To find the value of $a$ (the y-intercept) in the regression equation of $y$ on $x$, we can use the following formula:

$y = a + bx$

where:

• $a$ is the y-intercept.
• $b$ is the slope of the regression line and is calculated by the formula:

$b = \frac{SS_{xy}}{SS_{xx}}$

Given Data

• $SS_{xx} = 546$
• $SS_{xy} = -1484$
• Mean of $x$ values, $\bar{x} = 30$
• Mean of $y$ values, $\bar{y} = 76$

Step 1: Calculate the Slope, $b$

$b = \frac{SS_{xy}}{SS_{xx}} = \frac{-1484}{546}$

Calculating this gives:

$b \approx -2.72$

Step 2: Calculate the Intercept, $a$

Now we can use the formula for $a$:

$a = \bar{y} - b \cdot \bar{x}$

Substitute the values:

$a = 76 - (-2.72) \cdot 30$

Calculate this:

$a \approx 76 + 81.6 = 157.6$

Final Answer

The value of $a$, rounded to two decimal places, is:

$a = 157.6$

Would you like further details on any step?

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1. How does changing $SS_{xy}$ impact the intercept and slope?
2. Why is the slope important in interpreting regression results?
3. How does rounding impact regression accuracy?
4. What is the role of means in finding intercepts?
5. How is regression different from correlation?

Tip: In regression, always interpret the slope to understand the relationship between variables fully.