Presentation on theme: "Topics Today: Case I: t-test single mean: Does a particular sample belong to a hypothesized population? Thursday: Case II: t-test independent means: Are."— Presentation transcript:
1TopicsToday: Case I: t-test single mean: Does a particular sample belong to a hypothesized population?Thursday: Case II: t-test independent means: Are two sample means drawn from an identical population or different populations?
2Z-Test vs t-Test Z-Test T-Test Where population standard deviation and standard error are knownWhere sample size is > 30Where the normal curve is the model for the sampling distribution for determining the probability of obtaining our result under the null hypothesisT-TestWhere population standard deviation and standard error are not known and have to be estimated from the sample dataWhere the t distribution is model for sampling distribution for determining probability of our results under the null hypothesis
5Degrees of Freedom (dfs) t distributions differ according to their degrees of freedom (based on the sample size)For single sample case the t distribution is based on n-1 dfsFor 2 sample case the t distribution for differences is based on N-2 dfs (i.e., n -1 for each sample)
6Hypothesis Testing Using t-Tests Identify population of interestDraw samples (preferably probability samples)Set up null and alternative hypothesesSelect level of significance (e.g., .05, .01)Calculate sample statistic (e.g. mean)Calculate standard error of the sample statisticConvert observed mean to standard error pointsDetermine t-critical value (based on level of significance chosen)Compare t-observed value against the t-critical valueDecide: “Reject” or “Do not reject” null hypothesis
7Sampling Distribution of Means: Standard Errors, Critical Values, and Ps t DistributionSample Size =31df=30 = .05 or .01Two tailedTest-2se-1se+1se+2seCriticalValues-2.75se-2.04se+2.04se+2.75sep =< = .05 = outside 2.04 on either end< = .01 = outside of 2.57 on either end
8Sampling Distribution of Means: Standard Errors, Critical Values, and Ps t DistributionSample Size =31df=30 = .05 or .01One tailedtestu-2se-1se+1se+2seCriticalValuesse+2.457sep =< = .05< = .01
9Sampling Distribution of Means: Standard Errors, Critical Values, and Ps t DistributionSample Size =31df=30 = .05 or .01One tailedtestu-2se-1se+1se+2seCriticalValues-2.457sesep =< = .01< = .05
10Case I: t-TestDoes a particular sample belong to a hypothesized population?Draw single sample from populationCalculate sample statistic such as meanTest null hypothesis of no difference between sample and population mean
11Case I: Assumptions Scores randomly sampled from some population Scores in the population are normally distributed
12Example t-Test Single Sample: SAT Data: UCLA from OAS
13t-Test for Single Sample Designs Set Null Hypothesis:Set Alternative Hypothesis:Decide Significance Level: .01Compute Standard Error:Compute tobservedLocate tcritical with N-1df:DecideReject H0 if tobserved >= tcritical :Do Not Reject H0 if tobserved < tcritical :Conclude:
14UCLA: Sampling Distribution Picture t DistributionSample Size =10df=9 = .01One tailedtest3.95-2se-1se+1se+2se= 1000CriticalValues+2.82sep == < .01
15X - tcritical(.01/1,9) (sx) <= <= X+ tcritical(.01/1,9) (sx) 99% Confidence IntervalX - tcritical(.01/1,9) (sx) <= <= X+ tcritical(.01/1,9) (sx)(21.26) <= <= (21.26)<= <=<= <=Can feel 99% confident that this interval includes the population mean for the UCLA undergraduates. Since it does not include the “known” population mean of 1000 we can conclude that UCLA undergraduates have on average higher total SAT scores than the population of high school graduates taking the SAT.