Chapter 11: Hypothesis Testing - Sections 11.1 & 11.2

Welcome back to Professor Baker's Math Class! Today, we're diving into Chapter 11, focusing on the basics of hypothesis testing. We'll cover setting up hypotheses, understanding significance levels, and making informed decisions based on data. Remember, hypothesis testing is a powerful tool that allows us to make inferences about populations based on sample data.

Key Concepts

  • Null Hypothesis ($H_0$): A statement about the value of a population parameter that we assume to be true unless there is strong evidence to the contrary. For example, $H_0: \mu = 150$ (the population mean is 150).
  • Alternative Hypothesis ($H_a$ or $H_1$): A statement that contradicts the null hypothesis; it represents the claim we are trying to find evidence to support. For example, $H_a: \mu \neq 150$ (the population mean is different from 150).

It's important to formulate these hypotheses correctly. The null hypothesis often represents the status quo, while the alternative hypothesis reflects a change or effect we're investigating.

Types of Hypothesis Tests

The alternative hypothesis determines the type of test:

  • Two-Tailed Test: Used when the alternative hypothesis states that the population parameter is different from the value stated in the null hypothesis (e.g., $H_a: \mu \neq \mu_0$).
  • Right-Tailed Test: Used when the alternative hypothesis states that the population parameter is greater than the value stated in the null hypothesis (e.g., $H_a: \mu > \mu_0$).
  • Left-Tailed Test: Used when the alternative hypothesis states that the population parameter is less than the value stated in the null hypothesis (e.g., $H_a: \mu < \mu_0$).

Steps in Hypothesis Testing

  1. Determine the Null and Alternative Hypotheses: Clearly define $H_0$ and $H_a$ based on the research question.
  2. Specify the Significance Level ($\alpha$): This is the probability of rejecting the null hypothesis when it is actually true (Type I error). Common values are 0.05 (5%) and 0.01 (1%).
  3. Validate Assumptions and Compute the Test Statistic: Ensure the data meets the assumptions of the hypothesis test (e.g., quantitative data, random sample, normal distribution or large sample size). Calculate the appropriate test statistic (e.g., z-score or t-statistic). For example, when $\sigma$ is known, we use the z-statistic: $$z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$$ where:
    • $\bar{x}$ is the sample mean
    • $\mu_0$ is the population mean from the null hypothesis
    • $\sigma$ is the population standard deviation
    • $n$ is the sample size
  4. Determine the Critical Value(s) or P-value: Use the chosen significance level and the distribution of the test statistic to find the critical value(s) or calculate the P-value.
  5. Make a Decision: Compare the test statistic to the critical value(s), or compare the P-value to the significance level. Reject $H_0$ if the test statistic falls in the rejection region or if the P-value is less than $\alpha$.
  6. State the Conclusion: Interpret the decision in the context of the original research question.

Example

Suppose the average national reading level for high school sophomores is 150 words per minute with a standard deviation of 15. A local school board member wants to know if sophomore students at Lincoln High School read at a level different from the national average for tenth graders. The level of the test is to be set at 0.05. A random sample size of 100 tenth graders from Lincoln High School has been drawn, and the resulting average is 154 words per minute.

  • $H_0: \mu = 150$
  • $H_a: \mu \neq 150$
  • $\alpha = 0.05$

The test statistic would be calculated as such: $$z = \frac{154-150}{\frac{15}{\sqrt{100}}} = 2.67$$

Keep practicing, and you'll become a hypothesis testing pro in no time!