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SADC Course in Statistics Further ideas concerning confidence intervals (Session 06)

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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 how the width of the confidence interval varies with changes in the sample size, underlying variability and level of confidence specified construct a confidence interval for a population proportion explain the idea of a confidence interval for any population parameter explain the conditions under which an approximate conf. interval may be found

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To put your footer here go to View > Header and Footer 3 Review of ideas of previous session Practical 5 demonstrated that the width of the 95% confidence interval depends on: (i) sample size the larger the sample the narrower the interval (ii) variation the greater the variability the wider the interval Both these point to the need to use a large enough sample size

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To put your footer here go to View > Header and Footer 4 Effect of other levels of confidence Why use a 95% confidence interval? What will happen if we find a 99% C.I.? Previously, a 95% C.I. for the true mean number of persons per room = (5.1, 10.4) We can also find a 99% C.I. as: 7.7 t 9 (s/n) = 7.7 3.25(3.7/10) = 7.7 3.8 = (3.9, 11.5) Notice that the interval is now wider! What are the implications of this?

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To put your footer here go to View > Header and Footer 5 Other confidence intervals There is no reason why other levels of confidence cannot be used It is up to the user to specify the degree of confidence required However, must recognise that higher the confidence, wider the interval Only change in the calculation is the t- value used It would be unusual however to have intervals with less than 90% confidence

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To put your footer here go to View > Header and Footer 6 Approximation to C.I.s A 95% C.I. is approximately given by estimate [ 2 x standard error ] This holds provided that: The sample size is relatively large (e.g. n > 30 or so) When the sampling distribution of the estimate follows an approximate normal distribution This applies to any estimate (not just the mean) for which the above holds.

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To put your footer here go to View > Header and Footer 7 Further notes Note what a confidence interval is NOT It does not tell you where the data values are – it concerns the estimate, not the data Ideas of estimation and confidence intervals apply to other population characteristics, the proportion of persons living below the poverty line the total number of malnourished children in a country The maximum daily temperature during the growing season

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To put your footer here go to View > Header and Footer 8 Possible complications Return to cattle numbers used in Practical 3. Here the first of 50 samples of size 10 gave mean=113.0, std.dev.=187.5 for the number of cattle per household. Hence a 95% confidence interval for the true mean number of cattle per HH is: 113 t 9 (s/n) = 113 2.26(187.5/10) = 113 134.0 = (-21.0, 247.0) What does the negative number mean?

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To put your footer here go to View > Header and Footer 9 Possible reasons for complications In the above example, a negative number is not meaningful. This can arise because The assumed sampling distribution for the estimate was incorrect The sample size was too small (n=10) The variability in data was too high In this example, all of the above were true as you will see during the practical that follows.

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To put your footer here go to View > Header and Footer 10 Dealing with complications One possibility is to consider transforming the data. Another is to use the correct distribution for the data being used in the computation of the confidence interval. For example, if the estimate was a proportion, a C.I. based on the normal distribution is very approximate. Calculating an exact C.I. for a proportion is beyond the scope of this module. However, we now discuss the approximate approach.

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To put your footer here go to View > Header and Footer 11 Estimates and C.I. for a proportion An example will be used for illustration. In Kilindi district, 404 HHs were sampled. Of these, 73 were found to be female headed. What is your estimate of the true proportion () of HHs in Kilindi that are female headed? Answer: = As before when discussing the estimation of the population mean, we need an estimate of precision…

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To put your footer here go to View > Header and Footer 12 Standard error of a proportion For a sample selected using simple random sampling, the standard error of a proportion is: where is the true value of the population parameter, and n=sample size. is however unknown, so we use an estimate of, to compute the above expression, giving std. error ( ) =

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To put your footer here go to View > Header and Footer 13 Approx. C.I. for a pop n proportion Assuming that the sample size is large enough (here 404 is considered large), we can use the Central Limit Theorem to say that the estimate follows an approximate normal distribution. Hence the 95% C.I. for true is given by = 0.181 ± 1.96(0.019) = (0.144, 0.218)

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To put your footer here go to View > Header and Footer 14 Practical work follows to ensure learning objectives are achieved…

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