Confidence Intervals Dr. Amjad El-Shanti MD, PMH,Dr PH University of Palestine 2016

Confidence Interval We know that when we want to study a population, we select a random sample from this population and the importance of the sample is by how much it reflect information about the population. We studied the sample mean and the standard error (SE) for each sample, and we want to study how much these two values tell us about the likely values of the population mean which is unusually unknown.

Statistical estimation Population Random sample Parameters Statistics Every member of the population has the same chance of being selected in the sample estimation

Large sample Case: If we select different samples each one would give a different estimate, the difference being due to sampling variation. Imagine collecting many independent samples of the same size and calculating the sample mean of each one. A frequency distribution of these means could then be formed and it is usually normally distributed with mean which is usually the population mean and standard deviation would equal to б/ √n where б is unknown we use the estimated value sd/√n, sd = the standard deviation of the sample. This value as we said before it is the standard error of the sample mean and it measures how precisely the population means estimated by the sample mean. The larger the sample. The smaller the standard error.

Large sample Case: The interpretation of the standard error of a sample mean is similar to that of the standard deviation. Approximately 95% of the sample means obtained by repeated samplin would lie within 1.96 standard error above or below the population mean.

Large sample Case: As there is a 95% probability that the sample mean lies within 1.96 standard error above or below the population mean, there is a 95% probability that the interval between X-1.96(sd/√n) and X+1.96(sd/√n) contains the population mean. The interval from X-1.96(sd/√n) to X+1.96(sd/√n) represents likely values of the population mean and it is called the 95% confidence interval for the population mean and X-1.96(sd/√n) and X+1.96(sd/√n) are the upper and lower 95% confidence limits foe the population mean.  For Large Sample Size, The Size Confidence Interval is: C. I. = X (sd/ √n) The confidence intervals for percentages other than 95% are calculated in the same way using the appropriate percentage point Z of the standard normal distribution in place of 1.96 for example, the 99% confidence interval is C. I. = X (sd/ √n)

Interval estimation Confidence interval (CI) % 14% 2% z

Interval estimation Confidence interval (CI), interpretation and example  x= 41.0, SD= 8.7, SEM=0.46, 95% CI (40.0, 42), 99%CI (39.7, 42.1)

Statistical estimation Estimate Point estimate Interval estimate sample mean sample proportion confidence interval for mean confidence interval for proportion Point estimate is always within the interval estimate

Small Sample Size: For the previous calculation we always assumed a large sample size (n>25), If sample size is small, we have two problems: 1.The sample standard deviation (sd) is subject to sampling variation, may not be reliable to estimate б. 2.When the distribution in the population is not normal, the distribution of sample mean also not normal. – The second problem can solve by central limit theorem which says that whether the variable is normal or not normal the sample mean will tend to be normally distributed. But because of first problem we can’t use the normal distribution, instead we will use a distribution called the t-distribution and this is valid only when the population is normally distributed. If the population is not normally distributed we either use a transformation or nonparametric confidence interval.

Confidence Interval Using t-distribution: (X- μ)/ (sd/√n) is a t-distribution with (n-1) degree of freedom. The shape of t-distribution is a symmetrical bell shaped distribution with mean of zero but it is more spread out having longer tails.  For small sample size the confidence interval is: C.I.= X+t * (sd/√n) where t can be found from the table if we know the percentage and the degree of freedom. As the sample size increase the t- distribution approach the normal distribution.

Confidence Interval Example 1: As part of a malaria control program, it was planned to spray all houses in a rural area with insecticide and it was necessary to estimate the amount that would be required. Since it was not feasible to measure all houses a random sample of 100 houses was chosen and the spray able surface of each of these was measured. The mean spray able surface area of these 100 houses was 23.3 m 2 and the standard deviation was 5.9 m 2. It is unlikely that the mean surface area of all houses (μ). Its precession is measured by the standard error (б/√n) estimated by: s/√n = 5.9/ √100 = 0.6 m 2 - There is a 95% probability that the sample mean of 23.3 m 2 differs from the population mean by less than 1.96 s.e = 1.96 * 0.6= 1.2 m 2 The 95% confidence interval is : X * s/√n = = 22.1 to 24.5 m 2 We say that with confidence interval, the mean of the population house area be between 22.1 and 24.5 m 2

Confidence Interval Example 2: The following are the numbers of hours of relief obtained by six arthritic patients after receiving new drug: 2.2, 2.4, 4.9, 2.5, 3.7, 4.3 hours X = = S= 1.13 hours, n=6 d.f =n-1 = 6-1 =5 s.e= s/√n= 0.46 hours The 5 % point of the t distribution with 5 degree of freedom is 2.57 (from the table), and so the 95% confidence interval for the average number of hours of relief for arthritic patient in general is * 0.046= = 2.1 to 4.5 hours.