In-Depth Notes for Stat 151 Module 5: Hypothesis Tests for 1 Proportion

Quick Review of Sampling Distribution of p̂

  • Population Proportion (p): Ratio of successful outcomes (successes) to total elements in the population.
  • RULE 1: Mean of the sampling distribution of p̂ is equal to p:
    [
    \mu_{p̂} = p
    ]
  • RULE 2: Standard deviation of the sampling distribution of p̂:
    [
    \sigma_{p̂} = \sqrt{\frac{p(1-p)}{n}}
    ]
  • RULE 3: Sampling distribution of p̂ approximates normal if n is large and p is not close to 0 or 1.
    • Sample size is large if:
    • np ≥ 10 and n(1 – p) ≥ 10

Hypothesis Tests for Proportion

  • Hypothesis Test Defined: A method using sample statistics to validate claims about population parameters.
  • Example Scenario: Eric claims a coin is fair (p = 0.5). Amy checks this by sampling 1000 flips, finding 400 heads.
    • Questions posed:
    • Is the proportion of heads < 0.5 (invalidating Eric's claim)?
    • Could the difference between the expected proportion (0.5) and observed (0.4) be due to sampling error?

Procedures for Hypothesis Testing

  1. Define the variable and relevant parameters/assumptions.
  2. State the null hypothesis (H0) and alternative hypothesis (Ha).
  3. Gather data (sample) and calculate the test statistic.
  4. Assess evidence against H0 using p-value.
  5. Make a decision based on p-value.
  6. State the conclusion.

Understanding Hypotheses

  • Null Hypothesis (H0): A claim assumed to be true until evidence suggests otherwise, typically states “no effect.”
    • Written as: H0: parameter = hypothesized value
  • Alternative Hypothesis (Ha): Claim true only when H0 is rejected; it’s the hypothesis we seek evidence for.

Types of Hypotheses

  • Two-tailed Test:
    • H0: parameter = specific value vs Ha: parameter ≠ specific value.
  • Upper-tailed Test:
    • H0: parameter = specific value vs Ha: parameter > specific value.
  • Lower-tailed Test:
    • H0: parameter = specific value vs Ha: parameter < specific value.

Decision-Making in Hypothesis Testing

  • H0 is rejected only if sample evidence suggests H0 is false.
  • Two possible conclusions:
    • Reject H0 (accept Ha)
    • Do not reject H0 (not the same as accepting H0).
  • Analogy: Like a criminal trial—innocence presumed until doubt is raised (H0: innocent, Ha: guilty).

P-values and Decisions

  • p-value: Probability of observing a test statistic as extreme or more extreme than what was observed, assuming H0 is true.
    • High p-value = consistent with H0 (do not reject H0).
    • Low p-value = unlikely observed results under H0 (reject H0).

Alpha Levels (Significance Levels)

  • Determines the threshold for rejecting H0, often set at common values (0.10, 0.05, 0.01).
  • Statistical Significance: Results are significant if p-value < alpha; does not imply practical importance.

Conclusion from Hypothesis Tests

  • Always report the p-value along with conclusions from H0 testing.
  • Include confidence intervals where possible to provide context for your results.

Types of Errors in Hypothesis Testing

  • Type I Error: Rejecting H0 when it is actually true (false positive).
  • Type II Error: Failing to reject H0 when it is false (false negative).
  • Probability of Errors:
    • Type I error rate (α): P(H0 is rejected | H0 is true).
    • Type II error rate (β): P(H0 is not rejected | H0 is false).
    • Power of the test = 1 - β: Probability of correctly rejecting a false H0.

Example Contexts for Hypothesis Testing

  • Various scenarios were discussed, such as testing the proportion of Canadians eating chocolate or assessing on-time delivery rates.
  • Each example includes establishing H0 and Ha, determining the type of test, calculating p-values, and considering decision errors that could arise based on test results.

Using Confidence Intervals in Hypothesis Testing

  • Confidence intervals are related to hypothesis tests; if the null value is outside the confidence interval, H0 can be rejected.
  • A confidence level of C% correlates with a 1-α hypothesis test.
  • Example: Testing the claim about coin fairness through confidence intervals that reflect the proportion of heads observed.