The following observations are given : (1,4), (2, 8), (3, 2), (4,12), (5,10), (6,14), (7,16), (8,6), (9,18).
Estimate the value ofy when the value of x is 10 and also estimate the value ofx when the value of y = 5

Answer

To solve the problem, we will follow these steps:

### Step 1: Organize the Data
We have the following observations:
- \( (1, 4) \)
- \( (2, 8) \)
- \( (3, 2) \)
- \( (4, 12) \)
- \( (5, 10) \)
- \( (6, 14) \)
- \( (7, 16) \)
- \( (8, 6) \)
- \( (9, 18) \)

Let's create a table for these values of \( x \) and \( y \):

| \( x \) | \( y \) |
|---------|---------|
| 1 | 4 |
| 2 | 8 |
| 3 | 2 |
| 4 | 12 |
| 5 | 10 |
| 6 | 14 |
| 7 | 16 |
| 8 | 6 |
| 9 | 18 |

### Step 2: Calculate the Mean Values
Calculate the mean of \( x \) and \( y \):

\[ \text{Mean of } x ( \bar{x} ) = \frac{\sum x}{n} = \frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9}{9} = \frac{45}{9} = 5 \]

\[ \text{Mean of } y ( \bar{y} ) = \frac{\sum y}{n} = \frac{4 + 8 + 2 + 12 + 10 + 14 + 16 + 6 + 18}{9} = \frac{90}{9} = 10 \]

### Step 3: Calculate Deviations from the Mean
Now, we will calculate the deviations from the mean for both \( x \) and \( y \):

| \( x \) | \( y \) | \( x - \bar{x} \) | \( y - \bar{y} \) |
|---------|---------|-------------------|-------------------|
| 1 | 4 | 1 - 5 = -4 | 4 - 10 = -6 |
| 2 | 8 | 2 - 5 = -3 | 8 - 10 = -2 |
| 3 | 2 | 3 - 5 = -2 | 2 - 10 = -8 |
| 4 | 12 | 4 - 5 = -1 | 12 - 10 = 2 |
| 5 | 10 | 5 - 5 = 0 | 10 - 10 = 0 |
| 6 | 14 | 6 - 5 = 1 | 14 - 10 = 4 |
| 7 | 16 | 7 - 5 = 2 | 16 - 10 = 6 |
| 8 | 6 | 8 - 5 = 3 | 6 - 10 = -4 |
| 9 | 18 | 9 - 5 = 4 | 18 - 10 = 8 |

### Step 4: Calculate \( x^2 \), \( y^2 \), and \( xy \)
Now, we will calculate \( x^2 \), \( y^2 \), and \( xy \):

| \( x \) | \( y \) | \( x - \bar{x} \) | \( y - \bar{y} \) | \( (x - \bar{x})^2 \) | \( (y - \bar{y})^2 \) | \( (x - \bar{x})(y - \bar{y}) \) |
|---------|---------|-------------------|-------------------|------------------------|------------------------|-----------------------------------|
| 1 | 4 | -4 | -6 | 16 | 36 | 24 |
| 2 | 8 | -3 | -2 | 9 | 4 | 6 |
| 3 | 2 | -2 | -8 | 4 | 64 | 16 |
| 4 | 12 | -1 | 2 | 1 | 4 | -2 |
| 5 | 10 | 0 | 0 | 0 | 0 | 0 |
| 6 | 14 | 1 | 4 | 1 | 16 | 4 |
| 7 | 16 | 2 | 6 | 4 | 36 | 12 |
| 8 | 6 | 3 | -4 | 9 | 16 | -12 |
| 9 | 18 | 4 | 8 | 16 | 64 | 32 |

### Step 5: Calculate Summations
Now we will calculate the summations:

\[ \sum (x - \bar{x})^2 = 16 + 9 + 4 + 1 + 0 + 1 + 4 + 9 + 16 = 60 \]

\[ \sum (y - \bar{y})^2 = 36 + 4 + 64 + 4 + 0 + 16 + 36 + 16 + 64 = 240 \]

\[ \sum (x - \bar{x})(y - \bar{y}) = 24 + 6 + 16 - 2 + 0 + 4 + 12 - 12 + 32 = 80 \]

### Step 6: Calculate Regression Coefficients
The regression coefficient \( b_{yx} \) is given by:

\[ b_{yx} = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2} = \frac{80}{60} = \frac{4}{3} \approx 1.33 \]

### Step 7: Formulate the Regression Equation
Using the regression equation:

\[ y - \bar{y} = b_{yx}(x - \bar{x}) \]

Substituting the values:

\[ y - 10 = \frac{4}{3}(x - 5) \]

Rearranging gives:

\[ y = \frac{4}{3}x + 10 - \frac{20}{3} = \frac{4}{3}x + \frac{10}{3} \]

### Step 8: Estimate \( y \) when \( x = 10 \)
Substituting \( x = 10 \):

\[ y = \frac{4}{3}(10) + \frac{10}{3} = \frac{40}{3} + \frac{10}{3} = \frac{50}{3} \approx 16.67 \]

### Step 9: Estimate \( x \) when \( y = 5 \)
Using the regression equation of \( x \) on \( y \):

\[ x - \bar{x} = b_{xy}(y - \bar{y}) \]

Where \( b_{xy} = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (y - \bar{y})^2} = \frac{80}{240} = \frac{1}{3} \)

Thus, the equation becomes:

\[ x - 5 = \frac{1}{3}(y - 10) \]

Substituting \( y = 5 \):

\[ x - 5 = \frac{1}{3}(5 - 10) = \frac{1}{3}(-5) = -\frac{5}{3} \]

So,

\[ x = 5 - \frac{5}{3} = \frac{15}{3} - \frac{5}{3} = \frac{10}{3} \approx 3.33 \]

### Final Answers
- The estimated value of \( y \) when \( x = 10 \) is approximately **16.67**.
- The estimated value of \( x \) when \( y = 5 \) is approximately **3.33**.