Statistical Hypothesis Solver & Plotter
Perform one-sample, independent two-sample T-tests and Z-tests locally. View means, variances, exact p-values, and trace critical rejection regions dynamically on SVG curves.
Test Settings
FAIL TO REJECT Null Hypothesis (H₀)
Since the computed p-value (0.48567) is greater than your stated significance level (α = 0.05), we do not have sufficient statistical evidence to reject the null hypothesis.
SVG Standard Distribution Plotter
Statistical Hypothesis Formulations, Type Errors, and P-Values
What is Statistical Hypothesis Testing?
Hypothesis testing represents a standard mathematical framework utilized in data science, scientific research, and quality assurance workflows to determine if experimental sample metrics deviate significantly from baseline assumptions. We formulate two opposing statements:
- Null Hypothesis (H₀): Declares that no statistical difference or treatment effect exists. Any variation is due to pure chance.
- Alternative Hypothesis (H₁): Declares that a meaningful difference or treatment effect does exist.
To make a decision, we evaluate our data to compute a **test statistic** (a $t$-score or $Z$-score) which measures how many standard deviations our sample mean sits away from the null value.
Type I and Type II Errors
In scientific decisions, we establish a threshold of error called the **significance level (α)**—usually set at 5% (0.05). This bounds our risk of committing errors:
- Type I Error (False Positive): Occurs when we reject a null hypothesis that is actually true. Denoted by $\alpha$, it represents finding an effect where none exists.
- Type II Error (False Negative): Occurs when we fail to reject a null hypothesis that is actually false. Denoted by $\beta$, it represents missing a real effect.
T-test vs. Z-test Selection
Choose a **Z-test** when the overall population standard deviation ($\sigma$) is known, and the sample size is large ($N > 30$). Choose a **T-test** (Student's T-distribution) when the population variance is unknown, and we must estimate standard deviations directly from the sample data. T-distributions have heavier tails than normal curves to adjust for smaller sample sizes.