not biased) then anything we need to know about the population can be derived from that sample.

Public Opinion Polls

We all know about public opinion polls. During any election season, about a dotzen polling organizations publish poll numbers. You can look at any standard textbook for a general discussion on polls. Some of them would explain how and why the predictions made by various opinion polls in the presidential elections in 1936 (Franklin Roosevelt vs. Alfred Landon) and 1948 (Harry Truman vs. Thomas Dewey) went wrong. It happened because the contemporary sampling methods were not sophisticated enough, and the samples the polsters drew failed to represent the whole population.

These days, it is fairly common that about a dotzen polls, predicting opposite outcome during an election season. It happens because of their failure to collect a representative sample. Traditionally, such polls used the listing in the telephone books. In the recent past, the advent of cell phones have created a confussion in polling industry, because cell phones are not listed and some people do not own a land phone.

Clinical Studies

When a vaccine or a new drug is tested, the statistical methods used are interesting. Following are some of the the main points regarding the process:

1. We pick two samples to be called the control group and the treatment group. The two samples need not have the same size.
2. The treatment group receives the treatment, and the control group does not receive the treatment.
3. Both the groups are ignorant about who is receiving the treatment and who is not.
4. Finally, the two groups are compared. If the treatment group does better than the control group, then it is accepted that the treatment is working.

1.4 Sampling Methods

Random Sampling

Developing a "representative sample" is a real challenge for a statistician (rest is mathematics and is easy because it has already been worked out). If a statistician tries to pick a sample, his/her human bias is essentially bound to result in a "biased sample." Whatever method we use to select a sample, the selection of the sample members must be random. That means that mathematics and methods of chance must guide the selection of sample members. A sample picked in such a manner is called a random sample, and the method is called random sampling.

Another important concern regarding sampling is its cost.

We briefly describe two methods of random sampling here.

1. In the method of simple random sampling each member of the population has an equal chance of being selected in the sample.
2. The other method of sampling is called stratified sampling: First, divide the population into categories, called strata, and randomly select a sample from these strata. Then further divide the chosen strata into categories, called substrata, and select a random sample of substrata from each of those strata. The process is continued for a number of times.

Sample Size

The sample size required for statistical studies need not be very large, even when the population is large. In practice, it is often less than 1500. If you follow CNN polls or others, they normally sample from 700 to 1200 people.

The job of a statistician is to make inferences about a large population on the basis of a (small) sample.

1. Any numerical value computed from the sample data is called a statistic.
2. Any numerical value computed from the whole population data is called a parameter.
3. Unless the population is small, the actual values of a parameter is never known. However, because the samples are small, we can always compute the actual values of the statistics. The game here is to estimate the parameters by appropriate statistics.

Example. Suppose we want to understand the income distribution of the U.S. population, and we want to know the average income of the U.S. population.

1. Here the average U.S. income is a parameter.
2. Because it is almost impossible to compute the actual value of the parameter average U.S. income, we take a sample (say of size 1500) and compute the average income of the sample members. This sample average is a statistic.
3. It is reasonable to use this (statistic) sample average income as an estimate for the (parameter) average U.S. income.

Sampling Error

A statistic used to estimate a parameter is only an estimate. We do not expect that the value of the statistic is exactly equal to the value of the parameter. In the above example, we would not expect the sample average income to be exactly equal to the average U.S. income. The difference between the actual value of the parameter and the computed/observed value of the statistic used to estimate the parameter is called the sampling error.

There are two types of sampling errors:

1. Chance error: Because a sample is not the whole population, a statistic cannot be the exact value of the parameter. Given that other things are "perfect" and identical, two different samples will produce two different values of the statistics. You get different estimates (i.e., the value of the statistic) from different samples, for the same parameter. The error in estimation that arises this way is called the chance error. This error arises out of the sampling variability, and the choice of sample belongs to randomness or chance. Statisticians are comfortable with chance error for various reasons.
   1. First, the very nature of statistical methods makes this error unavoidable.
   2. Second, by increasing sample size this error can also be controlled.
   3. Finally, the statistician can tell us how often this error exceeds a tolerable limit.
      
      
2. Sampling bias: The error that arises from poor sampling is called sampling bias. Although many sophisticated methods of sampling are available, implementation is not easy. In any case, sampling bias can be eliminated by strictly and properly implementing the sampling methods. Of course, the cost of sampling is the casualty.