Study Notes on Hypothesis Testing and Types of Errors

Unit 4: Hypothesis Tests and Types of Errors

Overview

  • Topics Covered:

    • Hypothesis Tests and Types of Errors

    • P-Values and Tests of Means

    • Mean Differences and Difference in Means

    • Tests of Variance

    • Correlation and Independence

Conceptual Framework

  • Hypothesis Testing:

    • A statistical hypothesis is an assertion about an unknown population parameter value.

    • The process of hypothesis testing evaluates the validity of the claim by analyzing the difference between the sample statistic and the hypothesized population parameter value.

    • A hypothesis can test whether a significant difference exists among multiple populations concerning common parameters.

Inferential Statistics

  • Inferential Statistics: Focuses on estimating unknown parameters using sample statistics.

    • If a claim is made, the sample statistic should be close to the hypothesized parameter value, assuming it's correct.

    • This testing approach uses a test statistic.

    • Greater differences between the sample statistic and hypothesized parameters increase doubt about the hypothesis' validity.

Level of Significance

  • The level of significance is the probability threshold for concluding that an observed difference is statistically significant. It is determined before sampling.

    • Commonly denoted as α (alpha), indicating the risk of rejecting a true null hypothesis.

    • Examples include:

    • α = 0.05 (5%) for general research

    • α = 0.01 (1%) for stringent testing

    • α = 0.10 (10%) for less rigorous research.

General Procedure for Hypothesis Testing

  1. State the Null Hypothesis (H₀): Assumes no difference exists.

    • Null hypotheses often involve equality, e.g., H₀: μ = μ₁.

  2. State the Alternative Hypothesis (H₁): Contradicts the null hypothesis, indicating a difference exists.

  3. Select the level of significance (α).

  4. Select the appropriate test statistic based on sample size and type.

  5. Make a decision to accept or reject H₀ based on comparison with critical values.

Types of Hypotheses

  • Directional Hypothesis: Specifies the direction of the expected difference.

    • Example:

    • H₀: μ₁ = μ₂ (no difference)

    • H₁: μ₁ < μ₂ (mean of group 1 is less than group 2).

  • Non-Directional Hypothesis: Does not specify a direction.

    • Example:

    • H₀: μ = 75,000

    • H₁: μ ≠ 75,000 (mean not equal to 75,000).

Test Statistic

  • Formula: ext{Test statistic} = rac{ ext{Value of sample statistic} - ext{Value of hypothesized population parameter}}{ ext{Standard error of the sample statistic}}

Type I and Type II Errors

  • Type I Error (α): Rejecting the null when it is true (false positive).

    • Occurs when a claim of an effect is made in error.

    • Example: Concluding a new drug is more effective when it is not.

  • Type II Error (β): Not rejecting the null when it is false (false negative).

    • Occurs when a true effect is missed.

    • Example: Concluding no difference between two treatments when one exists.

P-values

  • P-values assist in deciding whether to reject the null hypothesis.

    • Small P-value (≤ 0.05): Reject H₀.

    • Large P-value (> 0.05): Do not reject H₀.

Large Sample Tests

  • Hypothesis Testing for Single Population Mean:

    • Null: H₀: μ = μ₀

    • Alternate: H₁: μ ≠ μ₀ (two-tailed).

    • If σ (standard deviation) is known, the z-test statistic applies:
      z = rac{ar{x} - μ₀}{σ/ ext{√}n}

Summary of Critical Values for Sample Statistic z

  • Critical Values: Summary for significance levels:

    • α = 0.10: z_{ ext{critical}} = ±1.28

    • α = 0.05: z_{ ext{critical}} = ±1.645

    • α = 0.01: z_{ ext{critical}} = ±2.33

    • α = 0.005: z_{ ext{critical}} = ±2.58

Worked Examples

  • Example 1: Testing lightbulb average life claim at 1% significance:

    • H₀: μ = 60 days

    • H₁: μ < 60 days

    • Use one-tailed test as H₁ specifies a direction.

  • Example 2: Difference between groups (e.g., mice diets) utilizes two-tailed tests.

Hypothesis Testing for Population Variances

  • Testing differences between population variances involves:

    • Formulating null (H₀: σ₁^2 = σ₂^2) and alternative hypotheses.

    • Using the F statistic for comparison:
      F = rac{s₁^2}{s₂^2}
      where s₁^2 and s₂^2 are sample variances.

Chi-Square Test for Independence

  • Assess relationships between two qualitative variables.

    • Steps include:

    1. State the hypotheses: H₀ (no association) vs H₁ (association exists).

    2. Create a contingency table.

    3. Calculate expected frequencies:
      E = rac{ ext{(Row total) x (Column total)}}{ ext{Grand total}}

    4. Compute the chi-square statistic:
      χ² = Σ rac{(O - E)²}{E}

    5. Compare with critical values for conclusion.

Rank Correlation

  • Spearman's Rank Correlation Coefficient:

    • Formula:
      p = 1 - rac{6Σd²}{n(n² - 1)}

    • Where d is the rank difference for each pair and n is the number of pairs.

    • Correction for tied ranks is applied if necessary.