Let's begin by extracting the coordinates of the points from the given graph:

Coordinates of the 10 points:

• Lower cluster (5 points):

  • (1, 2), (2, 2), (1, 3), (2, 3), (1, 4)

• Isolated point at the top:

  • (8, 9)

Now, the full list of coordinates for all 10 points is:

• (1, 2), (2, 2), (1, 3), (2, 3), (1, 4), (8, 9)

Part (a) - Regression Line for All 10 Points:

We need to find the equation of the regression line using the 10 points.

1. Find the mean of $x$ and $y$: $\bar{x} = \frac{1+2+1+2+1+8}{6}, \quad \bar{y} = \frac{2+2+3+3+4+9}{6}$
2. Calculate the slope ($m$) of the regression line: $m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$
3. Find the y-intercept ($b$): $b = \bar{y} - m\bar{x}$
4. Equation of the regression line: $y = mx + b$

Part (b) - Regression Line Without Point (8, 9):

Now, remove the point (8, 9) and repeat the same steps with the remaining 5 points:

• (1, 2), (2, 2), (1, 3), (2, 3), (1, 4)

Find the regression line equation using these 5 points.

Part (c) - Comparison of Results:

Finally, compare the regression line equations obtained in parts (a) and (b). You will likely notice that the slope and y-intercept change when the outlier point (8, 9) is removed, reflecting the impact of this outlier on the overall trend.

I will now calculate the results step by step.

Part (a) Calculation:

Using all points, I'll compute the regression line for you.### Part (a) - Regression Line for All 10 Points: The equation of the regression line using all 6 points is:

$y = 0.893x + 1.6$

Part (b) Calculation:

Now, I'll remove the point (8, 9) and calculate the regression line for the remaining 5 points.### Part (b) - Regression Line Without the Point (8, 9): After removing the point $(8, 9)$, the equation of the regression line is:

$y = -0.5x + 3.5$

Part (c) - Comparison of Results:

• With all 6 points (including $(8, 9)$): The regression line has a positive slope $0.893$, indicating a generally increasing trend.
• Without the point $(8, 9)$: The regression line has a negative slope $-0.5$, indicating a decreasing trend, which is significantly different from the first equation.

This shows that the point $(8, 9)$ acts as an outlier and strongly affects the direction of the trend when included.

Would you like further clarification or details on any step?

Related Questions:

1. How does an outlier like $(8, 9)$ affect the overall trend in a regression line?
2. What are the general steps for calculating a regression line manually?
3. Can you explain why the slope changed so much after removing the outlier?
4. How does the regression equation change when the points are symmetrically distributed?
5. How can we interpret the y-intercept in the context of these points?

Tip:

When calculating regression lines, always inspect for outliers, as they can heavily influence the slope and intercept.