Formulating Appropriate Null and Alternative Hypotheses on a Population Mean

Grade 11 - Statistics and Probability Lesson Notes

Objectives

  • At the end of the lesson the students should be able to:

    • Formulate the appropriate null and alternative hypotheses on a population mean.

    • Code: M11/12SP-IVb-1

Time Frame: Formulating Appropriate Null and Alternative Hypotheses on a Population Mean
Time Schedule

  1. Study the lesson: 28 minutes

    • Read carefully and understand the examples given.

    • Proceed to the activity on page 4 (FORMU-TAIL) once confident.

    • Take note of important information.

  2. Complete Activity (FORMU-TAIL): 10 minutes

    • Write answers only on a Β½ crosswise yellow paper.

    • Ensure answers are complete and legible.

  3. Submission of Work: 2 minutes

    • Submit your work to the assigned teacher or class president (serves as attendance).

Lesson: Formulating Appropriate Null and Alternative Hypotheses on a Population Mean

  • Definition: Hypothesis testing is the process of using statistical tests to determine whether an observed difference between two or more samples is statistically significant.

  • Practical Significance: Allows for evidence-based decision-making rather than relying on subjective feelings or assumptions.

  • Statistical Hypotheses: Statements about a population parameter that require evaluation.

Two Types of Hypotheses

  1. Null Hypothesis (𝐻₀ or 𝐻₀)

    • Represents no effect, no difference, or no change.

    • It is the default assumption that is subjected to testing.

    • Outcomes can result in either rejection or failure to reject the null hypothesis.

  2. Alternative Hypothesis (𝐻₁ or 𝐻ₐ)

    • Indicates that there is an effect, difference, or change.

    • It is tested against the null hypothesis and is supported if the null hypothesis is rejected.

Types of Alternative Hypothesis

  • Two-tailed Test

    • Nondirectional test that only predicts a difference between values without indicating direction (greater or smaller).

    • Involves an extreme test statistic in either tail (positive or negative) leading to rejection of the null hypothesis.

    • Symbolically represented with an inequality: β‰ .

  • One-tailed Test

    • Directional test that predicts not only the difference but also the expected direction of the effect.

    • The rejection region is situated within one tail of the distribution (may be a Right-tailed Test or Left-tailed Test).

    • Utilizes greater than (>) or less than (<) symbols.

Three Different Ways of Writing Hypotheses
Steps to State Null and Alternative Hypotheses

  1. Identify the parameter in a given problem.

  2. Determine the claim to be tested that appears in the null or alternative hypothesis.

  3. Translate the claim into mathematical symbols/notations.

  4. Formulate the null hypothesis (𝐻₀ or 𝐻₀) followed by the alternative hypothesis (𝐻₁ or 𝐻ₐ).

Examples of Hypotheses Formulation

  1. Example 1: Teacher A's Study on Mathematics Games

    • Scenario: Effects of mathematical games on student performance.

    • Class Size: 45 students.

    • Mean Score: 90.

    • Standard Deviation: 3.

    • Previous Study Values: πœ‡ = 85, 𝜎 = 5.

    • Hypotheses:

      • 𝐻₀: πœ‡ = 85

      • 𝐻₁: πœ‡ β‰  85 (Claim)

      • Type: Two-tailed test.

  2. Example 2: Piggery Owner's Belief on Organic Feeds

    • Previous Year’s Income: β‚±120,000.

    • Hypotheses:

      • 𝐻₀: πœ‡ = 120,000

      • 𝐻₁: πœ‡ > 120,000 (Claim)

      • Type: Right-tailed test.

  3. Example 3: Restaurant Waiting Time

    • Claim: Average waiting time is less than 20 minutes.

    • Hypotheses:

      • 𝐻₀: πœ‡ β‰₯ 20

      • 𝐻₁: πœ‡ < 20 (Claim)

      • Type: Left-tailed test.

  4. Example 4: Grade 11 Study Time Claim

    • Claim: Study time is at most 240 minutes/day on average.

    • Sample Size: 30 students, Mean = 300 minutes, SD = 90 minutes.

    • Hypotheses:

      • 𝐻₀: πœ‡ ≀ 240 (Claim)

      • 𝐻₁: πœ‡ > 240

      • Type: Right-tailed test.

  5. Example 5: Municipality's Net Worth Claim

    • Claim: Average net worth of families is at least β‚±730,000.

    • Sample Size: 50 families, Mean = β‚±860,000, SD = β‚±65,000.

    • Hypotheses:

      • 𝐻₀: πœ‡ β‰₯ 730,000 (Claim)

      • 𝐻₁: πœ‡ < 730,000

      • Type: Left-tailed test.

Important Note on Claim Language

  • If the claim includes "at most" or "at least," it is typically represented in the null hypothesis.

FORMU-TAIL Activity Instructions

  • Task: Formulate the null and alternative hypotheses for the following scenarios. Identify if they are one-tailed or two-tailed, specifying left or right direction for one-tailed tests.

  1. Smartphone Battery Manufacturer Claim: Mean life > 650 hours.

    • 𝐻₀:

    • 𝐻₁: _

    • Type: - tailed test.

  2. International Shipping Company Claim: Package arrives in < 8 business days.

    • 𝐻₀:

    • 𝐻₁: _

    • Type: - tailed test.

  3. Grocery Shoppers Report: Mean shoppers who never buy store brand = 300.

    • 𝐻₀:

    • 𝐻₁: _

    • Type: - tailed test.

  4. Principal's Claim of Above-Average Intelligence: Sample mean IQ = 113.

    • 𝐻₀:

    • 𝐻₁: _

    • Type: - tailed test.

  5. BYD Manufacturer Claim: Batteries last on average ≀ 350 hours.

    • 𝐻₀:

    • 𝐻₁: _

    • Type: - tailed test.