Let's find the required statistical measures step-by-step based on the given IQ and GPA data.
### 1. Calculate the Sum of Squares for IQ, \( SS(x) \):
The formula for sum of squares for IQ is given by:
[tex]\[ SS(x) = \sum (x_i - \bar{x})^2 \][/tex]
where \( x_i \) are the IQ values and \( \bar{x} \) is the mean IQ.
### 2. Calculate the Sum of Squares for GPA, \( SS(y) \):
The formula for sum of squares for GPA is given by:
[tex]\[ SS(y) = \sum (y_i - \bar{y})^2 \][/tex]
where \( y_i \) are the GPA values and \( \bar{y} \) is the mean GPA.
### 3. Calculate the Sum of Products, \( SS(xy) \):
The formula for sum of products is given by:
[tex]\[ SS(xy) = \sum (x_i - \bar{x})(y_i - \bar{y}) \][/tex]
### 4. Calculate the Correlation Coefficient, \( r \):
The formula for the correlation coefficient is given by:
[tex]\[ r = \frac{SS(xy)}{\sqrt{SS(x) \cdot SS(y)}} \][/tex]
Given the data:
[tex]\[
\begin{array}{ll}
IQ & GPA \\
117 & 3.9 \\
93 & 2.7 \\
102 & 2.9 \\
110 & 3.1 \\
88 & 2.4 \\
75 & 1.9 \\
\end{array}
\][/tex]
### Solution:
Using the given formulas and the data provided:
#### Sum of Squares for IQ, \( SS(x) \):
[tex]\[ SS(x) = 1173.5 \][/tex]
#### Sum of Squares for GPA, \( SS(y) \):
[tex]\[ SS(y) = 2.29 \][/tex]
#### Sum of Products, \( SS(xy) \):
[tex]\[ SS(xy) = 50.15 \][/tex]
#### Correlation Coefficient, \( r \):
[tex]\[ r = 0.97 \][/tex]
Therefore, the calculated values are:
- \( SS(x) = 1173.5 \)
- \( SS(y) = 2.29 \)
- \( SS(xy) = 50.15 \)
- The correlation coefficient, \( r \), is \( 0.97 \)
These values represent the squared deviations for IQ, GPA, their product, and the measure of the linear relationship between IQ and GPA, respectively.
