Find the most likely price in Mumbai (x) corresponding to the price of ₹ 70 at Kolkata (y) from the following data :
`{:(,"Mumbai", "Kolkata"),("Average price "," "67," "65),("Standard deviation "," "3 . 5 ," "2 . 5 ):}`
Correlation of coefficient = 0 . 5

Answer

To find the most likely price in Mumbai (x) corresponding to the price of ₹ 70 at Kolkata (y), we will use the regression line formula based on the provided data.

### Given Data:
- Average price in Mumbai (x̄) = 67
- Average price in Kolkata (ȳ) = 65
- Standard deviation in Mumbai (σx) = 3.5
- Standard deviation in Kolkata (σy) = 2.5
- Correlation coefficient (p) = 0.5
- Price in Kolkata (y) = 70

### Step-by-Step Solution:

1. **Write the Regression Equation**:
The regression line of y on x is given by:
\[ y - ȳ = p \cdot \frac{σy}{σx} (x - x̄) \]
Substituting the known values:
\[ y - 65 = 0.5 \cdot \frac{2.5}{3.5} (x - 67) \]

2. **Calculate the Coefficient**:
Calculate \( \frac{σy}{σx} \):
\[ \frac{σy}{σx} = \frac{2.5}{3.5} \approx 0.7143 \]
Now substitute this back into the regression equation:
\[ y - 65 = 0.5 \cdot 0.7143 (x - 67) \]
Simplifying further:
\[ y - 65 = 0.3571 (x - 67) \]

3. **Substitute y = 70**:
Now, we substitute \( y = 70 \):
\[ 70 - 65 = 0.3571 (x - 67) \]
This simplifies to:
\[ 5 = 0.3571 (x - 67) \]

4. **Solve for x**:
Divide both sides by 0.3571:
\[ x - 67 = \frac{5}{0.3571} \]
Calculate \( \frac{5}{0.3571} \):
\[ x - 67 \approx 14 \]
Therefore:
\[ x \approx 67 + 14 = 81 \]

5. **Final Result**:
The most likely price in Mumbai corresponding to the price of ₹ 70 at Kolkata is approximately ₹ 81.