Let's solve the given problem step-by-step.

1. Direction of the Relation Between $x$ and $y$

To determine the direction of the relation between $x$ and $y$, we can use the correlation coefficient, which gives us the strength and direction of a linear relationship between two variables.

2. Finding the Regression Model Equation $y = b_0 + b_1x$

We will calculate the slope $b_1$ and the intercept $b_0$ for the regression line.

First, let's summarize the given dataLet's solve the given problem step-by-step.

1. Direction of the Relation Between $x$ and $y$

To determine the direction of the relation between $x$ and $y$, we can use the correlation coefficient, which gives us the strength and direction of a linear relationship between two variables.

2. Finding the Regression Model Equation $y = b_0 + b_1x$

We will calculate the slope $b_1$ and the intercept $b_0$ for the regression line.

First, let's summarize the given

$x$	$y$
3	2
1	4
7	6
5	7
9	10
8	7

We need the following sums to compute the coefficients:

• $\sum x$
• $\sum y$
• $\sum xy$
• $\sum x^2$
• $n$ (number of data points)

Let's calculate these values: $\sum x = 3 + 1 + 7 + 5 + 9 + 8 = 33$ $\sum y = 2 + 4 + 6 + 7 + 10 + 7 = 36$ $\sum xy = (3 \cdot 2) + (1 \cdot 4) + (7 \cdot 6) + (5 \cdot 7) + (9 \cdot 10) + (8 \cdot 7) = 6 + 4 + 42 + 35 + 90 + 56 = 233$ $\sum x^2 = 3^2 + 1^2 + 7^2 + 5^2 + 9^2 + 8^2 = 9 + 1 + 49 + 25 + 81 + 64 = 229$ $n = 6$

Now we calculate the slope $b_1$ and the intercept $b_0$:

$b_1 = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$ $b_1 = \frac{6 \cdot 233 - 33 \cdot 36}{6 \cdot 229 - 33^2}$ $b_1 = \frac{1398 - 1188}{1374 - 1089}$ $b_1 = \frac{210}{285}$ $b_1 \approx 0.7368$

$b_0 = \frac{\sum y - b_1 \sum x}{n}$ $b_0 = \frac{36 - 0.7368 \cdot 33}{6}$ $b_0 = \frac{36 - 24.3}{6}$ $b_0 \approx 1.95$

Regression Model Equation

The regression model equation is: $y = 1.95 + 0.7368x$

Description of the Model

This regression model indicates that for each unit increase in $x$, the value of $y$ increases by approximately 0.7368 units. The intercept 1.95 suggests that when $x$ is 0, $y$ would be around 1.95.

Summary

1. Direction of the relation: Positive (as $x$ increases, $y$ also tends to increase).
2. Regression model equation: $y = 1.95 + 0.7368x$.

Would you like further details or have any questions?

Here are 5 questions you might ask next:

1. How do you interpret the slope in a regression model?
2. What is the significance of the intercept in a regression equation?
3. How can we determine if the regression model is a good fit for the data?
4. What are the assumptions underlying linear regression analysis?
5. How would you calculate the residuals for each data point?

Tip: Always plot the data points along with the regression line to visually assess the fit of the model.