Reading Students Casino

Casino Games Activity 6

8

Data Analysis What are some other data trends we can find with scatterplots? Year

0

1000

1750

1800

1930

1960

Population (billions)

0.3

0.32

0.8

1

2

3

Year

1974

1987

1990

1992

1995

1999

Population (billions)

4

5

5.2

5.4

5.7

6

The table above shows data for world population milestones compared to the year in which they occurred. A quick look at the data shows that while it took over 1800 years for the world’s population to reach 1 billion, it took only 130 years to add another 1 billion to the first billion. Because the rate of change in the population is not consistent, this data will not be linear. But what if we assumed it was? Would we get an error? Would the computer tell us it wasn’t? No. The computer will give you a linear equation for this data just like any other data. This is why you have to use common sense. You can’t rely solely on the computer/calculator without thinking. Here’s the linear line of best fit for the data above. As you can see, the computer had no problem giving me a linear equation, but it doesn’t show the trend in the data.

Finding a linear equation to fit the data is called linear regression, so it’s clear in this case that we need to use nonlinear regression. It’s more difficult to find tools to do nonlinear regressions. Graphing calculators can. Your job, no matter what the technology, is to choose an appropriate model. A situation like population growth is usually best described by an exponential model. Here’s how an exponential model fits the data.

nonlinear

linear

In the summer, shark attacks and ice cream sales are found to be correlated at a beach. Does this mean sharks hate when ice cream is sold, and therefore this causes them to bite swimmers? Maybe, but we can’t say for sure.

If you ever see a test question where it says “these things are correlated. What can we conclude?”, then eliminate any answer that says something about one thing causing the other.

Correlation never never never never means by itself that variable A causes variable B to change (or vice versa). There might be a causal relationship, but correlation alone is not enough to determine this.

If two things are correlated, does that mean one thing causes the other thing? Nooooooooooooooooooooooooooooo!!!!!!!!!!!!!!!!!!!!

A negative correlation is when variable A goes up, variable B goes down. When variable A goes down, variable B goes up.

A positive correlation is when variable A goes up, variable B also goes up. When variable A goes down, variable B also goes down.

What is correlation? Correlation is the term for when two things change together. When the temperature outside goes up, so do the sales of ice cream. There’s a correlation between temperature and ice cream sales. There’s a correlation between brown hair and brown eyes. There’s also a correlation between hours of TV watching and grades, except that the more a student watches TV, the lower their grades tend to be. This is an example of a negative correlation.

Why?

Draw a scatterplots with the following r values: 1. r = 1 2. r = -1 3. r = 0

Problem Set 14

The coefficient of correlation tells how linear the data are. In this picture, you can see that the most linear scatterplots have numbers closer to 1 and -1. The scatterplots with no relationships have numbers closer to 0.

the number is -1, the correlation is perfectly negative.

! The coefficient of correlation is a fancy term for a number that goes along with a correlation. ! If the number is 1, the correlation is perfect. If the number is zero, there is no correlation. If

How do we use correlation? In the last activity, we came up with linear lines of best fit using linear regression. One way to find out if this line of best fit will be useful at all is to determine how good the correlation is using something called the coefficient of correlation, sometimes shown as r. Once we have that, r 2 shows us how good the line of best fit will be for making predictions. Again, closer to 1 is better for r 2 .

10. Below, four data sets are presented for which a computer gave the same r-squared value and the same regression line. Write a paragraph describing what you see about the similarities and differences among the sets. Be sure to talk about each set individually. Does the line of best fit work for all of them? Would a nonlinear regression be more appropriate for any? Does the term “outlier” help to describe what’s going on with any of the sets? What does this say about the necessity of using common sense instead of just trusting that a computer or calculator knows what it’s doing?