What does the slope t-test check?

It tests the null hypothesis H0: β1 = 0 (the slope is zero, meaning X has no linear effect on Y) against the two-sided alternative H1: β1 ≠ 0. A small p-value (below your chosen alpha) provides evidence to reject H0 and conclude that X significantly predicts Y.

How is the p-value calculated?

The t-statistic is t = b1 / SE(b1), where b1 is the estimated slope and SE(b1) is its standard error. This t-statistic follows a t-distribution with n-2 degrees of freedom under H0. The two-tailed p-value is calculated from that distribution using the regularized incomplete beta function.

What is the F-test for?

For simple linear regression the F-test is equivalent to the slope t-test (F = t²). It tests the overall significance of the model — whether the regression explains significantly more variance than the mean-only model. F = MSR / MSE, where MSR is the mean regression sum of squares and MSE is the mean squared error.

What is R-squared vs. adjusted R-squared?

R-squared is the proportion of variance in Y explained by X (range 0–1). Adjusted R-squared penalizes for the number of predictors; for simple regression with one predictor it equals 1 - (1 - R²) * (n-1) / (n-2), and is always slightly lower than R².

What are the assumptions of simple linear regression?

Linearity (the true relationship is linear), independence of observations, homoscedasticity (constant error variance), and normally distributed residuals. The significance tests are most reliable when these hold, especially for small samples.

How many data points do I need?

At least 3 pairs are required so that degrees of freedom n-2 ≥ 1. The test has low power with fewer than 10 observations. Both X and Y must have the same number of values.

Linear Regression Significance Test

Name: Linear Regression Significance Test
Availability: InStock
Author: Nham Vu

Enter X and Y datasets to get slope, intercept, R-squared, t-statistic, F-statistic, p-value, and confidence intervals for a simple linear regression.

Enter Your Data

X (Predictor) Values

Y (Response) Values

Significance Level (alpha)

Load example dataset:

Enter X and Y data, then click Run Regression

Regression Equation

Slope (b1)

Intercept (b0)

R-squared

t-statistic

F-statistic

p-value

Full Results Table

Parameter	Estimate	Std. Error	t-value	p-value

ANOVA Table

Source	SS	df	MS	F	p-value

Slope Confidence Interval

Scatter Plot with Regression Line

Summary

Enter X and Y datasets to get slope, intercept, R-squared, t-statistic, F-statistic, p-value, and confidence intervals for a simple linear regression.

How it works

Enter the X (predictor) values as comma-separated numbers in the first field.
Enter the matching Y (response) values in the second field — same number of values as X, same order.
Select a significance level (alpha): 0.10, 0.05, or 0.01.
Click "Run Regression" to compute the OLS estimates: slope, intercept, standard errors, t-statistic, F-statistic, R-squared, and p-value.
The scatter plot with the fitted regression line renders automatically.
Copy the full results summary to the clipboard for a report or homework submission.

Use cases

Test whether advertising spend significantly predicts sales revenue.
Determine if study hours have a statistically significant effect on exam scores.
Check whether temperature significantly predicts electricity demand.
Validate regression homework answers before submission.
Quick OLS diagnostic before running full regression in R or Python.
Assess whether a numeric predictor significantly explains variation in a response variable.
Verify whether two economic time-series have a significant linear relationship.
Explore small datasets in data-science or econometrics coursework.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu