Explanatory | Response |
---|---|

### Summary

#### Best fitting values

#### Goodness of Fit

#### Slope significantly non-zero?

#### 95% Confidence Intervals

#### Intermediate Values

### Interpretation

The correlation coefficient is , indicating a **correlation** between the independent variable (\( x \)) and the dependent variable (\( y \)).

The coefficient of determination is , signifying that of the variation in \( y \) is explained by \( x \) in the regression model.

The slope of the regression line is , meaning that for each unit increase in \( x \), \( y \) is expected to increase by units.

The F-test statistic is with a P-value of . This result indicates that the regression model is statistically significant at the 5% level.

Assuming the true slope is zero (null hypothesis \(H_0: \beta=0\)), the probability of obtaining a test statistic as extreme as is . This means that under the null hypothesis of no linear relationship, only 0.1% of similarly collected samples would yield a test statistic as extreme.

**Conclusion**: , as the p-value () is than the significance level (0.05).

## Casual Idea behind Linear Regression

**Simple Linear Regression** is a fundamental statistical method for analyzing the relationship between two variables, often used to predict the value of one variable based on the other. Our tool makes it easy to perform this analysis using the **least squares method**, which finds the **regression line** that best fits your data.

When you input your data, the calculator provides a **step-by-step solution** by calculating the slope \(\beta\) and intercept \(\alpha\) of the **regression equation**:

\[ \hat{y} = \beta x + \alpha \]

This equation represents the line that best fits your data, allowing you to predict the dependent variable \(y\) for any given independent variable \(x\). The tool also generates a **prediction interval**, which gives you a range where future observations are likely to fall.

This tool also performs a full **regression analysis**. It evaluates the model's fit, provides R-squared values to show how well your data is explained by the model, and checks whether the slope is significantly different from zero. This helps ensure that your model is statistically significant and that the linear relationship is reliable.

In addition, you can test the assumptions of the linear model, such as linearity, homoscedasticity, and normality, ensuring that your analysis is robust. You can also visualize your data and the resulting **regression line**, making it easier to understand the relationship between your variables.

This tool simplifies the process of finding the **best-fitting equation** for your data, making it accessible to anyone looking to perform **Simple Linear Regression** without needing advanced statistical knowledge.

However, for a deeper dive into the linear regression, explore the mathematical introduction to linear regression. We break down the core equations, coefficients, and the mathematical model, offering a clear understanding of the mechanics driving this essential statistical tool.