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6. Significance tests - 6 Statistical Inference...

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6. Statistical Inference: Significance Tests Goal: Use statistical methods to test hypotheses such as “For treating anorexia, cognitive behavioral and family therapies have same mean weight change as placebo” ( no effect) “Mental health tends to be better at higher levels of socioeconomic status (SES)” (i.e., there is an effect) “Spending money on other people has a more positive impact on happiness than spending money on oneself.”
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Hypotheses : For statistical inference, these are predictions about a population expressed in terms of parameters (e.g., population means or proportions or correlations) for the variables considered in a study A significance test uses data to evaluate a hypothesis by comparing sample point 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) (Usually: “no effect”) Alternative hypothesis ( H a ): states that parameter value(s) falls in some alternative range of values (an “effect”)
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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) 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 is not necessarily true, but it is plausible) Process is analogous to American judicial system H 0 : Defendant is innocent H a : Defendant is guilty
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Significance Test for Mean Assumptions : Randomization, quantitative variable, normal population distribution (robustness?) Null Hypothesis : H 0 : µ = µ 0 where µ 0 is particular value for population mean (typically “no effect” or “no change” from a standard) Alternative Hypothesis : H a : µ µ 0 2-sided alternative includes both > and < H 0 value Test Statistic : The number of standard errors that 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.
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