Linear Regression, Part 1

Linear Correlation and Regression

Part 1: Association

In this module we will look at methods for studying the relationship between two variables. Many of the statistical ideas here are due to Sir Francis Galton, an English anthropologist who was interested in hereditary processes. Knowing that children bear some resemblance to their parents, Sir Galton was interested in how strong that resemblance was. While Galton inspired many statistical questions, the formal development of many of his ideas is due to his colleague Karl Pearson. Some of the data collected by Galton and Pearson are available, but the data sets are rather large for our purposes, so we will look at some test scores for a math class.

Consider the test data given below. For each test, compute the average (also known as the mean) and the standard deviation of the scores. What do you notice?

Student	Test 1 Score	Test 2 Score	Test 3 Score
Liza	70	75	60
Geoff	91	88	74
Kirsten	88	91	81
Jeremy	87	90	66
Ko	45	41	50
Clyde	81	80	95
Serpil	77	74	86
Shameeka	74	76	68
Zoe Ann	73	74	83
Yiannis	95	91	94
Alfred	69	70	88
Suzi	67	65	59
Ashya	65	63	69
Sreek	60	66	77
Lionel	83	81	75

Use your computer algebra system to draw the following scatter plots:

Test 2 scores versus Test 1 scores,
Test 3 scores versus Test 2 scores.

Click here for help creating a scatter plot with Maple.

We say that two variables are positively associated when larger values of one variable tend to occur with larger values of the other variable. When larger values of one variable occur with smaller values of the other variable, we say the variables are negatively associated. What do the scatter plots tell you about the association between a student's score on Test 1 and his or her score on Test 2? What about the association between a student's score on Test 2 and Test 3?

Consider relationship between a student's test average and the number of times he or she misses class during the term. Would you expect a positive or negative association between these variables? Why?

If a student scored an 82 on Test 1, what do you predict he would score on Test 2? If the student scored an 82 on Test 2, what would you predict for his score on Test 3? Which prediction do you expect to be more accurate? Why?

In the scatter plot of Test 2 scores versus Test 1 scores, the points appear to be tightly clustered along a line: there is a strong linear association between scores on the first two tests. In the second scatter plot, the clustering is looser (and the linear association is weaker). How is this observation related to the previous question?