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Compute the y-intercept if x-bar = 57, y-bar = 251, sx= 12, sy= 37 and r = 0.341.

This question aims to find the $y$-intercept from the equation of the line by first finding the slope coefficient. The point at which the graph line crosses the $y-axis$ is known as the $y$-intercept. Figure 1 illustrates the graphical concept of the $y$-intercept.

Figure 1

This question is based on the concept of line equation, where the equation of a line is given as:

\[ y = mx + c \]

Where the slope is represented by $m$ while the intercept of the line is represented by $c$. The slope is a numeric value that shows the line’s inclination and is equivalent to the $\tan$ of the line’s angle with the positive $x-axis$.

Expert Answer

The equation of the line is given as:

\[ \overline{y} = b_1 \overline{x} + b_0 \]

From the given values, we know that:

\[ \overline{x} = 57, \hspace{0.4in} \overline{y} = 251, \hspace{0.4in} s_x = 12, \hspace{0.4in} s_y = 37, \hspace{0.4in} r = 0.341 \]

To find the $y$-intercept, first, we have to find the slope coefficient.

For slope coefficient, the formula is given as:

\[ b_1 = r (\dfrac{s_y} {s_x}) \] 

By putting in the values, we get:

\[ b_1 = (0.341) (\dfrac{37} {12}) \]

 \[ b_1 = (0.341) (3.083) \]

 \[ b_1 = 1.051 \]

Now, the $y$-intercept coefficient is given as:

\[ b_o = \overline{y}\ -\ b_1 \overline{x} \]

By putting in the values, we get:

\[ b_o = 251\ -\ (1.051) (57) \]

 \[ b_0 = 251\ -\ 59.9 \]

 \[ b_0 = 191.9 \]

Numeric Result

The $y$-intercept of the line with a slope coefficient of $1.051$, $\overline{x} = 57$, and $\overline{y} = 251$ is $191.9$.

Example

Find the $y$-intercept if $\overline{x} =50$, $\overline{y} =240$, $s_x=6$, $s_y=30$ and $r=0.3$.

The equation of lines is given as:

\[ y = mx + c \]

From the given values, we know that:

\[ \overline{x} = 50, \hspace{0.4in} \overline{y} = 240, \hspace{0.4in} s_x = 6, \hspace{0.4in} s_y = 30, \hspace{0.4in} r = 0.3 \]

To find the $y$-intercept, we have to find the slope coefficient.

For slope coefficient, we have the formula given as:

\[ m = r (\dfrac{s_y} {s_x}) \] 

By putting in the values, we get:

\[ m = (0.3) (\dfrac{30}{6}) \]

\[ m = (0.3) (5) \]

\[ m = 1.5 \]

Now, the $y$-intercept coefficient is:

\[ c = y\ -\ mx \]

By putting in the values, we get:

\[ c = 240\ -\ (1.5) (50) \]

\[ c = 240\ -\ 75 \]

\[ c = 165 \]

Figure 2

Images/Mathematical drawings are created with Geogebra.

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