Inference for a Single MeanThe t-distributionsThe one-sample Confidence Interval (CI) for with Unknown The one-sample t testInference for a Single Mean - Dependent Two Samples (Matched Pairs)Example of matched pairs designMatched Pairs Confidence IntervalPaired t testComparing Two Means-Independent Two Samples with 1 =2Two Sample t-testsTwo Sample t Confidence IntervalComparing Two Means - Independent Two Samples w/ 1 = 2 Pooled EstimatorTest StatisticsHT and CIRemarksChapter 7 - Inference for DistributionsAnh DaoJuly 24th, 2009Chapter 7 - Inference for DistributionsInference for a Single MeanTo this point, when examining the mean of a population we have alwaysassumed that the population standard deviation (σ) was known.In practice this is seldom the case.We usually must estimate the population standard deviation with thesample standard deviation s.When we do this, the sampling distribution of the sample mean is nolonger normally distributed, because of the adjustment for estimating σwith s .Thus, instead of using the Z , the standard normal distribution, wemust use the appropriate t-distribution.Chapter 7 - Inference for DistributionsInference for a Single MeanThe t-distributionsAlthough there is only one Z -distribution, there are many, manyt-distributions.In fact, there is a different t-distribution for each sample size used.The shape of each t-distribution is very similar to the Z-distribution, butis slightly flatter.The larger the sample size, the closer the t-distribution is to theZ -distribution.Chapter 7 - Inference for DistributionsInference for a Single MeanThe t-distributionsThe way we distinguish between various t-distributions is by finding thedegrees of freedom, df , that correspond to the sample size.When we are looking at only one sample, the degrees of freedom arethe sample size minus one: df = n −1.We say that the one-sample t-statistictn−1=¯x − µs/√nhas the t distribution with n − 1 degrees of freedom.Chapter 7 - Inference for DistributionsInference for a Single MeanThe t-distributionsA table of t-distribution critical values can be found in Table D.Difference between Z and t tables:Make sure to get acquainted with this table and how it differs from theZ -table.Chapter 7 - Inference for DistributionsInference for a Single MeanThe t-distributionsExample of how the Z and t distributions compare:Here is a nice applet showing that: as sample size increases,t-distribution converges to normal distribution.http://www.stat.tamu.edu/ jhardin/applets/signed/T.htmlChapter 7 - Inference for DistributionsInference for a Single MeanThe t-distributionsWith the change from σ to s, and the change from z∗to t∗, the steps inproducing confidence intervals and hypothesis tests are the same aswe have seen previously.We find that s is calculated from the data using the formula:s =vuut1n − 1nXi=1(xi−¯x)2Chapter 7 - Inference for DistributionsInference for a Single MeanThe one-sample Confidence Interval (CI) for µ with Unknown σThe formula for a confidence interval for µ with unknown σ is¯x ± t∗s√nt∗is found in table D at the back of the book. It must correspond to theappropriate df = n − 1. It is easier to find the confidence level at thebottom of the table and go up to the correct dfChapter 7 - Inference for DistributionsInference for a Single MeanThe one-sample Confidence Interval (CI) for µ with Unknown σOne-sample t CI ExampleAn economist wants to determine the average amount that a family ofsize four in the United States spends on housing annually. He randomlyselects 85 families of size four and finds the amount they spent onhousing the previous year.The economist wishes to estimate the mean with 99% confidence.Chapter 7 - Inference for DistributionsInference for a Single MeanThe one-sample Confidence Interval (CI) for µ with Unknown σOne-sample t CI ExampleInformation we have:Sample Size: n = 85.Data: $6,789, $8,233, $4,784, . . ., $5,974 (85 numbers)Calculated from the data:¯x = $6, 219, s = $1,978Degree of Freedom: df = n − 1 = 85 − 1 = 84Chapter 7 - Inference for DistributionsInference for a Single MeanThe one-sample Confidence Interval (CI) for µ with Unknown σOne-sample t CI Example¯x ± t∗s√n= 6, 219 ± 2.6391, 978√85= (5652.82, 6785.18)t∗is found in Table D. We first go to the 99% confidence level at thebottom. Since 84 is NOT in the table, we then go down to 80 df (alwaysround down). Thus, t∗= 2.639.This is a 99% confidence interval for the true average amount a familyof four in the United States spends on housing annually.Chapter 7 - Inference for DistributionsInference for a Single MeanThe one-sample t testProcedures:1State the null hypothesis (H0).2State the alternative hypothesis (Haor H1).3State the level of significance (e.g., α = 0.05).4Calculate the test statistic (t statistic):t =¯x − µ0s/√n5Calculate p -valueChapter 7 - Inference for DistributionsInference for a Single MeanThe one-sample t testFor a one-sided test: (Ha: µ > µ0)p − value = P(T > t)For a one-sided test: (Ha: µ < µ0)p − value = P(T < t)For a two-sided test: (Ha: µ 6= µ0)p − value = P(T ≥ |t| or T ≤ −|t|) = 2P(T ≥ |t|)Because of the limited number of t-values given in Table D, it is morecommon to find a range for the p-value, rather than the exact value.Chapter 7 - Inference for DistributionsInference for a Single MeanThe one-sample t test6Reject or fail to reject H0based on the p-value.If the p-value is less than or equal to α, reject H0.It the p-value is greater than α, fail to reject H0.7State your conclusion.If H0is rejected, “There is significant statistical evidence that thepopulation mean is different from µ0.”If H0is not rejected, “There is NOT significant statistical evidencethat the population mean is different than µ0.”Notice that these last two steps are exactly the same as for the case where σis known.Chapter 7 - Inference for DistributionsInference for a Single MeanThe one-sample t testTV ExampleSuppose that the data collected from our class survey is a random samplefrom the entire university (which obviously is not). We wish to see if there isevidence that the average amount of television watched for students here ismore than 7 hours per week4Chapter 7 - Inference for DistributionsInference for a Single MeanThe one-sample t testSample Size: n = 38¯x: 8.05, s = 7.46Degree of Freedom:df = n − 1 = 37Chapter 7 - Inference for DistributionsInference for a Single MeanThe one-sample t
View Full Document
Unlocking...