Let's analyze and solve the given problems based on the table provided. The data shows values for $x$ and $y$:

(a) Find the correlation coefficient $r$

The formula for the correlation coefficient $r$ is:

$r = \frac{n\sum{xy} - \sum{x}\sum{y}}{\sqrt{\left(n\sum{x^2} - (\sum{x})^2\right)\left(n\sum{y^2} - (\sum{y})^2\right)}}$

Where:

• $n$ is the number of data points (in this case 4).
• $\sum{x}, \sum{y}$ are the sums of $x$ and $y$ values.
• $\sum{xy}$ is the sum of the products of $x$ and $y$ values.
• $\sum{x^2}, \sum{y^2}$ are the sums of the squares of $x$ and $y$ values.

I will calculate the correlation coefficient next.

(b) Describe the relation between $x$ and $y$

This asks for a general description of the relationship, which can be inferred from the correlation coefficient and how the values change together.

(c) Find the equation $y = ax + b$ of the regression line for $y$ on $x$

To find the regression line, we use the least squares method: $a = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}$ $b = \frac{\sum{y}\sum{x^2} - \sum{x}\sum{xy}}{n\sum{x^2} - (\sum{x})^2}$ where $a$ is the slope, and $b$ is the intercept.

(d) Describe what the coefficient $a$ represents

The coefficient $a$ represents the slope of the regression line. It indicates the rate of change of $y$ with respect to $x$.

(e) Describe what the constant $b$ represents

The constant $b$ represents the y-intercept of the regression line. It shows the value of $y$ when $x = 0$.

I'll calculate these values now.Here are the results:

(a) Correlation coefficient $r$

The correlation coefficient $r = 0.965$ (approximately). This indicates a strong positive correlation between $x$ and $y$.

(b) Describe the relation between $x$ and $y$

There is a strong positive linear relationship between $x$ and $y$, as the correlation coefficient is close to 1. This suggests that as $x$ increases, $y$ also increases significantly.

(c) Equation of the regression line

The regression equation is:

$y = 2.2x - 0.5$

(d) What does the coefficient $a$ represent?

The coefficient $a = 2.2$ represents the slope of the line. For every unit increase in $x$, $y$ increases by 2.2 units.

(e) What does the constant $b$ represent?

The constant $b = -0.5$ is the y-intercept. It indicates that when $x = 0$, $y$ would be $-0.5$.

Would you like further clarification or details on any part of this?

Here are 5 related questions:

1. How do we interpret the correlation coefficient in different ranges?
2. What if the correlation coefficient $r$ was negative?
3. Can we use this regression line to predict values of $y$ for $x$ values outside the given range?
4. How does the strength of correlation affect the reliability of predictions?
5. What would happen if there were more data points added to the set?

Tip: A strong correlation does not imply causation, even though it indicates a relationship between variables.