For the data given​ below, answer parts ​(a) through ​(f).
x
negative 18
negative 15
negative 13
negative 12
negative 8
y
102
122
120
132
142

Let's begin by understanding the given dataLet's begin by understanding the given 

- \( x = \{-18, -15, -13, -12, -8\} \)
- \( y = \{102, 122, 120, 132, 142\} \)

Now, let's address each part:

### Part (a): Create a scatter plot of the data
To create a scatter plot, we plot each pair \((x, y)\) on a coordinate plane. Each \(x\)-value is paired with a corresponding \(y\)-value to form a point.

### Part (b): Determine the correlation coefficient \(r\)
The correlation coefficient \(r\) measures the strength and direction of the linear relationship between \(x\) and \(y\). It ranges from -1 (perfect negative correlation) to 1 (perfect positive correlation). We'll calculate this value.

### Part (c): Describe the strength of the linear correlation
Using the correlation coefficient \(r\), we'll assess whether the relationship between \(x\) and \(y\) is strong, weak, or moderate.

### Part (d): Find the equation of the regression line
We need to calculate the linear regression line, typically in the form:
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept. This line best fits the given data points.

### Part (e): Plot the regression line on the scatter plot
We'll plot the regression line on the same graph as the scatter plot to visualize how well it fits the data.

### Part (f): Use the regression equation to predict the value of \(y\) when \(x = -10\)
Substitute \(x = -10\) into the regression equation found in part (d) to predict the corresponding \(y\)-value.

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I'll start by calculating the correlation coefficient \(r\), and the equation of the regression line. Let me compute these.### Part (b): The correlation coefficient \( r \) is approximately 0.96. This indicates a very strong positive linear relationship between \( x \) and \( y \).

### Part (d): The equation of the regression line is:
\[
y = 3.86x + 174.57
\]

Now, I'll move on to predicting \( y \) when \( x = -10 \) (Part f).### Part (f): When \( x = -10 \), the predicted value of \( y \) is approximately 136.

Would you like to see the scatter plot with the regression line (Parts a and e), or need any further details?

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Here are some related questions:

1. How is the correlation coefficient interpreted in other cases (e.g., when \( r \) is close to 0)?
2. Can we apply linear regression to non-linear data? If not, why?
3. What are the assumptions behind using a regression line?
4. How would the prediction change if the slope was negative?
5. What happens if we use more data points outside the current range of \( x \)?

**Tip:** The closer the correlation coefficient \( r \) is to 1 or -1, the stronger the linear relationship between the variables.

For the data given​ below, answer parts ​(a) through ​(f). x: -18, -15, -13, -12, -8; y: 102, 122, 120, 132, 142.

Learn how to create a scatter plot, calculate the correlation coefficient, and find the regression line for the data points x: -18, -15, -13, -12, -8 and y: 102, 122, 120, 132, 142. This problem covers key statistical concepts such as correlation and linear regression, suitable for Grades 10-12.

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