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 Y 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: .