4. Define correlation. Calculate and analyze the correlation coefficient between the number of study hours and the number of sleeping and interpret it.

Number of Study Hours 

2

4

6

8

10

Number of Sleeping Hours

10

9

8

7

6

Also find coefficient of determination and interpret it.

### Definition of Correlation
Correlation is a statistical measure that expresses the extent to which two variables are linearly related. It shows the strength and direction of a linear relationship between two variables.

### Calculation of Correlation Coefficient
The correlation coefficient, \( r \), between two variables can be calculated using the formula:
\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]

Using the provided data### Definition of Correlation
Correlation is a statistical measure that expresses the extent to which two variables are linearly related. It shows the strength and direction of a linear relationship between two variables.

### Calculation of Correlation Coefficient
The correlation coefficient, \( r \), between two variables can be calculated using the formula:
\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]

Using the provided 

- Number of Study Hours: \([2, 4, 6, 8, 10]\)
- Number of Sleeping Hours: \([10, 9, 8, 7, 6]\)

Let's calculate it step-by-step:

1. **Sum of the Study Hours (x) and Sleeping Hours (y):**
   \[ \sum x = 2 + 4 + 6 + 8 + 10 = 30 \]
   \[ \sum y = 10 + 9 + 8 + 7 + 6 = 40 \]

2. **Sum of the products of Study Hours and Sleeping Hours (xy):**
   \[ \sum xy = (2 \times 10) + (4 \times 9) + (6 \times 8) + (8 \times 7) + (10 \times 6) = 20 + 36 + 48 + 56 + 60 = 220 \]

3. **Sum of the squares of Study Hours and Sleeping Hours:**
   \[ \sum x^2 = 2^2 + 4^2 + 6^2 + 8^2 + 10^2 = 4 + 16 + 36 + 64 + 100 = 220 \]
   \[ \sum y^2 = 10^2 + 9^2 + 8^2 + 7^2 + 6^2 = 100 + 81 + 64 + 49 + 36 = 330 \]

4. **Number of observations (n):**
   \[ n = 5 \]

Now, substituting these values into the correlation coefficient formula:
\[ r = \frac{5(220) - (30)(40)}{\sqrt{[5(220) - (30)^2][5(330) - (40)^2]}} \]
\[ r = \frac{1100 - 1200}{\sqrt{[1100 - 900][1650 - 1600]}} \]
\[ r = \frac{-100}{\sqrt{200 \times 50}} \]
\[ r = \frac{-100}{\sqrt{10000}} \]
\[ r = \frac{-100}{100} \]
\[ r = -1 \]

### Interpretation of the Correlation Coefficient
- **Correlation Coefficient (r):** -1
  - This value indicates a perfect negative linear relationship between the number of study hours and the number of sleeping hours. As study hours increase, sleeping hours decrease at a consistent rate.

### Coefficient of Determination
The coefficient of determination, \( r^2 \), represents the proportion of the variance in the dependent variable that is predictable from the independent variable.

\[ r^2 = (-1)^2 = 1 \]

### Interpretation of the Coefficient of Determination
- **Coefficient of Determination (r²):** 1
  - This means that 100% of the variability in the number of sleeping hours can be explained by the number of study hours.

Would you like details or have any questions?

Here are some questions you might want to ask:
1. What does a correlation coefficient of -1 signify?
2. How is the coefficient of determination interpreted?
3. Can you explain how these values were calculated?
4. What does it mean when the correlation coefficient is negative?
5. How does the correlation coefficient relate to linear regression?

**Tip:** The correlation coefficient ranges from -1 to 1, where values close to these extremes indicate a strong relationship.

Learn how to calculate and interpret the correlation coefficient and coefficient of determination using study hours and sleeping hours data. Understand the relationship between these variables with a step-by-step explanation.

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