What is the null hypothesis?

A tentative assumption about a population parameter.

Hypothesis Testing Essentials

Hypothesis testing is a statistical method to determine if a statement about a population parameter should be rejected based on sample data. It involves setting up competing hypotheses and using evidence to make a decision.

Core Principles

Hypothesis testing evaluates statements about population parameters.
Null Hypothesis (H₀): A tentative assumption about a parameter.
Alternative Hypothesis (Hₐ or H₁): The opposite of the null hypothesis.
The procedure uses sample data to test competing statements.
Equality always resides in the null hypothesis.
Tests can be one-tailed (lower or upper) or two-tailed.
Errors are possible due to reliance on sample data.
Two main approaches: p-value and critical value.

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 (z).
Step 5 (p-value approach): Compute the p-value.
Step 5 (critical value approach): Determine the critical value.
Step 6: Apply the rejection rule (compare p-value to α or test statistic to critical value).
Step 7: Draw a conclusion about rejecting or failing to reject H₀.

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): A statement that contradicts the null hypothesis; it is what we suspect might be true.
Level of Significance (α): The probability of making a Type I error (rejecting H₀ when it is true).
Test Statistic: A value calculated from sample data used to test a hypothesis.
p-value: The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming H₀ is true.
Critical Value: The boundary value that separates the rejection region from the non-rejection region.
Type I Error: Rejecting the null hypothesis when it is actually true.

Timeline

Early 20th Century: Development of hypothesis testing framework by Ronald Fisher, Jerzy Neyman, and Egon Pearson.
Mid-20th Century: Widespread adoption and refinement of hypothesis testing in various scientific fields.
Late 20th Century: Increased use of computational methods and software for hypothesis testing.
Present Day: Continued application and evolution of hypothesis testing, including Bayesian approaches.

People

Ronald Fisher: Pioneering statistician, contributed to the foundations of hypothesis testing.
Jerzy Neyman: Developed the Neyman-Pearson lemma, crucial for hypothesis testing theory.
Egon Pearson: Collaborated with Neyman on hypothesis testing, defining Type I and Type II errors.

Quiz

What is the null hypothesis?: A tentative assumption about a population parameter.
Which approach compares the p-value to the level of significance?: p-value Approach