Statistics and Probability-Part 9

Correlating Data A ___________________ ____________ shows information about two variables that are paired in some way and can help you see whether a relationship exists between the variables. By looking for a trend in the scatter plot, you can see that a relationship exists. If the points on a scatter plot are close to forming a straight line, there is a ___________ correlation. If the points are scattered all over and do not follow a pattern, there is a __________ correlation. Correlations can also be positive or negative. A negative correlation exists if the values of one variable increase while the values of the other decrease. A scatter plot with a strong correlation does not imply that a cause-effect relationship between the variables exists. Other kinds of analysis must be done to determine whether a change in one variable causes a change in the other.

In 1962, researchers studied 100 newborn babies to see whether there was a relationship between the length of the body and the circumference of the head. The table shows the results of this study. Most tallies appear along the diagonal that extends from the lower left to the upper right corner of the graph.

In relation to the diagonal, where do you find the babies with large heads relative to their lengths? As the body lengths increase, what happens to the head sizes of the babies? The scatter plot displays a picture of the information in the table.

Explain why the axes do not start at (0, 0) Explain why the axes do not start at (0, 0). What information seems to get lost in the scatter plot when compared with the table? Are there babies for whom body lengths are almost equal to head sizes?

To show body lengths, head circumferences, and the number of babies studied, a three- dimensional graph can be drawn.

What information does the dark bar in the graph represent What information does the dark bar in the graph represent? Check this information against the data in the table. What are the advantages and disadvantages of a three-dimensional graph?

Scatter plots are often made to investigate the relationship between two variables. The points on a scatter plot may look like a “cloud.” -When the cloud is long and slender, there is a strong correlation between the two variables. -When the correlation is strong, knowing something about one of the variables helps you know how the other variable will behave. -If the cloud of points is very scattered or in a circle, no underlying relationship exists between the variables. -Knowing something about one of the variables cannot tell you about the other. In this case, there is no correlation between the two variables.

Describe the correlation in the scatter plots Describe the correlation in the scatter plots. Indicate whether there seems to be no correlation, a weak correlation, or a strong correlation.

For which of the above scatter plots can you best predict the value of y when x=4? Explain your reasoning. What do you think is meant by the phrase “negative correlation”? Which of the scatter plots on the previous page show a negative correlation?

For each of the following cases, decide whether there is no correlation, a strong correlation, or a weak correlation between the two variables mentioned. If there is a correlation, is it positive or negative? • a person’s height and pulse rate • the number of hours of sports training per week and pulse rate • the height of a dinosaur and the length of its tail • results on a math test and a science test • temperature outside in the summer and kilowatt hours of electricity used • number of children per household and number of television sets per household • number of hours students study and their grade point averages

In a cause-effect relationship between two variables, a change in one variable directly causes a change in the other. If you expect a strong correlation between the two variables, can you be sure that a cause-effect relationship exists? Why or why not? Do you think there is a cause-effect relationship in any of the situations in the problems above? Explain your reasoning.