Hypotheses and P-Values

Learn standardized test statistics, p-values and decisions.

Lesson 2 of 32 parts Practice Flashcards

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Standardized Test Statistics

Objective: Explain standardized test statistics, apply the stated method, and check its conditions.

A standardized mean-test statistic compares estimate minus null value with the estimate’s SE. Its reference distribution and validity depend on design and modeling conditions; an estimated mean SE often leads to a t procedure. A large magnitude alone does not give a p-value without the reference model and alternative. Specify the reference distribution and alternative before obtaining a p-value.

Formulas

Test statistic = (estimate−null value)/SE

Worked examples

Try itA mean test has x̄=49, null mean μ₀=7, and estimated SE=6. Find its standardized test statistic.Show answer

Step 1: Use (estimate−null value)/SE. Step 2: Compute (49−7)/6. Step 3: The statistic is 7; its reference distribution depends on the test conditions.

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Standardized Test Statistics

Objective: Explain standardized test statistics, apply the stated method, and check its conditions.

Formulas

Test statistic = (estimate−null value)/SE

Worked examples

Try itA mean test has x̄=49, null mean μ₀=7, and estimated SE=6. Find its standardized test statistic.Show answer

Step 1: Use (estimate−null value)/SE. Step 2: Compute (49−7)/6. Step 3: The statistic is 7; its reference distribution depends on the test conditions.

P-Values and Decisions

Objective: Explain p-values and decisions, apply the stated method, and check its conditions.

A p-value is the null-model probability of results at least as incompatible with the null as observed, according to the specified statistic and alternative. Compare it with preselected α. Failing to reject is not proof of equality, and rejecting is not certainty that a particular alternative is true. A p-value is not P(H₀ is true).

Formulas

Reject H₀ when p≤α under the stated decision rule

Worked examples

Try itA synthetic test reports p=0.025 with α=0.05. Which decision follows the stated rule?Show answer

Step 1: Compare the reported p-value with the preselected alpha. Step 2: 0.025 is at most 0.05. Step 3: Reject H₀ when p≤α; otherwise fail to reject it. Neither decision is proof of certainty.

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