Consider the following data :
`{:(x,5,7,8,4,6),(y,2,4,3,2,4):}`
What is the regression equation of y on x?

Options

A. `y=0.6+0.4x`
B. `y=0.7+0.3x`
C. `y=6+5x`
D. `y=4+9x`

Explanation

To find the regression equation of \( y \) on \( x \) using the given data, we will follow these steps:

### Step 1: Calculate the Means of \( x \) and \( y \)

The data provided is:
- \( x = (5, 7, 8, 4, 6) \)
- \( y = (2, 4, 3, 2, 4) \)

**Calculating \( \bar{x} \) (mean of \( x \)):**
\[ \bar{x} = \frac{\sum x}{n} = \frac{5 + 7 + 8 + 4 + 6}{5} = \frac{30}{5} = 6 \]

**Calculating \( \bar{y} \) (mean of \( y \)):**
\[ \bar{y} = \frac{\sum y}{n} = \frac{2 + 4 + 3 + 2 + 4}{5} = \frac{15}{5} = 3 \]

### Step 2: Calculate \( \sum xy \) and \( \sum x^2 \)

We will create a table to calculate \( xy \) and \( x^2 \):

| \( x \) | \( y \) | \( x^2 \) | \( xy \) |
|---------|---------|-----------|----------|
| 5 | 2 | 25 | 10 |
| 7 | 4 | 49 | 28 |
| 8 | 3 | 64 | 24 |
| 4 | 2 | 16 | 8 |
| 6 | 4 | 36 | 24 |
| **Total** | | **190** | **94** |

**Calculating \( \sum xy \) and \( \sum x^2 \):**
\[ \sum xy = 10 + 28 + 24 + 8 + 24 = 94 \]
\[ \sum x^2 = 25 + 49 + 64 + 16 + 36 = 190 \]

### Step 3: Calculate the Regression Coefficient \( b_{yx} \)

The formula for the regression coefficient \( b_{yx} \) is:
\[ b_{yx} = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} \]

Substituting the values:
- \( n = 5 \)
- \( \sum xy = 94 \)
- \( \sum x = 30 \)
- \( \sum y = 15 \)
- \( \sum x^2 = 190 \)

\[ b_{yx} = \frac{5 \cdot 94 - 30 \cdot 15}{5 \cdot 190 - 30^2} \]
\[ = \frac{470 - 450}{950 - 900} \]
\[ = \frac{20}{50} = 0.4 \]

### Step 4: Write the Regression Equation

The regression equation of \( y \) on \( x \) is given by:
\[ y - \bar{y} = b_{yx}(x - \bar{x}) \]

Substituting the values we have:
\[ y - 3 = 0.4(x - 6) \]

Expanding this:
\[ y - 3 = 0.4x - 2.4 \]
\[ y = 0.4x + 0.6 \]

### Final Regression Equation
The regression equation of \( y \) on \( x \) is:
\[ y = 0.4x + 0.6 \]

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