Let's go step by step through this problem.

Data Set:

The data points are as follows:

• $(x_1, y_1) = (2, 3)$
• $(x_2, y_2) = (3, 4)$
• $(x_3, y_3) = (5, 5)$
• $(x_4, y_4) = (7, 7)$

1. Does the data appear linear?

Looking at the data, it seems that the values of $y$ increase in a way that suggests a linear relationship with $x$. This is reinforced by the relatively constant rate of change between consecutive points.

Thus, the answer is likely Yes, the data does appear linear.

2. Best-fitting line (Linear Regression Equation):

To find the equation of the best-fitting line $\hat{y} = mx + b$, we use the following formulas for the slope $m$ and the intercept $b$:

$m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$ $b = \frac{\sum y - m(\sum x)}{n}$

Where:

• $n$ is the number of data points (here, $n = 4$)
• $\sum x$ is the sum of the x-values
• $\sum y$ is the sum of the y-values
• $\sum xy$ is the sum of the products of corresponding x and y values
• $\sum x^2$ is the sum of the squares of the x-values

Step-by-Step Calculations:

• $\sum x = 2 + 3 + 5 + 7 = 17$
• $\sum y = 3 + 4 + 5 + 7 = 19$
• $\sum xy = (2)(3) + (3)(4) + (5)(5) + (7)(7) = 6 + 12 + 25 + 49 = 92$
• $\sum x^2 = 2^2 + 3^2 + 5^2 + 7^2 = 4 + 9 + 25 + 49 = 87$

Now, we can compute the slope $m$:

$m = \frac{4(92) - (17)(19)}{4(87) - 17^2} = \frac{368 - 323}{348 - 289} = \frac{45}{59} \approx 0.7627$

Next, we calculate the intercept $b$:

$b = \frac{19 - 0.7627(17)}{4} = \frac{19 - 12.9659}{4} = \frac{6.0341}{4} \approx 1.5085$

Thus, the equation of the best-fitting line is approximately: $\hat{y} = 0.7627x + 1.5085$

3. Sketching the Graph:

On the graph provided:

• Plot the points $(2, 3), (3, 4), (5, 5), (7, 7)$.
• Draw the regression line with the equation $\hat{y} = 0.7627x + 1.5085$.

Let me know if you'd like more details or have any questions! Here are 5 related questions you might want to explore:

1. How does the residual between observed and predicted $y$ values look for each point?
2. What does the slope tell us about the relationship between $x$ and $y$?
3. How would removing or adding more data points affect the regression line?
4. What is the correlation coefficient for this data set?
5. Could this line be used for extrapolation beyond $x = 7$? Why or why not?

Tip: Always check if there are any outliers or anomalies in the data before fitting a regression line.