22 27 recap natural questions hypothesis testing

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RecapNatural QuestionsHypothesis TestingExampleExample: GPAsI believe that Cal is populated by smart students, who work hard,and thus have high GPAs. I think the average GPA of all Calstudents is 3.0. My office mate thinks that the professors aredifficult graders, and thus the GPA is lower than 3.0.1State the hypotheses:Null:The average GPA of a Cal student is 3.0Alt.:The average GPA of a Cal student is less than 3.02Gather evidence: Take a SRS of size 1,000 students from Cal,and record their GPAs.Observe an average GPA of 2.7.3Compare evidence to null hypothesis: Calculate a teststatistic.23 / 27
RecapNatural QuestionsHypothesis TestingExampleExample: GPAs (cont.)4Decide whether or not to reject the null hypothesis: Let’spretend we reject, and conclude that the difference betweenthe observed GPA (2.7) and the hypothesized GPA (3.0) isnot due to chance.Another SRS of size 1,000 students from Cal is done, andobserves an average GPA of 2.5.Should this SRS reject the null hypothesis? Is this evidencestronger or weaker than the evidence from the first SRS? Why?24 / 27
RecapNatural QuestionsHypothesis TestingExampleStrength of EvidenceTo quantify the difference between these two evidence, we useobserved levels of significanceAll observed levels of significance are between 0% and 100%(or 0 and 1)Lower level =stronger evidence against null hypothesisHigher level =weaker evidence against null hypothesisCalled the “p-value”Representsthe chance the test statistic is as or more extremethan the one observed,assuming the null hypothesis is trueDoes not representthe chance the null hypothesis is true25 / 27
RecapNatural QuestionsHypothesis TestingExampleStrength of Evidence (cont.)Customarily:If p-value<5%, evidence is(statistically) significant. “Thep-value =and thus we reject the null hypothesis due tostatistically significant evidence.”If p-value<1%, evidence ishighly (statistically)significant. “The p-value =and thus we reject the nullhypothesis due to highly (statistically) significant evidence.”26 / 27
RecapNatural QuestionsHypothesis TestingExampleImportant TakeawaysHypothesis tests answer questions about the makeup of theworldAll conclusions refer to thenull hypothesisWe cannot prove the null hypothesis. We can only fail todisprove it.With box models, hypothesis tests decide whether differencebetween observed and expected are due to chanceSome evidence is stronger than others, quantified byp-valueP-valuesdo notrepresent the chance that the null hypothesisis trueNext time: Developing hypotheses, calculating test statistics,finding p-values.27 / 27

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