Welcome to Chapter 11, Part 1: Hypothesis Testing!

In this section, we'll be embarking on an exciting journey into the realm of hypothesis testing. Hypothesis testing is a crucial tool in statistics, allowing us to make informed decisions based on data. We will be focusing on tests involving single samples. Get ready to challenge assumptions and draw conclusions!

What is a Hypothesis?

A hypothesis is simply a statement or claim about a characteristic of one or more populations. Think of it as a question we want to answer using data. A hypothesis test provides a structured procedure to test this claim.

The Null and Alternative Hypotheses

Every hypothesis test involves two key hypotheses:

  • Null Hypothesis (H0): This is a statement about the value of a population parameter that we assume to be true unless we have sufficient evidence to reject it. It often represents the status quo or a previously established belief.
  • Alternative Hypothesis (Ha or H1): This is a statement that contradicts the null hypothesis. It represents what we are trying to find evidence to support.

Here's an example: Suppose a potato chip manufacturer wants to ensure their 10-ounce bags are filled correctly.

  • H0: $\mu = 10$ (The average weight of the bags is 10 ounces)
  • Ha: $\mu \neq 10$ (The average weight of the bags is not 10 ounces)

Types of Hypothesis Tests

The alternative hypothesis determines the type of test we conduct:

  • 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., $\mu \neq 10$).
  • 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., $\mu > 10$).
  • 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., $\mu < 10$).

Testing a Hypothesis About a Mean ($\sigma$ Known)

When testing a hypothesis about a population mean with a known population standard deviation ($\sigma$), we use the z-test.

Assumptions:

  1. The data is quantitative.
  2. The data is obtained via a random sample of size $n$.
  3. The population is normally distributed, or the sample size $n$ is large ($n > 30$).
  4. The population standard deviation $\sigma$ is known.

Test Statistic:

The test statistic is calculated as:

$$z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$$

Where:

  • $\bar{x}$ = the sample mean
  • $\mu_0$ = the population mean (from the null hypothesis)
  • $\sigma$ = the known population standard deviation
  • $n$ = the sample size

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.

H0: $\mu = 150$

Ha: $\mu \neq 150$

z = (154 - 150)/(15/$\sqrt{100}$) = 2.67

Since 2.67 > 1.960, we reject H0.

Keep practicing, and you'll master hypothesis testing in no time! Good luck!