1、Meaning and use of confidence intervals(Session 05)第1頁，共14頁。Learning ObjectivesBy the end of this session, you will be able to explain the meaning of a confidence intervalexplain the role of the t-distribution in computing a confidence interval for the population meancalculate a confidence interval

2、for the population mean using sample datastate the assumptions underlying the above calculation2第2頁，共14頁。Revision on standard errorsRecall from the previous session that The standard error provides a measure of the precision of the sample meanthe formula s/n gives the standard error of the mean when

3、 simple random sampling is usedA low standard error indicates that the sample mean has high precision, i.e. the sample mean is a “good” estimate of the population mean3第3頁，共14頁。Standard errors more generallyWhenever sample data is used to find an estimate of a popn parameter, it should be accompanie

4、d by a measure of its precision!The formula s/n applies only when using as an estimate of the population mean . Formulae will differ for other estimates, depending on how the sample was selected.The higher the standard error, the less precise is the estimate - but how high should it be before we sta

5、rt to get worried about our estimate?4第4頁，共14頁。Confidence Interval for Instead of using a point estimate, it is usually more informative to summarise using an interval which is likely (i.e. with 95% confidence) to contain .This is called an interval estimate or a confidence interval (C.I.)For exampl

6、e, we could report that the mean landholding size of HHs in Kilindi district in Tanzania is 7.62 acres with 95% confidence interval (6.95, 8.28), i.e. there is a 95% chance that the interval (6.95,8.28) includes the true value .5第5頁，共14頁。Finding the Confidence IntervalThe 95% confidence limits for (

7、lower and upper) are calculated as:andwhere tn-1 is the 5% level for the t-distribution with (n-1) degrees of freedom.Statistical tables and statistical software give t-values.6第6頁，共14頁。t-values for computation of 95% C.I.P 10 5 2 = 16.3112.731.822.924.306.9632.353.184.5442.132.783.7552.022.573.3661

8、.942.453.1471.892.363.0081.862.312.9091.832.262.82101.812.232.76201.722.092.53301.702.042.46401.682.022.42601.672.002.39 1.641.962.337第7頁，共14頁。Correct interpretation of C.I.sIf we sampled repeatedly and found a 95% C.I. each time, only 95% of them would include the true , i.e. there is a 95% chance

9、that a single interval includes .8第8頁，共14頁。An example (persons per room)In Practical 3, the first of 50 samples of size 10 gave mean=7.7, std.dev.=3.7 for the number of persons per room.Hence a 95% confidence interval for the true mean number of persons per room:7.7 t9 (s/n) = 7.7 2.26(3.7/10)= 7.7

10、2.64= (5.1, 10.4)Can you interpret this interval? Write down your answer. We will then discuss.9第9頁，共14頁。Underlying assumptionsThe above computation of a confidence interval assumes that the data have a normal distribution.More exactly, it requires the sampling distribution of the mean to have a nor

11、mal distribution.What happens if data are not normal?Not a serious problem if sample size is large because of the Central Limit Theorem (see Session 4)10第10頁，共14頁。Using the Central Limit TheoremRecall this theorem says that the sampling distribution of the mean has a normal distribution, for large s

12、ample sizes.So even when data are not normal, the formula for a 95% confidence interval will give an interval whose “confidence” is still high - approximately 95%.Better attach some measure of uncertainty than worry about exact confidence level.11第11頁，共14頁。Practical work follows Note: The formula on slide 6 for a confidence interval applies when estimation of