Meaning and use of confidence intervals

1. Meaning and use of confidence intervals (Session 05)

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. 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. Standard errors more generally… • Whenever sample data is used to find an estimate of a popn 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. 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. Finding the Confidence Interval The 95% confidence limits for  (lower and upper) are calculated as: and where tn-1 is the 5% level for the t-distribution with (n-1) degrees of freedom. Statistical tables and statistical software give t-values.

7. t-values for computation of 95% C.I. P 10 5 2  = 1 6.31 12.7 31.8 2 2.92 4.30 6.96 3 2.35 3.18 4.54 4 2.13 2.78 3.75 5 2.02 2.57 3.36 6 1.94 2.45 3.14 7 1.89 2.36 3.00 8 1.86 2.31 2.90 9 1.83 2.26 2.82 10 1.81 2.23 2.76 20 1.72 2.09 2.53 30 1.70 2.04 2.46 40 1.68 2.02 2.42 60 1.67 2.00 2.39  1.64 1.96 2.33

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. 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  2.64 = (5.1, 10.4) Can you interpret this interval? Write down your answer. We will then discuss.

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. 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. 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. Practical work follows …