Confidence Intervals, z-Test, Student's t-Test & Chi-Square Tests
“Constructing intervals, rejection regions, p-value decision rules, and two-sample statistical tests.”
1.The Formal Classical Testing Framework
§ 5.1.Statistical inference bridges empirical sample data to broader population truths through rigorous confidence intervals and hypothesis tests.
The Four Outcomes of Hypothesis Testing:
1. True Negative (Correct): H0 is True and we Fail to Reject H0 (Confidence level 1 - alpha).
2. Type I Error (False Alarm, alpha): H0 is True but we Reject H0 (Significance level alpha).
3. Type II Error (Missed Detection, beta): H0 is False but we Fail to Reject H0.
4. True Positive (Correct, Power): H0 is False and we Reject H0 (Power of test 1 - beta).
Selecting the Appropriate Test Statistic:
- Population variance sigma known, large sample: Use z-test.
- Population variance sigma unknown, normal population: Use Student's t-test.
- Comparing categorical frequency counts against expected ratios: Use Chi-Square test.
2. Worked Practice Problem: One-Sample z-Test & Confidence Interval
Two-tailed tests place alpha/2 probability mass in EACH tail (e.g. 0.025 in each tail for alpha = 0.05).
Problem: A benchmark latency test on n = 36 query evaluations yields a sample mean latency of X_bar = 42.0 ms. The population standard deviation is known to be sigma = 6.0 ms. Construct a 95% confidence interval for the true mean latency mu, and test H0: mu = 40.0 ms vs H1: mu != 40.0 ms at alpha = 0.05.
Step 1: Compute the Standard Error (SE) of the sample mean:
SE = sigma / sqrt(n) = 6.0 / sqrt(36) = 6.0 / 6 = 1.0 ms.
Step 2: Find the 95% critical value z_{alpha/2}:
For alpha = 0.05, alpha/2 = 0.025 => z_{0.025} = 1.96.
Step 3: Construct the 95% Confidence Interval:
CI = X_bar ± z_{alpha/2} SE = 42.0 ± 1.96 1.0 = 42.0 ± 1.96
95% CI = [ 40.04 ms, 43.96 ms ].
Step 4: Compute the test statistic for H0: mu = 40.0 ms:
z_stat = (X_bar - mu_0) / SE = (42.0 - 40.0) / 1.0 = 2.0 / 1.0 = +2.0.
Step 5: Apply decision rule:
For a two-tailed test at alpha = 0.05, the critical rejection region is |z| >= 1.96.
Since |z_stat| = 2.0 >= 1.96, we REJECT H0.
Step 6: Compute p-value:
p-value = 2 P(Z >= 2.0) = 2 (1 - Phi(2.0)) = 2 (1 - 0.9772) = 2 0.0228 = 0.0456.
Since p-value = 0.0456 < 0.05, the result is statistically significant at alpha = 0.05.
Conclusion: True mean latency is significantly different from 40.0 ms; 95% CI is [40.04, 43.96] ms.
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