By using the data `barx=25, bary=30, b_(yx)=1.6 and b_(xy)=0.4` find
(a) The regression equation y on x.
(b) What is the most likely value of y when x=60?
(c ) What is the coefficient of correlation between x and y ?

Answer

To solve the given problem step by step, we will find the regression equation of y on x, the most likely value of y when x = 60, and the coefficient of correlation between x and y.

### Step 1: Find the regression equation of y on x

The regression equation of y on x is given by the formula:

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

Where:
- \( \bar{y} = 30 \) (mean of y)
- \( \bar{x} = 25 \) (mean of x)
- \( b_{yx} = 1.6 \) (regression coefficient of y on x)

Substituting the values into the equation:

\[ y - 30 = 1.6(x - 25) \]

Expanding this:

\[ y - 30 = 1.6x - 40 \]

Adding 30 to both sides:

\[ y = 1.6x - 10 \]

### Step 2: Find the most likely value of y when x = 60

Now, we substitute \( x = 60 \) into the regression equation we found:

\[ y = 1.6(60) - 10 \]

Calculating this:

\[ y = 96 - 10 = 86 \]

So, the most likely value of y when x = 60 is **86**.

### Step 3: Find the coefficient of correlation between x and y

The coefficient of correlation \( r \) can be calculated using the relationship between the regression coefficients:

\[ r^2 = b_{yx} \cdot b_{xy} \]

Where:
- \( b_{xy} = 0.4 \) (regression coefficient of x on y)

Substituting the values:

\[ r^2 = 1.6 \cdot 0.4 \]

Calculating this:

\[ r^2 = 0.64 \]

To find \( r \), we take the square root:

\[ r = \sqrt{0.64} = 0.8 \]

Thus, the coefficient of correlation between x and y is **0.8**.

### Summary of Results
(a) The regression equation of y on x is \( y = 1.6x - 10 \)
(b) The most likely value of y when x = 60 is **86**.
(c) The coefficient of correlation between x and y is **0.8**.