Step 1: Analyze the Data for Linearity

We are given the data points:

$x$	$y$
2	3
5	3
6	5
7	7

Visual Inspection

The relationship between $x$ and $y$ does not immediately appear perfectly linear because there is some deviation in $y$ as $x$ increases. However, to determine this conclusively, we can calculate the regression line and see how well the data fits.

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Step 2: Equation of the Best-Fitting Line

1. Calculate the Slope ($m$)

The slope of the line $m$ can be calculated as:

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

2. Calculate the Y-Intercept ($b$)

The intercept $b$ is given by:

$b = \frac{\sum y - m \sum x}{n}$

Summation Values:

$x$	$y$	$x^2$	$xy$
2	3	4	6
5	3	25	15
6	5	36	30
7	7	49	49

• $\sum x = 20$
• $\sum y = 18$
• $\sum x^2 = 114$
• $\sum xy = 100$
• $n = 4$

Slope Calculation:

$m = \frac{4(100) - (20)(18)}{4(114) - (20)^2} = \frac{400 - 360}{456 - 400} = \frac{40}{56} \approx 0.714$

Intercept Calculation:

$b = \frac{\sum y - m \sum x}{n} = \frac{18 - (0.714)(20)}{4} = \frac{18 - 14.28}{4} = \frac{3.72}{4} \approx 0.93$

Best-Fitting Line:

$\hat{y} = 0.714x + 0.93$

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Step 3: Sketch the Graph

To sketch the graph, we will:

1. Plot the data points: $(2,3), (5,3), (6,5), (7,7)$.
2. Draw the regression line $\hat{y} = 0.714x + 0.93$.

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Let me generate the graph for you. matplotlib.pyplot as plt import numpy as np

Data points

x = np.array([2, 5, 6, 7]) y = np.array([3, 3, 5, 7])

Regression line: y = 0.714x + 0.93

m = 0.714 b = 0.93 x_line = np.linspace(1, 8, 100) y_line = m * x_line + b

Plotting the data points

plt.scatter(x, y, color='blue', label='Data points')

Plotting the regression line

plt.plot(x_line, y_line, color='red', label='Regression line (y = 0.714x + 0.93)')

Adding labels and title

plt.title('Data Points and Regression Line') plt.xlabel('x') plt.ylabel('y') plt.axhline(0, color='black', linewidth=0.5, linestyle='--') plt.axvline(0, color='black', linewidth=0.5, linestyle='--') plt.legend() plt.grid()

Show the graph

plt.show()The graph above shows the data points (blue) and the best-fitting regression line (red) given by $\hat{y} = 0.714x + 0.93$.

Summary:

1. Does the data appear linear? Yes, it is reasonably linear as the data points align closely with the regression line.
2. Equation of the line: $\hat{y} = 0.714x + 0.93$.

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Do you have any questions or need more details? Here are five related questions for further exploration:

1. How is the slope of the regression line affected by outliers in the data?
2. What is the interpretation of the slope and intercept in this context?
3. How can you calculate the correlation coefficient to measure the strength of the linear relationship?
4. What would happen to the regression line if an additional point was added at $(10, 10)$?
5. How can you predict $y$ values for new $x$ values using this regression equation?

Tip: Always calculate the correlation coefficient $r$ to better understand the strength of the linear relationship.