STAT101_Chap6 - 6 Statistical Inference Significance Tests...

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6. Statistical Inference: Significance Tests Goal: Use statistical methods to test hypotheses such as “Mental health tends to be better at higher levels of socioeconomic status (SES)” (i.e., there is an effect) “For treating anorexia, cognitive behavioral and family therapies have same mean weight change as placebo” ( no effect)
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Hypotheses : Predictions about a population expressed in terms of parameters A significance test uses data to summarize evidence about a hypothesis by comparing sample estimates of parameters to values predicted by the hypothesis. We answer a question such as, “If the hypothesis were true, would it be unlikely to get data such as we obtained?”
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Five Parts of a Significance Test Assumptions about type of data (quantitative, categorical), sampling method (random), population distribution (e.g., normal, binary), sample size (large enough?) Hypotheses : Null hypothesis ( H 0 ): A statement that parameter(s) take specific value(s) (Often: “no effect”) Alternative hypothesis ( H a ): states that parameter value(s) in some alternative range of values p. 1 examples?
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Test Statistic : Compares data to what null hypo. H 0 predicts, often by finding the number of standard errors between sample point estimate and H 0 value of parameter P -value ( P ): A probability measure of evidence about H 0. The probability (under presumption that H 0 true) that the test statistic equals observed value or value even more extreme in direction predicted by H a . The smaller the P -value, the stronger the evidence against H 0. Conclusion : If no decision needed, report and interpret P- value If decision needed, select a cutoff point (such as 0.05 or 0.01) and reject H 0 if P-value ≤ that value
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The most widely accepted cutoff point is 0.05, and the test is said to be “ significant at the .05 level” if the P-value ≤ 0.05. If the P -value is not sufficiently small, we fail to reject H 0 (then, H 0 not necessarily true, but it is plausible) Process is analogous to American judicial system H 0 : Defendant is innocent
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Significance Test for Mean Assumptions : Randomization, quantitative variable, normal population distribution Null Hypothesis : H 0 : µ = µ 0 where µ 0 is particular value for population mean (typically no effect or change from a standard) Alternative Hypothesis : H a : µ µ 0 ( 2-sided alternative includes both > and <) Test Statistic : The number of standard errors the sample mean falls from the H 0 value 0 where / y t se s n se μ - = =
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When H 0 is true, the sampling dist of the t test statistic is the t distribution with df = n - 1. P-value : Under presumption that H 0 true, probability the test statistic equals observed value or even more extreme (i.e., larger in absolute value), providing stronger evidence against H 0 – This is a two-tail probability, for the two-sided H a Conclusion : Report and interpret P -value. If needed, make decision about H 0
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Example : Anorexia study (revisited) Weight measured before and after period of treatment y = weight at end – weight at beginning
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