Confidence Interval Module One of the key concepts of statistics enabling statisticians to make incredibly accurate predictions is called the Central Limit Theorem. The Central Limit Theorem is defined in this way: · For samples of a sufficiently large size, the real distribution of means is almost always approximately normal. · The distribution of means gets closer and closer to normal as the sample size gets larger and larger, regardless of what the original variable looks like (positively or negatively skewed). · In other words, the original variable does not have to be normally distributed. · This is because, if we as eccentric researchers, drew an almost infinite number of random samples from a single population (such as the student body of NMSU), the means calculated from the many samples of that population will be normally distributed and the mean calculated from all of those samples would be a very close approximation to the true population mean. It is this very characteristic that makes it possible for us, using sound probability based sampling techniques, to make highly accurate statements about characteristics of a population based upon the statistics calculated on a sample drawn from that population. · Furthermore, we can calculate a statistic known as the standard error of the mean (abbreviated s.e.) that describes the variability of the distribution of all possible sample means in the same way that we used the standard deviation to describe the variability of a single sample. We will use the standard error of the mean (s.e.) to calculate the statistic that is the topic of this module, the confidence interval. The formula that we use to calculate the standard error of the mean is: s.e. = s / √N – 1 where s = the standard deviation calculated from the sample; and N = the sample size. So the formula tells us that the standard error of the mean is equal to the standard deviation divided by the square root of the sample size minus 1. This is the preferred formula for practicing professionals as it accounts for errors that may be a function of the particular sample we have selected. THE CONFIDENCE INTERVAL (CI) The formula for the CI is a function of the sample size (N). For samples sizes ≥ 100, the formula for the CI is: CI = (the sample mean) + & - Z(s.e.). Let’s look at an example to see how this formula works. * Please use a pdf doc. “how to solve the problem”, I have provided for you under the “notes” link. Example 1 Suppose that we conducted interviews with 140 randomly selected individuals (N = 140) in a large metropolitan area. We assured these individuals that their answers would remain confidential, and we asked them about their law-breaking behavior. Among other questions the individuals were asked to self-report the number of times per month they exceeded the speed limit. One of the objectives of the study was to estimate (make an inference about) the average nu.