Math Problem Statement

x|2|4|6|8

Solution

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

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!

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Math Problem Analysis

Mathematical Concepts

Linear equations
Correlation coefficient
Slope-intercept form

Formulas

Slope formula
Linear correlation coefficient formula

Theorems

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Suitable Grade Level

Grades 9-12

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