Hypothesis Testing Calculator

What is Hypothesis Testing?

Hypothesis testing is a statistical method for making decisions based on sample data. You start with a claim (hypothesis) about a population parameter and use data to determine whether there’s enough evidence to reject it.

The Hypotheses

Null Hypothesis (\( H_0 \))

The default assumption, typically stating “no effect” or “no difference.”

Example: \( H_0: \mu = 50 \) (the population mean equals 50)

Alternative Hypothesis (\( H_a \))

What you’re trying to find evidence for.

Example: \( H_a: \mu \neq 50 \) (the population mean differs from 50)

Test Types

Two-Tailed Test

Tests if the parameter differs from the hypothesized value in either direction.

$$H_a: \mu \neq \mu_0$$

Left-Tailed Test

Tests if the parameter is less than the hypothesized value.

$$H_a: \mu < \mu_0$$

Right-Tailed Test

Tests if the parameter is greater than the hypothesized value.

$$H_a: \mu > \mu_0$$

The P-Value

The p-value is the probability of observing results at least as extreme as the sample, assuming the null hypothesis is true.

• Small p-value → strong evidence against \( H_0 \)
• Large p-value → weak evidence against \( H_0 \)

Decision Rules

Using P-Value

• If p-value \( \leq \alpha \): Reject \( H_0 \)
• If p-value \( > \alpha \): Fail to reject \( H_0 \)

Using Critical Values

• If test statistic falls in rejection region: Reject \( H_0 \)
• Otherwise: Fail to reject \( H_0 \)

Significance Level (\( \alpha \))

The threshold for rejecting \( H_0 \), typically 0.05 (5%). This is the probability of making a Type I error (rejecting a true null hypothesis).

Types of Errors

Type I Error (\( \alpha \))

Rejecting \( H_0 \) when it’s actually true (false positive).

Type II Error (\( \beta \))

Failing to reject \( H_0 \) when it’s actually false (false negative).

Power = \( 1 – \beta \) (probability of correctly rejecting a false \( H_0 \))

Test Statistics

Z-Test (\( \sigma \) known)

$$z = \frac{\bar{x} – \mu_0}{\sigma / \sqrt{n}}$$

T-Test (\( \sigma \) unknown)

$$t = \frac{\bar{x} – \mu_0}{s / \sqrt{n}}$$
$$

Proportion Test

$$z = \frac{\hat{p} – p_0}{\sqrt{p_0(1-p_0)/n}}$$

Common Misconceptions

1. P-value is NOT the probability \( H_0 \) is true
2. Failing to reject \( H_0 \) doesn’t prove \( H_0 \) is true
3. Statistical significance ≠ practical significance

Steps for Hypothesis Testing

1. State the hypotheses (\( H_0 \) and \( H_a \))
2. Choose significance level (\( \alpha \))
3. Calculate the test statistic
4. Find the p-value
5. Make a decision
6. State the conclusion in context