Null vs Alt

Introduction to Hypothesis Testing

• This section covers the basics of hypothesis testing as part of inferential statistics.

• The goal of hypothesis testing is to make inferences about a population based on a sample.

Inferential Statistics Overview

• In inferential statistics, a large population is analyzed through small samples.

• A population's key numeric properties include:   - Population average ($ext{MEU}$)   - Population proportion   - Population standard deviation

• Example Property of Interest: Population average ($ext{MEU}$).

Sampling

• Due to difficulty measuring the entire population, a sample is taken.

• Sample statistics that can be calculated include:   - Sample average ($\bar{X}$)   - Sample proportion   - Sample standard deviation

Making Inferences

• The purpose of inferential statistics is to infer the unknown population parameter from the known sample statistic:   - Based on a sample average ($\bar{X}$), we make inferences about the population average ($ext{MEU}$).

• Tools used for inference:   - Confidence intervals (e.g., being 95% confident that $ext{MEU}$ lies between 2.1 and 3.2).   

Introduction to Hypothesis Testing

• Hypothesis testing is a method to test claims or hypotheses about a population parameter based on sample data.

• The focus is on answering a specific question about the population, typically in binary form (e.g., is ext{MEU} < 5 or $ext{MEU} ightarrow 5$?).

Setting Up Hypotheses

• Hypothesis testing involves two main hypotheses:   - Null Hypothesis ($H_0$)   - Alternative Hypothesis ($H_A$ or $H_1$)

Null Hypothesis ($H_0$)

• Symbol: $H_0$

• Always includes an equality component (e.g., equals, less than or equal to, or greater than or equal to).

• Example: If examining $ext{MEU}$:   - $H_0: ext{MEU} ightarrow 5$ (Using the parameter interested in).

Alternative Hypothesis ($H_A$)

• Symbol: $H_A$ or $H_1$

• The alternative hypothesis contradicts the null hypothesis and may have no equal sign.

• Example: If null hypothesis is $H_0: ext{MEU} ightarrow 5$, then alternative:   - H_A: ext{MEU} < 5 .

Types of Hypothesis Tests

• There are three types of hypothesis tests:   1. Right-tailed test   2. Left-tailed test   3. Two-tailed test

Right-tailed Test

• Structure:   - Null Hypothesis: $H_0: ext{MEU} ightarrow 40$   - Alternative Hypothesis: H_A: ext{MEU} < 40

• The symbol points to the right indicating a right-tailed test.

Left-tailed Test

• Structure:   - Null Hypothesis: $H_0: P ightarrow 0.75$   - Alternative Hypothesis: H_A: P < 0.75

• The symbol points to the left indicating a left-tailed test.

Two-tailed Test

• Structure:   - Null Hypothesis: $H_0: ext{MEU} = 85$   - Alternative Hypothesis: $H_A: ext{MEU} eq 85$

• Neither side indicates direction, thus testing both:   - Greater than and Less than.

Important Considerations in Setting Hypotheses

• Key factors include:   - Always use the same population parameter for both hypotheses.   - Ensure that both hypotheses contradict each other:     - If $H_0$ is greater than or equal to a value, then $H_A$ must be less than that value.     - Use equalities in $H_0$ and non-equalities in $H_A$.

Specific Examples of Hypothesis Formulation

• Example Situation 1: Testing driver's test pass percentage *   - Null Hypothesis: $H_0: P ightarrow 0.50$ (50% pass on first try)   - Alternative Hypothesis: H_A: P > 0.50 (more than 50% pass on first try).

• Example Situation 2: Testing lesson time *   - Null Hypothesis: $H_0: ext{MEU} ightarrow 35$ (time is greater or equal to 35 minutes)   - Alternative Hypothesis: H_A: ext{MEU} < 35 (time is fewer than 35 minutes).

• Example Situation 3: Testing students' heights *   - Null Hypothesis: $H_0: ext{MEU} = 64$ (checking average height equals 64 inches)   - Alternative Hypothesis: $H_A: ext{MEU} eq 64$ (the mean does not equal 64).

Conclusion

• The lessons will continue with applications and resolving hypotheses using the null and alternative statements established.

• Students are encouraged to practice formulating hypotheses based on presented scenarios.