**MATH 111 –
STATISTICS – DR. NARDO**

**CORRELATION &
LINEAR REGRESSION EXAMPLE**

In 1862, the German musicologist Ludwig von Köchel made a chronological list of the musical works of Wolfgang Amadeus Mozart. This list is the source of the K-numbers or Köchel numbers that now accompany Mozart’s works. For example, the Sinfonia Concertante in E-Flat Major is K.364. The table below details these variables for a sample of Mozart’s works.

K-number X |
1 |
75 |
155 |
219 |
271 |
351 |
425 |
503 |
575 |
626 |

Year |
1761 |
1771 |
1772 |
1775 |
1777 |
1780 |
1783 |
1786 |
1789 |
1791 |

Here the K-number is the explanatory variable X because we will use the K-number to predict the year. The year is the response variable Y; it is the variable about which predictions are made. In the calculator, put the X’s in list one and the Y’s in list two.

The picture below can be interpreted as a scatterplot (if each axis is labeled with the appropriate variable, if values are put on tick marks to make the scale concrete, and if a title is at the top).

INITIAL OBSERVATIONS

The data appear to have a linear relationship with positive association.

WOULD IT BE PRUDENT TO USE LEAST SQUARES LINE (LINEAR
REGRESSION)

TO MAKE PREDICTIONS IN THIS CASE?

Yes, I would use the least squares line (linear regression) to make predictions. We can use this line to predict values of the response variable/Y/year from values of the explanatory variable/X/K-number.

GEOMETRIC REASON

Since the data points cluster tightly along a line, the variables have a linear relationship.

ANALYTIC REASON

Since the correlation coefficient (r = +0.9830315371) is extremely close to positive one, the variables have a linear relationship (with positive association).

USE THE LEAST SQUARES LINE TO PREDICT THE YEAR FOR K.364

The equation of the least squares line is:

_{
}.

Substituting *x* = 362 and rounding yields:

_{
}.