Thank you for the correction. The new data table is:

\[
\begin{array}{c|c|c|c|c}
x & 2 & 4 & 6 & 8 \\
y & 3 & 5 & 7 & 9 \\
\end{array}
\]

Let's solve each question step-by-step using the updated dataThank you for the correction. The new data table is:

\[
\begin{array}{c|c|c|c|c}
x & 2 & 4 & 6 & 8 \\
y & 3 & 5 & 7 & 9 \\
\end{array}
\]

Let's solve each question step-by-step using the updated 

### Step-by-Step Solutions:

#### 11. **Finding the y-intercept:**

The equation of a line in slope-intercept form is:

\[
y = mx + b
\]

To find the y-intercept (\(b\)), we need to calculate the slope first.

#### 12. **Finding the slope (m):**

The slope (\(m\)) is calculated by:

\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's use two points from the table, (2, 3) and (4, 5):

\[
m = \frac{5 - 3}{4 - 2} = \frac{2}{2} = 1
\]

So, the slope \(m = 1\).

#### 13. **Finding the equation of the line:**

Substitute the slope \(m = 1\) into the equation \(y = mx + b\).

Using one of the points, for example, \((2, 3)\):

\[
3 = 1 \cdot 2 + b \implies b = 3 - 2 = 1
\]

So, the equation of the line is:

\[
y = 1x + 1 \implies y = x + 1
\]

#### 14. **Finding the linear correlation coefficient (r):**

To find the linear correlation coefficient \(r\), we can use the formula:

\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]

Where:
- \(n = 4\) (number of data points)
- \(\sum x = 2 + 4 + 6 + 8 = 20\)
- \(\sum y = 3 + 5 + 7 + 9 = 24\)
- \(\sum xy = (2 \cdot 3) + (4 \cdot 5) + (6 \cdot 7) + (8 \cdot 9) = 6 + 20 + 42 + 72 = 140\)
- \(\sum x^2 = 2^2 + 4^2 + 6^2 + 8^2 = 4 + 16 + 36 + 64 = 120\)
- \(\sum y^2 = 3^2 + 5^2 + 7^2 + 9^2 = 9 + 25 + 49 + 81 = 164\)

Plugging these values into the formula gives:

\[
r = \frac{4(140) - (20)(24)}{\sqrt{[4(120) - (20)^2][4(164) - (24)^2]}}
\]

Let's compute this step-by-step. 

### Detailed Calculations for \(r\):

\[
r = \frac{560 - 480}{\sqrt{(480 - 400)(656 - 576)}}
\]

\[
r = \frac{80}{\sqrt{(80)(80)}}
\]

\[
r = \frac{80}{80} = 1
\]

Thus, the correlation coefficient \(r = 1\), indicating a perfect linear relationship.

### Answer Summary:
1. **Y-intercept (b):** 1
2. **Slope (m):** 1
3. **Equation of the line:** \(y = x + 1\)
4. **Correlation coefficient (r):** 1

Would you like more details on any part of the solution, or do you have further questions?

### Related Questions:
1. How do you interpret the correlation coefficient value?
2. What happens to the slope if all \(y\)-values increase by 2?
3. How do you derive the formula for a line using two points?
4. Can there be multiple lines of best fit for a given data set?
5. How do you calculate the standard error of the estimate for a regression line?

### Tip:
To check if your slope calculation is correct, pick different points and verify if the slope remains consistent across all pairs!

Learn how to analyze a linear relationship using given data points (x=2, 4, 6, 8 and y=3, 5, 7, 9). This problem covers finding the y-intercept, slope, equation of the line, and correlation coefficient. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation