To find the correlation coefficient and the strength of the model, we will follow these steps:
1. List the data:
[tex]\[
\text{Hours Spent Studying} = [1, 2, 3, 4, 5]
\][/tex]
[tex]\[
\text{Test Scores} = [72, 80, 90, 82, 95]
\][/tex]
2. Calculate the mean of the hours and scores:
[tex]\[
\text{Mean of Hours} = \frac{1 + 2 + 3 + 4 + 5}{5} = 3.0
\][/tex]
[tex]\[
\text{Mean of Scores} = \frac{72 + 80 + 90 + 82 + 95}{5} = 83.8
\][/tex]
3. Calculate the numerator for the correlation coefficient:
The numerator is obtained by summing the products of the differences of each value from their respective means:
[tex]\[
\sum ((\text{Hours}_i - \text{Mean Hours}) \cdot (\text{Scores}_i - \text{Mean Scores}))
\][/tex]
[tex]\[
= [(1-3.0)(72-83.8) + (2-3.0)(80-83.8) + (3-3.0)(90-83.8) + (4-3.0)(82-83.8) + (5-3.0)(95-83.8)]
\][/tex]
[tex]\[
= 48.0
\][/tex]
4. Calculate the denominator for the correlation coefficient:
The denominator is the product of the square root of the sum of squared differences from the mean for hours and scores:
[tex]\[
\sqrt{\sum (\text{Hours}_i - \text{Mean Hours})^2 \cdot \sum (\text{Scores}_i - \text{Mean Scores})^2}
\][/tex]
[tex]\[
= \sqrt{[(1-3.0)^2 + (2-3.0)^2 + (3-3.0)^2 + (4-3.0)^2 + (5-3.0)^2] \cdot [(72-83.8)^2 + (80-83.8)^2 + (90-83.8)^2 + (82-83.8)^2 + (95-83.8)^2]}
\][/tex]
[tex]\[
= 56.63920903402518
\][/tex]
5. Calculate the correlation coefficient (Pearson's r):
[tex]\[
r = \frac{\text{Numerator}}{\text{Denominator}} = \frac{48.0}{56.63920903402518} = 0.8474694618557385
\][/tex]
6. Evaluate the strength of the model:
- A correlation coefficient [tex]\( |r| > 0.7 \)[/tex] is considered a "strong" correlation.
- Since [tex]\( |0.8474694618557385| \approx 0.847 \)[/tex], which is greater than 0.7, the strength of the model is "strong."
Answers:
1. The correlation coefficient is [tex]\( 0.8474694618557385 \)[/tex].
2. The strength of the model is "strong."
