The following data gives the number of hours 10
 students spent studying and their corresponding grades on their midterm exams.

Hours Spent Studying	0
1
1.5
2
3
3.5
4
4.5
5.5
6
Midterm Grades	60
63
69
72
75
78
81
84
87
90

Step 3 of 3 :  Calculate the correlation coefficient, r. Round your answer to three decimal places.

To calculate the correlation coefficient \( r \), we use the following formula:

\[
r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{\left[n\sum x^2 - (\sum x)^2\right]\left[n\sum y^2 - (\sum y)^2\right]}}
\]

Where:
- \( x \) represents the hours spent studying,
- \( y \) represents the midterm grades,
- \( n \) is the number of data points.

### Step 1: Organize the data
| Hours (x) | Grades (y) |
|:---------:|:----------:|
|     0     |     60     |
|     1     |     63     |
|    1.5    |     69     |
|     2     |     72     |
|     3     |     75     |
|    3.5    |     78     |
|     4     |     81     |
|    4.5    |     84     |
|    5.5    |     87     |
|     6     |     90     |

### Step 2: Calculate the sums needed
We will need the following values for the formula:
- \( \sum x \) (sum of hours),
- \( \sum y \) (sum of grades),
- \( \sum xy \) (sum of the products of x and y),
- \( \sum x^2 \) (sum of squares of x),
- \( \sum y^2 \) (sum of squares of y),
- \( n \) (number of data points, which is 10).

Let's calculate these step by step.The correlation coefficient \( r \) is approximately 0.993, rounded to three decimal places. This indicates a very strong positive correlation between the hours spent studying and the midterm grades.

Would you like more details or have any further questions?

Here are some related questions to explore:
1. What does a correlation coefficient of 0.993 imply in terms of data relationship?
2. How do you interpret a negative correlation coefficient?
3. Can the correlation coefficient indicate causation?
4. How would you calculate the line of best fit from this data?
5. How does changing one data point affect the correlation coefficient?

**Tip:** A correlation coefficient close to 1 or -1 suggests a strong linear relationship, but it doesn't imply causation!

The following data gives the number of hours 10 students spent studying and their corresponding grades on their midterm exams.

Hours Spent Studying: 0, 1, 1.5, 2, 3, 3.5, 4, 4.5, 5.5, 6
Midterm Grades: 60, 63, 69, 72, 75, 78, 81, 84, 87, 90

Calculate the correlation coefficient, r. Round your answer to three decimal places.

Learn how to calculate the correlation coefficient (r) for the given data set of study hours and midterm grades. This example uses the correlation coefficient formula to find r = 0.993, indicating a strong positive linear relationship. Suitable for Grades 10-12.

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