MAT 1000

Mathematics in Today's World

MAT 1000 Topics

1. Statistics

Organize, summarize, and describe data

2. Probability

Looking for patterns in uncertain events

3. Mathematics of Information

Using math to transmit, store, and protect information

Introduction to Statistics

1. What is statistics?

2. The structure of data

3. Two methods to collect data

What is Statistics?

Three ways to think of statistics:

• numbers plus context

• decision making under

uncertainty

• the science of data

What is Statistics?

Numbers plus contextStatistics often relies on mathematics to manipulate numbers, but unlike in math, the numbers statisticians consider have a context.

The goal in doing statistics is to answer practical problems.

What is Statistics?

Decision making under uncertaintyPolicy makers, business leaders, scientists and many other people often face difficult decisions.

Statistics can help people to make informed decisions.

What is Statistics?

The science of dataScientists study particular aspects of the world: biologists study living creatures, astronomers study the universe, and so on.

Statisticians are scientists, and the particular aspect of the world that they study is data.

What is Statistics?

The science of dataStatisticians ask questions like:

• How can data be collected?•How can data be interpreted?• How can data be summarized?

As we will see, the methods used to collect, interpret, and summarize data vary depending on the circumstances.

So if statistics is the science of data, then what is data?

The structure of data

Data are attributes or characteristics of people or objects.

We call the people or objects the individuals.

The characteristics or attributes are called variables.

The structure of data

Example

If we are studying students at Wayne State, we might collect data on their age, GPA, and major.

The individuals are the students, the variables are: age, GPA, and major.

The structure of data

Example

If we are studying the economies of major cities in the U.S., we might collect data on their populations, median household incomes, and unemployment rates.

The individuals are the cities, the variables are: population, median household incomes, and unemployment rates.

The structure of data

Individuals may be people, animals, or concrete things (like cars or houses).

They can also be abstract things like days or countries.

Variables are of two kinds:

• Numeric (GPA, high temperature)

• Non-numeric (a student’s major)

We will usually consider only numeric variables.

The structure of data

Data should be collected to address some particular question.

Let’s consider some specific questions.

For each question, we will determine the individuals and variables we will need to study.

The structure of data

Question: Does hypnosis affect the immune system?

What kind of data would be relevant to addressing this question?

The individuals we should study are people.

For these individuals, we would need to measure the strength of their immune system. We can use white blood cell counts. Our variables could be white blood cell counts before hypnosis and after hypnosis.

The structure of data

Question: In K-12 education, will decreased class sizes lead to improved student learning?

Here we want to know what happens to students when they are in a smaller class. So our individuals are K-12 students.

For each student, we want to know how large their classes are.

We’d also like to know what they are learning and how well. This could be done using scores on a standardized test.

The structure of data

Question: What is the unemployment rate in America?

This is a question about Americans of working age, so these will be our individuals.

For these individuals, the variable we’d like to know is quite simple: are they employed or not?

Two methods to collect data

Observational study: the researchers performing the study merely observe the individuals.

Experiment: the researchers attempt to modify, influence, or affect the individuals they are studying.

Which method we choose depends on the question being asked.

Two methods to collect data

Experiments are the best way to study cause and effect relationships.

However sometimes there are practical or ethical considerations that prevent researchers from using experiments.

When it is not possible to use an experiment, then we can use an observational study.

Two methods to collect data

How does the government compute the unemployment rate?

“The government conducts a monthly survey called the Current Population Survey (CPS) to measure the extent of unemployment in the country…Each month, highly trained and experienced Census Bureau employees contact the 60,000 eligible sample households and ask about the labor force activities (jobholding and job seeking) or non-labor force status of the members of these households.”

Source: http://www.bls.gov/cps/cps_htgm.htm

Two methods to collect data

What are the individuals?

What variables are collected?

Is this an observational study or an experiment?

Two methods to collect data

Does hypnosis affect the immune system?

A professor recruits 65 college students. Each student’s white blood cell count is measured. Students are randomly assigned to one of two groups.

Students in the first group are hypnotized. Nothing is done to students in the second group (this is called a control group).

A week later, the professor re-measures the white blood cell counts of all 65 students. The hypnotized group showed a larger jump in white blood cell count. Does hypnosis affect the immune system?

Two methods to collect data

What are the individuals?

What variables are collected?

Is this an observational study or an experiment?

Two methods to collect data

What are the heart diseases risks of weight gain for women?

A study started in 1976 with 115,818 women aged 30 to 55 years and without a history of previous coronary heart disease (CHD).Each woman’s weight (body mass) was determinedEach woman was asked her weight at age 18.The cohort of women were followed for 14 years.The number of CHD (fatal and nonfatal) cases were counted (1292 cases).

Two methods to collect data

Results: compare those who gained less than 11 pounds (from age 18 to current age) to the others.11 to 17 lbs: 25% more likely to develop heart disease17 to 24 lbs: 64% more likely24 to 44 lbs: 92% more likelymore than 44 lbs: 165% more likely

“Weight, weight change, and coronary heart disease in women.” W.C. Willett, et. al., vol. 273(6), Journal of the American Medical Association, Feb. 8, 1995.

Reported in Science News, Feb. 18, 1995, p. 108

Two methods to collect data

What are the individuals?

What variables are collected?

Is this an observational study or an experiment?

Two methods to collect data

What are the individuals?

Who will the conclusion be applied to?Specific? General?

What variables are collected?

Attributes of the individuals.

Is this an observational study or an experiment?

Are the researchers assigning individuals to get different treatments?