The next topic I want to discuss is confidence levels, and here basically we'll just give some definitions and then some examples. So confidence levels in intervals relate to or express the degree of uncertainty associated with a single statistic. For example, confidence level is the percentage of all possible samples expected to include the true population parameter. For example, a 95% confidence level implies that 95% of the parameters fall within some specified range or intervals. So the confidence interval is a measure of the reliability, if you like, that, for example, the means and standard deviations of the population, when estimated from a particular sample. And the range then, is expected to contain the population property with some confidence level and usually we say that these are bounded by a lower confidence level, abbreviated as LCL, and an upper confidence level, abbreviated as UCL. And we get this for random variables from the Central Limit Theorem, which we previously had. And the Central Limit Theorem states that if we have many samples of N items which are taken from a normal distribution, which has a mean value of U and a variance M. Then the sample means themselves are normally distributed with a mean value equal to the mean value of the population as a whole and a standard deviation equal to the standard deviation of the population divided by the square root of the number of items taken in the samples. So therefore, we can say that the probability that a given average value say, X bar. Exceeds some confidence limit L is given by the probability that X is greater than L. Or the probability that X is greater than the normalized variable, L minus mu divided by sigma divided by square root of M. Which is the standard deviation of the sample means. The probability that the average value is between the lower confidence limit and the upper confidence limit is probability. X between LCL and UCL is given by this expression here. And the confidence limits for the mean of a normal distribution, if the standard deviation is known, is given by this expression, which is the top expression in the extract from the reference handbook here. And very commonly the most common level used is the 95% confidence level which occurs at 1.96 standard deviations from the mean value as shown here. So more generally the distribution is given in this table in the handbook Z alpha two. For example, 95% confidence level, which I just stated, occurs at 1.96 standard deviations from the mean, which is approximately here. So that contains 95% of the samples. And similarly, if we want to use the 80% confidence level, that occurs at 1.2 standard deviations or 99 percent level occurs at approximately 2.6 standard deviations, etc. For any arbitrary confidence level, we can just look up the value of Z alpha two in this table in the reference handbook. Now we can also use a confidence interval for other things. For example, the difference between two mean variables, whose variance is unknown which is given in the top equation here, or not known, the second equation here. We can also find a confidence interval for a variance, which is given by this expression. Or more useful possibly, we can get the sample size that we need to ensure that the confidence interval has some specified width. And this is given by this equation here which I've reproduced over here. So let's do an example on that. The response time for a ping to travel back and forth over an internet router Is normally distributed with a standard deviation of 25 milliseconds. But then, a new cable is installed with the same standard deviation. How many samples do we have to take to ensure that the 95% confidence interval for the new bing ping time has a width of not more that ten milliseconds. Which of these alternatives is it? So, in this case, the number of samples is given by that expression from the previous slide. So in this case we can simply calculate that from the table. We know that 95% occurs at two standard deviations from the mean. So its two times 1.96 times 25, the standard deviation of the entire population in the original cable, divided by the difference between the mean value and the previous mean value which is ten milliseconds. Computing that out the answer is 96.04. However, to be sure that gives us the true confidence interval we have to round this up to the next higher number, so the answer is 97, which is C. And this concludes my discussion of confidence levels and intervals.