computation of Karl Pearson Correlation Coefficient and interpretation of result.

Question

The following table presents the values of X and Y sets of observations:

X	12	15	16	17	14
Y	110	124	134	139	124

a.      Use the information provided to compute Karl Pearson Correlation Coefficient and interpret the result.

b.     Compute the value of coefficient of determination and interpret its value.

Answer 1

Part (a)

The Karl Pearson correlation coefficient is defined as:

(∑(X_i - X*)∑(Y_i - Y*))/(√∑(X_i - X*)²∑(Y_i - Y*)²)

where X* denotes the mean of the X values and Y* denotes the mean of the Y values.

X*=(12+15+16+17+14)/5=14.8

Y*=(110+124+134+139+124)/5=126.2

Numerator:

∑(X_i - X*)∑(Y_i - Y*) = ∑(X_i - 14.8)∑(Y_i - 126.2)

= [(12-14.8)+...+(14-14.8)]·[(110-126.2)+...+(124-126.2)]

= 84.2

Denominator:

√∑(X_i - X*)²∑(Y_i - Y*)² = √∑(X_i - 14.8)²∑(Y_i - 126.2)²

= √[(12-14.8)²+...+(14-14.8)²]·[(110-126.2)²+...+(124-126.2)²]

=85.7

Then the Karl Pearson correlation coefficient is 84.2/85.7 ≈ 0.98.

Part (b)

The coefficient of determination R² is the square of the answer to part (a), so R²=0.98²≈0.96. R² is always a number between 0 and 1, and a number close to 1 indicates a strong relationship between the independent and dependent variables. Equivalently, the data points are very close to the regression line.