Hypothesis Testing with One Sample

Chapter 7: Hypothesis Testing with One Sample

Slide 1: Overview

  • Title: Elementary Statistics Seventh Edition Chapter 7

  • Focus: Introduction to hypothesis testing using sample statistics to draw conclusions about population parameters.

Slide 2: Chapter Outline

  • 7.1 Introduction to Hypothesis Testing

  • 7.2 Hypothesis Testing for the Mean

  • 7.3 Hypothesis Testing for the Mean (Repeated)

  • 7.4 Hypothesis Testing for Proportions

  • 7.5 Hypothesis Testing for Variance and Standard Deviation

Section 7.1: Introduction to Hypothesis Testing

  • Understanding hypothesis tests and their application in statistics.

Objectives of Section 7.1

  1. Gain a practical understanding of hypothesis tests.

  2. Learn to state null and alternative hypotheses.

  3. Identify Type I and Type II errors and interpret levels of significance.

  4. Determine whether to use one-tailed or two-tailed tests and find P-values.

  5. Make and interpret decisions based on test results.

  6. Write claims for hypothesis tests.

Hypothesis Tests (Introduction)

  • Definition: A hypothesis test uses sample statistics to test a claim regarding the population parameter.

    • Example: If a manufacturer claims its hybrid car has a mean gas mileage of 50 miles per gallon, how can it be tested?

Statistical Hypothesis

  • A statement about a population parameter, with two hypotheses:

    • Null Hypothesis (H0): Represents the claim.

    • Alternative Hypothesis (Ha): The complement of the null hypothesis.

  • If one hypothesis is false, the other must be true.

Stating a Hypothesis: Definitions

  • Null Hypothesis (H0): Contains a statement of equality (e.g., =, ≤, ≥).

  • Alternative Hypothesis (Ha): Contains a statement of strict inequality (e.g., <, >).

  • Examples:

    • H0: μ ≥ k, Ha: μ < k

    • H0: μ ≤ k, Ha: μ > k

Writing Hypotheses

  • Convert claims about population parameters into mathematical statements.

  • Always start with the assumption that H0 is true,

    • Analyze sampling distribution based on this assumption.

Types of Errors in Hypothesis Testing

  • Type I Error: Incorrectly rejecting the null hypothesis when it is true.

  • Type II Error: Failing to reject the null hypothesis when it is false.

Identifying Types of Errors

  • Example: If the USDA limit for salmonella in ground beef is p = 0.075:

    • Type I Error: Rejecting H0 when contamination is below the limit.

    • Type II Error: Not rejecting H0 when contamination exceeds the limit.

  • A Type II error carries a greater risk, as it may result in public health issues.

Level of Significance

  • The pre-defined threshold for the probability of making a Type I error, denoted as alpha (α).

    • Common Values: 0.05, 0.01, 0.10.

  • The probability of making a Type II error is denoted as beta (β).

Statistical Tests

  1. State H0 and Ha, and set the level of significance.

  2. Take a random sample and calculate sample statistics.

  3. The test statistic (z, t, etc.) is compared to the standard based on H0.

  4. Decision-making based on test statistic analysis.

P-Values

  • Definition: The probability of obtaining a sample statistic as extreme or more extreme than the observed value if H0 is true.

Nature of the Test

  • Types of hypothesis tests:

    • Left-tailed test:

      • Alternative hypothesis (Ha) indicates a less-than condition.

    • Right-tailed test:

      • Ha indicates a greater-than condition.

    • Two-tailed test:

      • Ha indicates a not-equal condition.

Example: Identifying the Nature of a Test

  1. A school's claim: Proportion of students involved in activities is p.

  • H0: p = p0, Ha: p ≠ p0 (Two-tailed test)

  1. Car dealership's claim: Mean time for oil change is < 15 minutes.

  • H0: Mean ≥ 15, Ha: Mean < 15 (Left-tailed test)

  1. Company's claim: Mean furnace life is > 18 years.

  • H0: Mean ≤ 18, Ha: Mean > 18 (Right-tailed test)

Making a Decision Based on P-value

  • Compare P-value with α to determine:

    1. If P-value ≤ α, reject H0.

    2. If P-value > α, fail to reject H0.

Example: Interpreting Decisions

  1. Rejection of H0 supports the claim against student participation.

  2. Failing to reject H0 suggests insufficient evidence against the school’s claim on student involvement.

Steps for Hypothesis Testing

  1. Formulate and state hypotheses.

  2. Set the level of significance.

  3. Determine the sampling distribution.

  4. Calculate the test statistic.

  5. Find the P-value and draw conclusions.

  6. Write your final interpretation in context.

Strategies for Hypothesis Testing

  • Your strategy should reflect whether you are attempting to support or refute a claim. Always align hypotheses with your research goals to facilitate a proper conclusion.

Example: Writing Hypotheses with Claims

  • Investigate new surgical treatment recovery times:

    • Support Claim: Ha: Mean recovery < 96 hours, H0: Mean ≥ 96 hours.

    • Opposing Claim: H0: Mean recovery ≤ 96 hours, Ha: Mean > 96 hours.

Objectives of Section 7.1

  • Gain a practical understanding of hypothesis tests.

  • Learn to state null and alternative hypotheses.

  • Identify Type I and Type II errors and interpret levels of significance.

  • Determine whether to use one-tailed or two-tailed tests and find P-values.

  • Make and interpret decisions based on test results.

  • Write claims for hypothesis tests.