To solve this problem, let's break down the steps:

1. The Equation of the Line

You mentioned a straight line with:

• $x$-intercept at $x = 1$
• Passing through points $(2, 2)$ and $(6, 10)$

Step 1: Find the slope of the line.

The slope $m$ is calculated using the two points $(2, 2)$ and $(6, 10)$: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 2}{6 - 2} = \frac{8}{4} = 2$

Step 2: Find the equation of the line.

Using the point-slope form $y - y_1 = m(x - x_1)$ with $(2, 2)$ and slope $m = 2$: $y - 2 = 2(x - 2)$ $y - 2 = 2x - 4$ $y = 2x - 2$

Thus, the equation of the line is: $y = 2x - 2$

2. Observed and Predicted Values

Next, let's calculate the observed and predicted values at the given points.

2.1. Observed Value at $x = 5$:

From the table of given points, the observed value at $x = 5$ is $y = 7$.

2.2. Predicted Value at $x = 3$:

Using the line equation $y = 2x - 2$, substitute $x = 3$: $y = 2(3) - 2 = 6 - 2 = 4$ So, the predicted value at $x = 3$ is $y = 4$.

3. Sum of the Residuals

The residual for each point is the difference between the observed $y$-value and the predicted $y$-value from the line.

$x$	Observed $y$	Predicted $y = 2x - 2$	Residual (Observed - Predicted)
6	10	$2(6) - 2 = 10$	$10 - 10 = 0$
5	7	$2(5) - 2 = 8$	$7 - 8 = -1$
4	3	$2(4) - 2 = 6$	$3 - 6 = -3$
3	6	$2(3) - 2 = 4$	$6 - 4 = 2$
2	2	$2(2) - 2 = 2$	$2 - 2 = 0$

Sum of residuals: $0 + (-1) + (-3) + 2 + 0 = -2$ So, the sum of the residuals is $-2$.

4. Sum of the Squares of the Deviations

To find the sum of the squares of the residuals, we square each residual value.

Residual	Squared Residual
0	0
-1	1
-3	9
2	4
0	0

Sum of squared residuals: $0 + 1 + 9 + 4 + 0 = 14$ So, the sum of the squares of the deviations is $14$.

Summary of Results:

1. Observed value at $x = 5$: $7$
2. Predicted value at $x = 3$: $4$
3. Sum of the residuals: $-2$
4. Sum of the squares of the deviations: $14$

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Would you like further clarification on any of the steps? Here are some related questions:

1. How do you find the equation of a line given two points?
2. What is the significance of the residual in regression analysis?
3. How are residuals used to measure the accuracy of a model?
4. How do we interpret the sum of residuals when it’s non-zero?
5. Why do we square residuals when calculating deviations?

Tip: When summing residuals, a value close to zero indicates that the line is a good fit for the data.