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:

 

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            Substituting x = 362 and rounding yields:          

 

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