ch 11 Hypothesis Testing

Chapter 11: Hypothesis Testing

1. Introduction

  • Hypothesis Testing: Evaluates claims about population parameters based on sample statistics.

  • Objective: Determine if observed sample data significantly differs from the claimed population parameter, leading to a decision to reject or not reject the null hypothesis.

  • Key Concept: Unlike confidence intervals, hypothesis testing doesn’t prove claims but assesses them against a threshold of evidence, known as the significance level.

2. The Rationale Behind Hypothesis Testing

  • Definition of a Hypothesis: A claim or assumption about a population parameter.

  • Two Hypotheses:

    • Null Hypothesis (H₀): The claim to be tested, stating no effect or no difference (e.g., population mean is equal to a specific value).

    • Alternative Hypothesis (H₁ or Ha): The opposite of H₀, indicating a potential effect or difference.

2.1 Null and Alternative Hypotheses

  • Example of H₀: For population mean μ, a null hypothesis could be stated as H₀: μ = μ₀, where μ₀ represents the claimed mean.

  • Properties of Null Hypothesis:

    • Represents the status quo.

    • Always contains an equality sign (=).

    • Cannot be proven true, only not rejected.

  • Alternative Hypothesis H₁: Opposes H₀ and includes inequality signs (#, >, <).

2.2 Types of Hypothesis Test

  • Three types of tests based on H₁:

    1. Two-tailed: H₁: μ # μ₀

    2. Lower one-tailed: H₁: μ < μ₀

    3. Upper one-tailed: H₁: μ > μ₀

  • Example:

    • Manufacturer claims battery lasts 4 hours: H₀: μ = 4; H₁: μ # 4 (two-tailed).

2.3 Level of Significance and Errors

  • Significance Level (α): Probability of rejecting a true null hypothesis, commonly set at 0.01, 0.05, or 0.10.

  • Errors:

    • Type I Error: Rejecting a true null hypothesis (probability α).

    • Type II Error (β): Failing to reject a false null hypothesis.

  • Power of the Test: Probability of correctly rejecting a false null hypothesis (B = 1 - β).

3. Hypothesis Testing Methodology

  • Steps in Hypothesis Testing:

    1. State H₀ and H₁

    2. Choose α (significance level)

    3. Determine the statistical technique and test statistic

    4. Find critical value(s) or p-value

    5. Make a statistical decision: Reject H₀ or Do Not Reject H₀

    6. Interpret the decision in context

3.1 The Critical Value Approach

  • Critical Value(s): Points that separate rejection from non-rejection regions.

  • Rejection Region: Values that lead to rejection of the null hypothesis.

  • Non-rejection Region: Values that do not lead to rejection.

4. z-test for Population Mean μ (σ known)

  • Test Statistic Formula:
    zt=xˉ−μσ/nz_t = \frac{\bar{x} - \mu}{\sigma /\sqrt{n}}

  • Procedure Using Critical Value Approach:

    1. Set null and alternative hypotheses.

    2. Choose significance level.

    3. Calculate z-test statistic from sample data.

    4. Determine critical values based on z-tables.

    5. Compare test statistic with critical values for decision.

    6. Provide conclusion.

5. t-test for Population Mean μ (σ unknown)

  • Use when the population standard deviation is unknown and the sample size is small.

  • Test Statistic Formula:
    tt=xˉ−μs/nt_t = \frac{\bar{x} - \mu}{s/\sqrt{n}}

  • Procedure: Similar steps as with the z-test but use t-distribution for critical values.

6. z-test for Population Proportion

  • Used for outcomes categorized as success or failure.

  • Test Statistic Formula:
    z<em>t=p</em>s−pp(1−p)nz<em>t = \frac{p</em>s - p}{\sqrt{\frac{p(1-p)}{n}}}

7. Parametric vs. Non-Parametric Tests

  • Parametric Tests: Require assumptions regarding the population distribution (e.g., z-test, t-test).

  • Non-Parametric Tests: Do not require such assumptions (e.g., Chi-Square tests).

8. Goodness-of-Fit Test

  • Evaluates if observed data matches expected data based on a specific distribution.

  • Chi-Square Test:

    • Typical use for categorical data.

    • Test Statistic Formula:
      χ2=∑(O−E)2E\chi^2 = \sum \frac{(O - E)^2}{E}