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.