Hypothesis Testing and Confidence Intervals

Posted on Dec 9, 2024 in Mathematics

Confidence Interval (CI): A range of values likely to contain the population parameter.
Margin of Error (E): The amount added and subtracted from the sample mean to form the CI.
Critical Value (t_α/2): Value from the t or z distribution for the desired confidence level.
Degrees of Freedom (df): Number of independent values; df = n – 1 for t-tests.
Standard Error (SE): Variability measure of the sample mean; SE = s / √n (s = sample standard deviation, n = sample size).
P-value: Probability of a result as extreme as observed, assuming the null hypothesis is true.
Test Statistic (t or z): Computed value to determine hypothesis test outcome.

For t-distribution (σ unknown): CI = x̄ ± (t_α/2 × SE)
For z-distribution (σ known): CI = x̄ ± (z_α/2 × SE)
TI-84:
- t-interval: STAT → TESTS → TInterval
- z-interval: STAT → TESTS → ZInterval

1-Sample t-Test (Population Mean):
- Null: H₀: μ = μ₀
- Alternative: H₁: μ ≠ μ₀
- Test Statistic (t): t = (x̄ – μ₀) / (s / √n)
- TI-84: STAT → TESTS → TTest
Paired t-Test:
- Null: H₀: μ_d = 0 (no difference)
- Alternative: H₁: μ_d ≠ 0
- TI-84: STAT → TESTS → TTest, select “Paired”
2-Sample t-Test:
- Null: H₀: μ₁ = μ₂
- Alternative: H₁: μ₁ ≠ μ₂
- TI-84: STAT → TESTS → 2-SampTTest
1-Proportion z-Test:
- Null: H₀: p = p₀
- Alternative: H₁: p ≠ p₀
- Test Statistic (z): z = (p̂ – p₀) / √[(p₀(1 – p₀)) / n]
- TI-84: STAT → TESTS → 1-PropZTest

State Hypotheses: Define null (H₀) and alternative (H₁).
Set Significance Level (α): Common values are 0.05, 0.01.
Compute Test Statistic (t or z).
Find P-value or Critical Value.
Make a Decision:
- Reject H₀ if p-value ≤ α or test statistic exceeds critical value.
- Fail to reject H₀ if p-value > α.

(Examples 1-5 from original document, rewritten for brevity and clarity, would be included here)