Mean of x= 25
Mean of y =20
`sigma_x = 4`
` sigma_y=3`
` r = 0.5`
`b_(yx) = square`
`b_(xy)= square`
When x = 10
`y-square=square(10 - square)`
`:. y = square`

Answer

To solve the problem step by step, we will follow the instructions provided in the video transcript.

### Step 1: Calculate the regression coefficient \( b_{yx} \)

The formula for the regression coefficient \( b_{yx} \) is given by:

\[ b_{yx} = r \cdot \frac{\sigma_y}{\sigma_x} \]

Where:
- \( r = 0.5 \)
- \( \sigma_y = 3 \)
- \( \sigma_x = 4 \)

Substituting the values:

\[ b_{yx} = 0.5 \cdot \frac{3}{4} = \frac{5}{10} \cdot \frac{3}{4} = \frac{15}{40} = \frac{3}{8} \]

### Step 2: Calculate the regression coefficient \( b_{xy} \)

The formula for the regression coefficient \( b_{xy} \) is:

\[ b_{xy} = r \cdot \frac{\sigma_x}{\sigma_y} \]

Substituting the values:

\[ b_{xy} = 0.5 \cdot \frac{4}{3} = \frac{5}{10} \cdot \frac{4}{3} = \frac{20}{30} = \frac{2}{3} \]

### Step 3: Write the regression equation of \( y \) on \( x \)

The regression equation of \( y \) on \( x \) is given by:

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

Where:
- \( \bar{y} = 20 \)
- \( \bar{x} = 25 \)

Substituting the values:

\[ y - 20 = \frac{3}{8} (x - 25) \]

### Step 4: Substitute \( x = 10 \) into the regression equation

Now we substitute \( x = 10 \) into the regression equation:

\[ y - 20 = \frac{3}{8} (10 - 25) \]

Calculating the right side:

\[ y - 20 = \frac{3}{8} \cdot (-15) = -\frac{45}{8} \]

### Step 5: Solve for \( y \)

Now, we solve for \( y \):

\[ y = 20 - \frac{45}{8} \]

To combine these terms, convert 20 into eighths:

\[ 20 = \frac{160}{8} \]

Thus:

\[ y = \frac{160}{8} - \frac{45}{8} = \frac{115}{8} \]

### Final Answer

The value of \( y \) when \( x = 10 \) is:

\[ y = \frac{115}{8} \]

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