Chapter 9 Lessons 11-21: Diving into Hypothesis Testing

Welcome to a deeper dive into Chapter 9! This section focuses on hypothesis testing, a cornerstone of statistical inference. Let's break down the key concepts with clarity and examples.

Key Concepts:

  • Hypothesis Testing: A method to make decisions or judgments based on evidence. We start with a statement we want to investigate.
  • Hypothesis: A statement that something is true. For example, "The average weight of pretzels is 454g."

Null and Alternative Hypotheses

In hypothesis testing, we define two opposing hypotheses:

  • Null Hypothesis ($H_0$): The hypothesis to be tested. It often represents a statement of no effect or no difference. For example: $H_0: \mu = \mu_0$, where $\mu$ is the population mean and $\mu_0$ is a specific value.
  • Alternative Hypothesis ($H_a$): A hypothesis considered as an alternative to the null hypothesis. It represents what we are trying to find evidence for.

Types of Tests Based on the Alternative Hypothesis:

  • Two-Tailed Test: Used when the alternative hypothesis states that the population mean is different from a specified value. $H_a: \mu \neq \mu_0$.
  • Left-Tailed Test: Used when the alternative hypothesis states that the population mean is less than a specified value. $H_a: \mu < \mu_0$.
  • Right-Tailed Test: Used when the alternative hypothesis states that the population mean is greater than a specified value. $H_a: \mu > \mu_0$.

Examples:

Let's solidify these concepts with examples.

  1. Quality Assurance (Pretzel Bags): A company produces 454-g bags of pretzels. The quality assurance department performs a hypothesis test to ensure the packaging machine is working properly.

    • $H_0: \mu = 454$ (The mean weight is 454g)
    • $H_a: \mu \neq 454$ (The mean weight is not 454g - Two-tailed test)

  2. Taller Young Women: We want to determine if today's women in their 20s are taller, on average, than women in their 20s a half-century ago (62.6 inches).

    • $H_0: \mu = 62.6$ (The mean height is 62.6 inches)
    • $H_a: \mu > 62.6$ (The mean height is greater than 62.6 inches - Right-tailed test)

  3. Poverty and Dietary Calcium: We want to determine if the average adult with an income below the poverty level gets less than the recommended adequate intake (RAI) of calcium of 1000 mg.

    • $H_0: \mu = 1000$ (The mean calcium intake is 1000 mg)
    • $H_a: \mu < 1000$ (The mean calcium intake is less than 1000 mg - Left-tailed test)

Additional Resources:

Keep practicing, and you'll become a hypothesis testing pro! Good luck with your studies.