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Unformatted text preview: Revised on 11/9/2011 19:08 a11/p11 Chapter 6 Statistical Inference: Significance Tests Definition: A hypothesis is a statement (sentence) with a subject (a parameter), object (a number) and a verb (one of the following: =, ≠, <, >). Hypotheses come in pairs: • The Null hypothesis (Ho): is a hypothesis of no change (status quo). It represents our current knowledge about the value of the parameter. • Some examples of are Ho: μ = 5, or Ho: π = 0.4. • The Alternative Hypothesis (Ha): is a statement of change (a claim, conjecture, guess) in the value of the parameter. • Some examples are Ha: μ > 2 or Ho: μ < 6 or Ha: μ ≠ 10 • More examples Ha: π ≠ 0.9, or Ha: π < 0.4 or Ha: π > 0.6 Six Steps of Hypothesis Testing: 1. Specify the assumptions • Define the random variable (population) • Define the distribution of the population • Define the parameter of the distribution (known and unknown) • Specify data collection method and type of data • Specify the sample size. 2. Specify the hypotheses : • Equality is ALWAYS with Ho. • Always in pairs (Ho and Ha) • Ho is the hypothesis of the current knowledge (statusquo) • Ha is the hypothesis of change. • When one of the two hypotheses is true the other one MUST be false. 3. Specify the test statistic : • For testing hypotheses about the population mean, μ, the test statistic is always ( 1) ~ / n X T t S n μ-- = , where μ is the value of μ specified by Ho. • For testing hypotheses about the population proportion, π, the test statistic is always ~ (0,1) (1 ) p Z N n π π π- =- STA6126 Chapter 5, Page 1 of 12 4. Calculate the p-value : • The p-value = Probability of observing what is observed or more extreme, assuming Ho is true . • “More extreme” means “the values of the test statistic that support the alternative hypothesis.” • Always look at Ha to determine what is meant by “more extreme.” • If Ha: μ > μ then p-value = ( ) P X x ≥ = P(T (df) ≥ T cal ) • If Ha: μ < μ then p-value = ( ) P X x ≤ = P(T (df) ≤ T cal ) • If Ho: μ ≠ μ then p-value = ( ) P X x μ μ- ≥- = 2× P(T (df) ≥ |T cal | ) • If Ha: π > π then p-value = P(Z ≥ Z cal ) • If Ha: π < π then p-value = P(Z ≤ Z cal ) • If Ho: π ≠ π then p-value = 2 × P(Z ≥ |Z cal | ) 5. Make a decision: • Either “reject Ho” or “Do not reject Ho.” • Decision Rule is ALWAYS “ Reject Ho if p-value ≤ α” 6. Write a conclusion: • Explain the decision in layman’s language • with no technical jargon (except specifying the p-value or α). 6.1 Tests for Population Means 1. Assumptions: a) SRS b) Quantitative variable c) Normal population d) μ = mean of the population 2. Hypotheses: Ho: μ = μ , where μ is a number ; the value of μ specified by Ho....
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