Stat 226 - Section 6.2

Stat 226 - Section 6.2 - Statistical Inference Section 6.2...

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Section 6.2 Tests of Significance Section 6.2 1 Statistical Inference Confidence intervals are one of the two commonly used tools in statistical inference. Confidence intervals are especially useful when the goal is to estimate a population parameter and include information about the precision of the estimate. A second goal in statistical inference is to assess the evidence provided by the data in favor of some claim about the population. Section 6.2 2 Significance Tests A significance test is a formal procedure for comparing observed data with a hypothesis about the population. The plausibility of the hypothesis is evaluated against the information provided by the data. The hypothesis statement is always about the population parameters and always consists of two components: Null hypothesis Alternative hypothesis The results of the test are often expressed in terms of a probability that describes how well the data matches the hypothesis. Section 6.2 3 Example - Ink Cartridges The black ink cartridges for a particular type of inkjet printer have a labeled volume of 10 mL of ink. The manufacturing company periodically tests its filling machine for accuracy. A recent test of 100 ink fills had a mean volume of 9.971 mL. We will assume that the standard deviation of all ink fills is 0.2 mL. Section 6.2 4 Ink Inference In this case, we are especially interested in knowing something specific about the population mean ink volume μ . Does μ = 10 mL as labeled? This is an ideal setting for a significance test. We want to investigate if the sample provides evidence that the labeled volume is incorrect. The labeled volume will serve as our null hypothesis . How likely is a sample mean of 9.971 mL if the true population mean is 10 mL? Section 6.2 5 General Test Procedure The general procedure for a significance test consists of five steps. 1. Specify a level of significance α . 2. State the null hypothesis and the alternative hypothesis. 3. Calculate the test statistic. 4. Find the p -value. 5. State the conclusion in the context of the problem. Section 6.2 6
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Goal of Significance Testing In significance testing, or hypothesis testing, we examine observed data for evidence against the null hypothesis. The “evidence” is expressed as a probability of observing the sample result, assuming that the null hypothesis is true . If the probability is small, then there is substantial evidence against the null hypothesis. Section 6.2 7 Stating Hypotheses We will skip step 1 for now and move to step 2. 1. Specify a level of significance α . 2. State the null hypothesis and the alternative hypothesis. 3. Calculate the test statistic. 4. Find the p -value. 5. State the conclusion in the context of the problem.
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