Chapter 9: Hypothesis Tests

9.1 - Developing Null and Alternative Hypotheses

• Hypothesis testing determines if a statement about the value of a population parameter should be rejected.

• Null Hypothesis (H0): A tentative assumption about a population parameter.

• Alternative Hypothesis (Ha): The opposite of H0, representing what the researcher aims to support.

• The procedure uses sample data to test these competing statements.

• Development of hypotheses is context-dependent; understanding the situation is crucial.

• In some scenarios, identifying Ha first simplifies hypothesis formulation.

• Practice improves hypothesis formulation skills.

Alternative Hypothesis as a Research Hypothesis

• Many tests seek to gather evidence supporting a research hypothesis through Ha.

• Formulating Ha first can clarify the goal of the research.

• Examples:

  • New teaching method: H0: New method is no better; Ha: New method is better.

  • Sales force bonus plan: H0: Bonus plan won't increase sales; Ha: Bonus plan will increase sales.

  • New drug: H0: New drug does not lower blood pressure more; Ha: New drug lowers blood pressure more.

Null Hypothesis as an Assumption to be Challenged

• Sometimes researchers begin with a belief that a population parameter's statement is true.

• Hypothesis testing helps challenge this assumption and identify statistical evidence for rejection.

• Example:

  • Soft drink volume: H0: Label is correct (µ ≥ 67.6 oz.); Ha: Label is incorrect (µ < 67.6 oz.).

Summary of Forms for Null and Alternative Hypotheses

• The null hypothesis always contains an equality.

• Common forms for testing a population mean (µ0):

  1. One-tailed, lower-tail: H0: µ ≥ µ0; Ha: µ < µ0

  2. One-tailed, upper-tail: H0: µ ≤ µ0; Ha: µ > µ0

  3. Two-tailed: H0: µ = µ0; Ha: µ ≠ µ0

9.2 - Type I and Type II Errors

• Type I Error: Rejecting H0 when it is true (false positive).

• Probability of Type I error is known as the level of significance.

• Tests controlling only for Type I errors are called significance tests.

• Type II Error: Accepting H0 when it is false (false negative).

• Control for Type II error is more complex; statisticians prefer stating "do not reject H0" instead.

p-Value Approach to Hypothesis Testing

• The p-value reflects the probability calculated from test statistics, indicating sample support for H0.

• If p-value ≤ α (significance level), reject H0.

• Interpretation of p-values:

  • < 0.01: Overwhelming evidence for Ha

  • 0.01-0.05: Strong evidence for Ha

  • 0.05-0.10: Weak evidence for Ha

  • 0.10: Insufficient evidence for Ha

9.3 - Population Mean: σ Known

• Steps for conducting a hypothesis test when σ is known:

  1. Formulate H0 and Ha

  2. Determine one- or two-tailed test based on hypotheses

  3. Calculate the test statistic using: z = (x̄ − µ0) / (σ/√n)

  4. Determine p-value and use or compare test statistic with critical value

• Test statistic distribution for standardization is the normal standard distribution table.

9.4 - Population Mean: σ Unknown

• When σ is unknown, use sample standard deviation (s) and apply Student's t-distribution with n - 1 degrees of freedom.

• The rejection rules for both p-value and critical value approaches remain unchanged, now using t distribution.

Population Mean: Two-tailed Hypothesis Tests

• Two-tailed tests involve H0 = µ0, evaluating both tails of the distribution.

• Procedures for two-tailed are similar as one-tailed until interpreting the test statistic.

• Steps:

  1. Compute test statistic z

  2. For z > 0, compute corresponding area; z < 0, less than the statistic.

  3. Double the tail area for p-value.

  4. Reject H0 if p-value ≤ α.

• Confidence Interval Approach:

  • Determine confidence intervals around x̄. Reject H0 if µ0 falls outside the confidence interval.