STA6126 Chapter 6

# STA6126 Chapter 6 - Revised on 11:06 AM Chapter 6...

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STA6126 Chapter 5, Page 1 of 12 Revised on 8/22/2011 11:06 AM 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 0 ( 1) ~ / n X Tt Sn , where μ 0 is the value of μ specified by Ho. For testing hypotheses about the population proportion, π, the test statistic is always 0 00 ~ (0,1) (1 ) p ZN n 

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STA6126 Chapter 5, Page 2 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 0 then p-value = () P X x = P(T (df) T cal ) If Ha: μ < μ 0 then p-value = P X x = P(T (df) T cal ) If Ho: μ ≠ μ 0 then p-value = 00 P X x  = P(T (df) ≥ |T cal | ) If Ha: π > π 0 then p-value = P(Z Z cal ) If Ha: π < π 0 then p-value = P(Z Z cal ) If Ho: π ≠ π 0 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 0 , where μ 0 is a number ; the value of μ specified by Ho. Ha: μ ≠ μ 0 or Ha: μ < μ 0 or Ha: μ > μ 0 3. Test Statistic: 0 ( 1) ~ / n X Tt Sn . Remember that when the df is large, t approaches Z.
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STA6126 Chapter 6 - Revised on 11:06 AM Chapter 6...

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