Chapter 9 Notes: Testing a Claim

Null and Alternate Hypotheses

• Null Hypothesis (H0): Statement being tested; usually claims "no effect," "no difference," or no change.

• Alternate Hypothesis (Ha): Claim we seek evidence for; usually involves an "effect," "difference," or change.

Test Statistics

• Standardize the estimate to assess how far it is from the parameter.

• General form of test statistics used in hypotheses testing.

P-value

• Probability of obtaining sample result or a more extreme result under the null hypothesis.

• Smaller p-value indicates stronger evidence against H0.

Statistical Significance

• If p-value ≤ alpha (commonly 0.05), results are statistically significant.

Plan for Significance Test

1. Hypotheses: State H0 and Ha.

2. Conditions: Check conditions for the test.

3. Calculations: Compute test statistic, find p-value.

4. Interpretation: Use p-value to state conclusion in context.

Errors in Hypothesis Testing

• Type I Error: Rejecting H0 when it is true (false positive).

• Type II Error: Accepting H0 when it is false (false negative).

• Probability of Type I error = alpha.

Tests about Population Proportion

One-Proportion Z-Test

1. Hypotheses: H0: p = p0; Ha: p < p0, p > p0, or p ≠ p0.

2. Conditions:

   • Random sample.

   • Sample size < 10% of population.

   • np0 ≥ 10 and n(1-p0) ≥ 10.

3. Test Statistic:
   z = (p̂ - p0) / sqrt[(p0(1-p0))/n].

4. Conclusion: If p < alpha, reject H0; otherwise fail to reject H0.

Tests about Population Mean

One-Sample T-Test

1. Hypotheses: H0: μ = μ0; Ha: μ < μ0, μ > μ0, or μ ≠ μ0.

2. Conditions:

   • Random sample.

   • Sample size < 10% of population.

   • Normality (given or n ≥ 30).

3. Test Statistic:
   t = (x̄ - μ0) / (s/sqrt(n)).

4. Conclusion: If p < alpha, reject H0; otherwise fail to reject H0.

Paired Differences T-Test

• Used for comparing two treatments in paired data.

• Perform one-sample t analysis on the differences between paired observations (e.g., pre-test vs post-test scores).