Hypothesis Testing Overview

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1. Hypothesis Testing– One Sample Tests
   1. Introduction
   2. Null & Alternate Hypotheses
   3. Significance Level
   4. Type I & Type II Errors
   5. Power of Hypothesis Test
   6. Two-Tailed & One-Tailed Tests of Hypothesis
   7. Procedure for Hypothesis Testing of Means
   8. Hypothesis Testing of Means
   9. Hypothesis Testing of Proportion of Success </aside>

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1. Hypothesis Testing– Two Sample Tests
   1. Introduction
   2. Hypothesis Testing for Difference between Means
      1. When σ Is Known
      2. When σ Is Not Known
   3. Paired Difference Test
   4. Hypothesis Testing for Difference between Proportions </aside>

1. Hypothesis Testing: Detailed Explanation

   1. Introduction to Hypothesis Testing
      1. Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. It involves testing an assumption (hypothesis) to determine whether there is enough statistical evidence to support it.
   2. Key Terms
      1. Population: The entire group about which conclusions are drawn.
      2. Sample: A subset of the population used for analysis.
      3. Hypothesis: A statement or claim about a population parameter.
         1. Null Hypothesis ($H_0$): Assumes no effect or no difference; the default or status quo hypothesis.
         2. Alternative Hypothesis ($H_a$): Contradicts the null hypothesis; it represents the claim being tested.
   3. Significance Level ($\alpha$): The probability of rejecting the null hypothesis when it is true (Type I error). Common values: $0.05$, $0.01$, $0.10$.

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2. Types of Hypotheses

   1. Null Hypothesis ($H_0$):
      1. Example: $H_0: \mu = \mu_0$ (The population mean is equal to a specific value.)
   2. Alternative Hypothesis ($H_a$):
      1. Example: $H_a: \mu \neq \mu_0$ (The population mean is not equal to the specific value.)
      2. Types:
         1. Two-tailed test: Tests for differences in both directions ($H_a: \mu \neq \mu_0$).
         2. One-tailed test: Tests for a difference in a specific direction ($H_a: \mu > \mu_0$ or $H_a: \mu < \mu_0$).

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3. 3. Errors in Hypothesis Testing

4. Type I Error ($$\alpha$$):

   • Rejecting $$H_0$$ when it is true.
   • Example: Convicting an innocent person.

5. Type II Error ($$\beta$$):

   • Failing to reject $$H_0$$ when it is false.
   • Example: Letting a guilty person go free.

6. Power of a Test:

7. The probability of correctly rejecting $$H_0$$ when it is false ($$1 - \beta$$).