Question

A 2-column table with 6 rows. The first column is labeled hours spent studying with entries 1, 2, 3, 4, 5. The second column is labeled test score with entries 72, 80, 90, 82, 95.
Claudia records the hours she spent studying and her test scores for 5 tests.

What is the correlation coefficient?



What is the strength of the model?

Answer 1

To find the correlation coefficient, we can use the formula for Pearson's correlation coefficient \( r \). The entries in the table are as follows:

- Hours spent studying (X): 1, 2, 3, 4, 5
- Test scores (Y): 72, 80, 90, 82, 95

### Step 1: Calculate Means
Let's calculate the means of X and Y.

\[
\bar{X} = \frac{1 + 2 + 3 + 4 + 5}{5} = \frac{15}{5} = 3
\]
\[
\bar{Y} = \frac{72 + 80 + 90 + 82 + 95}{5} = \frac{419}{5} = 83.8
\]

### Step 2: Calculate the Numerator and Denominator
Next, we need to calculate the numerator and denominator for the correlation formula.

1. **Numerator:**
\[
\text{Numerator} = \sum{(X_i - \bar{X})(Y_i - \bar{Y})}
\]

Let's calculate each part:

- For \( X_1 = 1, Y_1 = 72 \): \( (1 - 3)(72 - 83.8) = (-2)(-11.8) = 23.6 \)
- For \( X_2 = 2, Y_2 = 80 \): \( (2 - 3)(80 - 83.8) = (-1)(-3.8) = 3.8 \)
- For \( X_3 = 3, Y_3 = 90 \): \( (3 - 3)(90 - 83.8) = (0)(6.2) = 0 \)
- For \( X_4 = 4, Y_4 = 82 \): \( (4 - 3)(82 - 83.8) = (1)(-1.8) = -1.8 \)
- For \( X_5 = 5, Y_5 = 95 \): \( (5 - 3)(95 - 83.8) = (2)(11.2) = 22.4 \)

Now sum these results:
\[
\text{Numerator} = 23.6 + 3.8 + 0 - 1.8 + 22.4 = 48
\]

2. **Denominator:**
\[
\text{Denominator} = \sqrt{\sum{(X_i - \bar{X})^2} \sum{(Y_i - \bar{Y})^2}}
\]

Calculate \( \sum{(X_i - \bar{X})^2} \):
- \( (1 - 3)^2 = 4 \)
- \( (2 - 3)^2 = 1 \)
- \( (3 - 3)^2 = 0 \)
- \( (4 - 3)^2 = 1 \)
- \( (5 - 3)^2 = 4 \)
\[
\sum{(X_i - \bar{X})^2} = 4 + 1 + 0 + 1 + 4 = 10
\]

Calculate \( \sum{(Y_i - \bar{Y})^2} \):
- \( (72 - 83.8)^2 = 138.24 \)
- \( (80 - 83.8)^2 = 14.44 \)
- \( (90 - 83.8)^2 = 38.44 \)
- \( (82 - 83.8)^2 = 3.24 \)
- \( (95 - 83.8)^2 = 125.44 \)
\[
\sum{(Y_i - \bar{Y})^2} = 138.24 + 14.44 + 38.44 + 3.24 + 125.44 = 319.8
\]

Now calculate the denominator:
\[
\text{Denominator} = \sqrt{10 \cdot 319.8} = \sqrt{3198} \approx 56.6
\]

### Step 3: Calculate the Correlation Coefficient
Now we compute \( r \):
\[
r = \frac{\text{Numerator}}{\text{Denominator}} = \frac{48}{56.6} \approx 0.847
\]

### Interpretation of the Correlation Coefficient
The correlation coefficient \( r \approx 0.847 \) indicates a strong positive correlation between hours spent studying and test scores.

### Strength of the Model
The strength of the correlation can be categorized as follows:
- **0 to 0.3:** Weak
- **0.3 to 0.7:** Moderate
- **0.7 to 1.0:** Strong

Since \( r \approx 0.847 \) falls in the range of strong positive correlation, we can conclude that there is a strong positive relationship between the hours spent studying and test scores.

### Final Summary
- **Correlation Coefficient**: \( r \approx 0.847 \)
- **Strength of the Model**: Strong positive correlation.