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1、Statistics II10thweek/Probability and Statistics(IV)Hypothesis Test We make a hypothesis about the whole population,and test it using the observed random samples Example:freezing point of salty water Hypothesis:the freezing point of salty water is not 0 The whole population:an imaginary pool of free
2、zing salty water Random samples:experiments of freezing salty water2Hypothesis Test Hypothesis test:statistical test for a hypothesis Null hypothesis 0:=0,the freezing point of salty water is 0 Null distribution:a distribution of the whole population when the null hypothesis is true Alternative hypo
3、thesis:0,the freezing point is not 0 We either reject or not-reject the null hypothesis based on evidence against the null hypothesis3Hypothesis Test4Formulate and Select Appropriate Statistics(z,t,F,chi-square etc.)Choose Level of SignificanceCollect Data and Calculate Test StatisticDetermine Proba
4、bilityAssociated with Test Statistic(-value)Determine Critical Value of Test StatisticCompare with Level of SignificanceDetermine if Calculated Test Statistic falls into(Non)Rejection RegionReject or Do not Reject Level of Significance Type I error occurs when the sample results lead to the rejectio
5、n of the null hypothesis when it is in fact true The probability of type I error()is also called the level of significance Most common significance levels used are 1%level,5%level and 10%level The higher the significance level()used for testing a hypothesis,the higher the probability of rejecting a
6、null hypothesis when it is true5ExampleObserved freezing point of salty water =0.31,0.67,0.61,2.07,1.31,=0.99,=0.70.0.Is it because 0 or because of random effect while =0?We will answer this question using =/6 H0:=0 vs.HA:0 The null distribution of t T(4),and t-statistic is 3.15 The probability that
7、-3.15(and more exceeding value)is observed is 0.0170.00.10.20.3-4-2042Example70.017P-Value P-value:the probability that a statistic exceeding the observed one(toward the alternative hypothesis)is from the null distribution In the example,p=0.034 because both 3.15 are extreme toward HA:080.00.10.20.3
8、-4-20420.0170.017Significance Significance level A threshold of p-value to reject or not to reject the null hypothesis People conventionally use 1%or 5%In the example,p=0.034 or 3.4%.Two possible conclusions With a 5%significance level,the freezing point of salty water is significantly different fro
9、m 0,or equally 0is rejected With a 1%significance level,the freezing point of salty water is not significantly different from 0,or equally 0cannot be rejected9SummaryStatistics is a tool to say something about the whole population from the observed random samples in a scientific wayThe“something”is“
10、mean”in many casesSummaryPr /=/(1)T-statisticsReferencesProbability and Stochastic Processes:A Friendly Introduction to Electrical and Computer Engineers(3rd edition),Yates and Goodman,WileyProbability,Statistics,and Random Processes for Electrical Engineering(3rd edition),Leon-Garcia,Pearson International Edition.