1. SADC Course in Statistics Meaning and use of confidence intervals (Session 05)

2. To put your footer here go to View > Header and Footer 2 Learning Objectives By the end of this session, you will be able to explain the meaning of a confidence interval explain the role of the t-distribution in computing a confidence interval for the population mean calculate a confidence interval for the population mean using sample data state the assumptions underlying the above calculation

3. To put your footer here go to View > Header and Footer 3 Revision on standard errors Recall from the previous session that The standard error provides a measure of the precision of the sample mean the formula s/n gives the standard error of the mean when simple random sampling is used A low standard error indicates that the sample mean has high precision, i.e. the sample mean is a good estimate of the population mean

4. To put your footer here go to View > Header and Footer 4 Standard errors more generally… Whenever sample data is used to find an estimate of a pop n parameter, it should be accompanied 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 start to get worried about our estimate?

5. To put your footer here go to View > Header and Footer 5 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 example, 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.

6. To put your footer here go to View > Header and Footer 6 Finding the Confidence Interval The 95% confidence limits for (lower and upper) are calculated as: and where t n-1 is the 5% level for the t- distribution with (n-1) degrees of freedom. Statistical tables and statistical software give t-values.

7. To put your footer here go to View > Header and Footer 7 t-values for computation of 95% C.I. P 10 5 2 = 16.3112.731.8 22.924.306.96 32.353.184.54 42.132.783.75 52.022.573.36 61.942.453.14 71.892.363.00 81.862.312.90 91.832.262.82 101.812.232.76 201.722.092.53 301.702.042.46 401.682.022.42 601.672.002.39 1.641.962.33

8. To put your footer here go to View > Header and Footer 8 Correct interpretation of C.I.s If 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 that a single interval includes.

9. To put your footer here go to View > Header and Footer 9 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 t 9 (s/n) = 7.7 2.26(3.7/10) = 7.7 2.64 = (5.1, 10.4) Can you interpret this interval? Write down your answer. We will then discuss.

10. To put your footer here go to View > Header and Footer 10 Underlying assumptions The 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 normal 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)

11. To put your footer here go to View > Header and Footer 11 Using the Central Limit Theorem Recall this theorem says that the sampling distribution of the mean has a normal distribution, for large sample 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.

12. To put your footer here go to View > Header and Footer 12 Practical work follows … Note: The formula on slide 6 for a confidence interval applies when estimation of is of interest. Different assumptions on the data, and interest in other population parameters, will lead to different confidence intervals.