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Hypothesis Testing Evaluation
A Comprehensive Step-by-Step Comparative Analysis: Z-Test vs. Student's t-Test
The operational mathematical distinction between a Z-test and a Student's t-test relies entirely on the state of population parameter knowledge: a Z-test requires a known, rigid population standard deviation ( σ ), while a t- test handles an unknown population variance by substituting the sample standard deviation ( s ) and adjusting the probability curve for estimation uncertainty.
The Core Scenario
A university claims that the true historical average score on its comprehensive mathematics entrance exam is 70 points. A researcher suspects the true performance level has drifted. She extracts a random sample of n = 36 students, yielding a sample mean score of X̄ = 73.5 points. The test is evaluated at a 5% significance level (α = 0.05) using a two-tailed distribution design.
Case 1: The Z-Test (Population Variance σ is Known)
Condition: Historical parameters confirm that the population standard deviation is fixed at σ = 9.
State the Hypotheses: Null Hypothesis ( H 0 ): μ = 70 (The population mean is exactly 70) Alternative Hypothesis ( H 1 ): μ ≠ 70 (The population mean is not 70 [Two-tailed]) Determine the Critical Boundaries: Splitting the error probability symmetrically ( α/2 = 0.025 in each tail) across the standard normal distribution yields critical value bounds of Zcrit = ±1.. Calculate the Test Statistic: First, determine the Standard Error of the mean (SE):
SE = σ / &sqrt;n = 9 / &sqrt;36 = 9 / 6 = 1.
Standardize the sample mean displacement into a calculated Z-score:
Z = (X̄ - μ) / SE = (73.5 - 70) / 1.5 = 3.5 / 1.5 = 2.
Execute the Decision Rule: Since the calculated metric Z = 2.33 exceeds the absolute threshold boundary of 1.96 , it falls squarely inside the upper rejection region. (Two-tailed P-value = 0.0198 ≤ 0.05). Statistical Conclusion: Reject H 0. There is sufficient evidence to declare that the true mean score differs from 70.
Comparative Metric Summary
Evaluated Parameter Case 1: Z-Test Frame Case 2: Student's t-Test Frame
Variance Prerequisite
Population Parameter Known ( σ = 9 )
Population Parameter Unknown, Sample Estimated ( s = 9 )
Sampling Curve (^) Standard Normal Distribution ( Z ) Student's t -Distribution ( df = 35 )
Critical Value Bounds
±1.960 (Fixed boundary) ±2.030 (Expands as sample size shrinks)
Computed Test Metric
Zcalc = 2.33 tcalc = 2.
Resulting P-Value (^) 0.0198 0.0257 (Reflects higher uncertainty margin)
Inference Decision (^) Reject H 0 (Highly Significant) Reject H 0 (Significant)
