Consider the following data set then fill in the blank below with the appropriate percent
rounded to two decimal places. DESMOS PROBLEM
X
(predictor variable)
Y
(response variable)
8 11
6 15
10 19
2 10
6 9
_______________ percent of the proportion of variability in y can be
explained by the linear relationship between x and y.


To determine what percentage of the variability in \( y \) (the response variable) can be explained by the linear relationship between \( x \) (the predictor variable), we need to find the **coefficient of determination**, denoted as \( R^2 \). This is typically computed from the linear regression model for the given data.

Steps:

1. **Find the linear regression equation** from the data points \((x, y)\).
2. **Compute \( R^2 \)** from the linear regression model.
3. **Convert \( R^2 \) to a percentage** and round it to two decimal places.

Given the dataset:
\[
\begin{aligned}
    x & : 8, 6, 10, 2, 6 \\
    y & : 11, 15, 19, 10, 9
\end{aligned}
\]
Let me calculate the regression line and \( R^2 \).Approximately **43.35%** of the proportion of variability in \( y \) can be explained by the linear relationship between \( x \) and \( y \), based on this data set.

Would you like further details on this calculation or any additional explanations?

Here are five related questions to deepen understanding:

1. What is the meaning of the coefficient of determination (\( R^2 \)) in regression analysis?
2. How does the value of \( R^2 \) relate to the strength of the linear relationship between variables?
3. What methods exist for improving the \( R^2 \) value in a regression model?
4. How does the interpretation of \( R^2 \) differ in nonlinear relationships?
5. Can an \( R^2 \) value ever be negative, and if so, what does that imply?

**Tip:** A higher \( R^2 \) indicates a stronger linear fit, but it doesn't necessarily mean the model is good — always check the residuals and other factors!

Consider the following data set then fill in the blank below with the appropriate percent rounded to two decimal places. DESMOS PROBLEM X (predictor variable) Y (response variable) 8 11 6 15 10 19 2 10 6 9 _______________ percent of the proportion of variability in y can be explained by the linear relationship between x and y.

Learn how to calculate the percentage of variability in y explained by x using linear regression and the coefficient of determination (R²). This problem covers key concepts like finding the linear regression equation and calculating R². Based on the dataset provided, the percentage is approximately 43.35%. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation