Chapter 8: Hypothesis Testing

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to make inferences about population parameters based on sample data. It involves formulating two competing hypotheses and using sample evidence to decide which hypothesis is more likely to be true.

• Hypothesis: A premise or claim that is tested using statistical methods.

• Null Hypothesis (H0): States that there is no difference between a population parameter and a specific value.

• Alternative Hypothesis (H1): Represents the claim to be tested; it states that there is a difference between a population parameter and a specific value.

Key Terms from Chapter 3

Understanding the distinction between population and sample statistics is essential for hypothesis testing.

Types of Hypotheses and Tests

There are three main ways to set up null and alternative hypotheses, which determine the type of statistical test performed:

1. Two-tailed test (Equal vs. Not Equal): : parameter = some value : parameter ≠ some value

2. Left-tailed test (Equal vs. Less Than): : parameter = some value : parameter < some value

3. Right-tailed test (Equal vs. Greater Than): : parameter = some value : parameter > some value

Formulating Hypotheses for Population Mean Tests

When testing claims about a population mean, hypotheses are formulated based on the context of the problem.

• Example: A coffee machine is designed to discharge a mean of 8 ounces per cup. To test if the true mean differs from 8: : μ = 8 : μ ≠ 8

• Example: A toy company suspects the mean time a toy captures children's attention is less than 25 minutes: : μ = 25 : μ < 25

• Example: An administrator believes this year's mean hourly wage for hospital clerical workers is greater than last year's H_0H_1$: μ > 8.25

Definitions and Key Concepts

Several important terms are used in hypothesis testing:

• Possible Outcomes:

  • Reject the null hypothesis (H0).

  • Fail to reject the null hypothesis.

• Statistical Test: Uses sample data to decide whether to reject H0.

• Test Value (Test Statistic): The numerical value calculated from sample data to perform the test.

• Level of Confidence (C): The probability that a statistical conclusion is correct (e.g., C = 90%).

• Level of Significance (α): The probability of making a Type I error; commonly α = 0.05 or α = 0.01.

Critical Region and Decision Making

The critical region is the range of values for the test statistic that leads to rejection of the null hypothesis.

• Critical Value (C.V.): The boundary value that separates the critical region from the noncritical region.

• One-tailed Test: The critical region is on one side of the mean (left or right).

• Two-tailed Test: The critical regions are on both sides of the mean.

Errors in Hypothesis Testing

Two types of errors can occur in hypothesis testing:

• Significance Level (α): Chosen by the researcher; as α increases, the probability of Type II error (β) decreases.

Test Statistics Formulas

Depending on the information available, different formulas are used:

• Z Test (Population Standard Deviation Known): Where standard error =

• T Test (Population Standard Deviation Unknown): Degrees of freedom:

Traditional (Critical Value) Method Steps

1. State the hypotheses and identify the claim.

2. Find the critical value(s) from the appropriate table.

3. Compute the test value.

4. Make the decision to reject or not reject the null hypothesis.

5. Summarize the results.

P-Value Method Steps

1. State the hypotheses and identify the claim.

2. Compute the test value.

3. Find the P-value.

4. Make the decision (compare P-value to α).

5. Summarize the results.

Interpreting P-Values

• If P-value ≤ α, reject the null hypothesis.

• If P-value > α, do not reject the null hypothesis.

• A small P-value indicates strong evidence against H0.

Examples of Hypothesis Testing

• Example (Z Test): Testing if the mean wait time at a restaurant is greater than 9 minutes, given sample mean, sample size, and population standard deviation.

• Example (T Test): Testing if the mean completion time for a new process is less than 75 minutes, using sample mean, sample standard deviation, and sample size.

Summary Table: Hypothesis Testing Methods

Important Cautions

• We never "accept" the null hypothesis; we only fail to reject it.

• Statistically significant results lead to rejection of the null hypothesis.

• Always interpret results in the context of the significance level and the type of test performed.

Additional info:

• Some examples and tables were inferred and expanded for clarity and completeness.

• Formulas and definitions were standardized for academic accuracy.