Hypothesis Testing Essentials

Hypothesis testing is a statistical method to determine if a statement about a population parameter is likely true or false, based on sample data. It involves formulating competing hypotheses (null and alternative) and using evidence to decide whether to reject the null hypothesis.

Core Principles

• Hypothesis testing evaluates statements about population parameters.
• Null hypothesis (H₀) is a tentative assumption, often stating no effect or difference.
• Alternative hypothesis (Hₐ or H₁) is the opposite of the null hypothesis.
• Hypothesis tests use sample data to challenge the null hypothesis.
• The equality part of a hypothesis always resides in the null hypothesis.
• Tests can be one-tailed (lower or upper) or two-tailed.
• Type I error (α) is rejecting H₀ when it is true.
• The level of significance (α) is the probability of a Type I error.
• Two main approaches: p-value and critical value.
• Both approaches aim to determine if sample evidence is strong enough to reject H₀.

Action Steps

• Step 1: State the null (H₀) and alternative (Hₐ) hypotheses.
• Step 2: Identify the test type (lower-tail, upper-tail, or two-tailed).
• Step 3: Specify the level of significance (α).
• Step 4: Collect sample data and compute the test statistic (e.g., z-score).
• Step 5 (p-value approach): Compute the p-value.
• Step 6 (p-value approach): Reject H₀ if p-value ≤ α.
• Step 5 (critical value approach): Determine the critical value(s).
• Step 6 (critical value approach): Apply the rejection rule (compare test statistic to critical value).
• Step 7: Draw a conclusion regarding H₀ and state it in the context of the problem.

Formulas

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

Key Terms

• Null Hypothesis (H₀): A statement about a population parameter that is assumed to be true until evidence suggests otherwise.
• Alternative Hypothesis (Hₐ or H₁): A statement that contradicts the null hypothesis; it is what we suspect might be true.
• Test Statistic: A value calculated from sample data used to test the hypotheses.
• p-value: The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming H₀ is true.
• Level of Significance (α): The probability threshold for rejecting the null hypothesis; typically set at 0.05 or 0.01.
• Critical Value: The boundary value(s) that define the rejection region(s) in the sampling distribution.
• Rejection Region: The set of values for the test statistic that would lead to rejecting the null hypothesis.
• Type I Error: Rejecting the null hypothesis when it is actually true.
• Type II Error: Failing to reject the null hypothesis when it is actually false.