Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

AP Statistics 9.1B Notes Interpreting P-Values The probability, computed assuming H0 is true, that the statistic (such as p̂ or x ) would take a value as extreme as or more extreme than the one actually observed is called the P-value of the test. The smaller the P-value, the stronger the evidence against H0 provided by the data. Small P-values are evidence against H0 because they say that the observed result is unlikely to occur when H0 is true. Large P-values fail to give convincing evidence against H0 because they say that the observed result is likely to occur by chance when H0 is true. Example For the job satisfaction study described in 9.1A Notes, the hypotheses are H0 : 0 Ha : 0 where is the mean difference in job satisfaction scores (self-paced – machine-paced) in the population of assemblyline workers at the company. Data from the 18 workers gave x 17 and sx 60. That is, these workers rated the self-paced environment, on average, 17 points higher. Researchers performed a significance test using the sample data and obtained a P-value of 0.2302. A. Explain what it means for the null hypothesis to be true in this setting. B. Interpret the P-value in context. C. Do the data provide convincing evidence against the null hypothesis? Explain. The conclusion of the job satisfaction study is not that H0 is true. The study looked for evidence against H 0 : 0 and failed to find strong evidence. That is all we can say. Failing to find evidence against H0 means only that the data are consistent with H0, not that we have clear evidence that H0 is true. Statistical Significance The final step in performing a significance test is to draw a conclusion about the competing claims you were testing. We will make one of two decisions based on the strength of the evidence against the null hypothesis (and in favor of the alternative hypothesis) – reject H0 or fail to reject H0. If our sample result is too unlikely to have happened by chance assuming H0 is true, then we’ll reject H0. Otherwise, we will fail to reject H0. Our conclusion in a significance test comes down to P-value small → reject H0 → conclude Ha (in context) P-value large → fail to reject H0 → cannot conclude Ha (in context) There is no rule for how small a P-value we should require in order to reject H0 – it’s a matter of judgment and depends on the specific circumstances. But we can compare the p-value with a fixed value that we regard as decisive, called the significance level. We write it as α, the Greek letter alpha. If we choose α = 0.05, we are requiring that the data give evidence against H0 so strong that it would happen less than 5% of the time just by chance when H0 is true. If we choose α = 0.01, we are insisting on stronger evidence against the null hypothesis, a result that would occur less often than 1 in every 100 times in the long run if H0 is true. When our P-value is less than the chosen α, we say that the result is statically significant. Statistically significant If the P-value is smaller than alpha, we say that the data are statistically significant at level α. In that case, we reject the null hypothesis H0 and conclude that there is convincing evidence in favor of the alternative hypothesis Ha. “Significant” in the statistical sense does not necessarily mean “important.” It means simply “not likely to happen just by chance.” The significance level α makes “not likely” more exact. Users of statistics have often emphasized standard significance levels such as 10%, 5%, and 1%.