The next topic I want to look at is the idea of hypotheses testing. And, here will we first look at some definitions and then test statistics and criteria, give some examples, and finally a few comments about some other tests. So, firstly the terminology and hypothesis testing. A statistical hypothesis is a claim about the value of single population characteristics or about the values of several population characteristics. Examples would be, the data come from a particular distribution. Say a normal distribution or a Gaussian distribution. Or, for example, the pipe diameter is equal to 85 millimeters. Or the pipe diameter is greater than 85 millimeters. Or less than 85 millimeters. Those would all be statistical hypotheses. The terminology is given in this extract, from the reference handbook, and first of all, we have the null hypothesis, denoted by H0. And the null hypothesis is a claim which is normally assumed to be true. For example, in a trial the null hypothesis might be that the accused is innocent would be the null hypothesis. And conversely the alternate hypotheses denoted by H1 is that the assertion is contradictory to the null hypotheses. In other words, we have to prove it. For example, the accused is guilty. So, those are shown in this extract here. So for example, the null hypothesis in this case is that the mean value is equal to some mean value Mu z. Mu 0, and the alternative hypothesis in this example, H1 is that the mean value is equal to some other value Mu 1. A test of a hypothesis is a method for using the sample data to decide whether the null hypothesis should be rejected or not. And we note that the answer is, itself, not absolutely certain. It has an associated confidence level with it. We have different types of error, type I error and type II error, which is shown in the extract here that alpha is the probability of a type I error, which is accepting that H0 is wrong when it's known to be true or a type II error, which is denoted by beta. Some typical tests and statistics and criteria, I should say, are given here. For various hypothesis. For example, the first hypothesis here, H0 is that the mean value is equal to some prescribed value, Mu0, and the alternative hypothesis is that Mu is not equal to that value. Another possibility, is that the null hypothesis, is that the mean value is equal Mu 0. And the alternative hypothesis, is that the mean value is less than that value, etcetera. These are, generally speaking, tested by computing this test statistic As shown here. Z0 = X- mu0 divided by the standard deviation divided by the square root of the number of samples and typical criteria for or rejection or acceptance that the magnitude of Z0 is greater than the magnitude of Z alpha over 2. For example, plus minus two standard deviations or 95% confidence limits, etcetera. So let me do an example on that and the example is that when it's working properly, a factory produces an average of 500 circuit boards a day with a standard deviation of 15. To test or determine if the plant is operating properly over some particular period, the production was measured for 20 consecutive days during which the average number of circuit boards produced was found to be 490. Which of the following statements is correct? There's at least a 5% probability that the plant is operating okay. There is at least a 95% probability that the plant is operating okay. There is at least a 5% probability that the plant is not operating okay. Or there is at least a 95% probability that the plant is not operating okay. Which of those statements is correct in this case? Well to find out, we compute our normalized variable Z which is x bar- mu divided by sigma, divided by square root of n. And in this case, we have x bar, the average number of circuit boards which was produced during this particular period is 490. The long term population mean is 500, so it's 490- 500 divided by the standard deviation of the population. Which is 15 divided by the square root of this particular sample size, which was 20 days. So computing that, we find that that is equal to -2.98. In other words, it's almost three standard deviations away from the mean. If we compare that with the values of confidence intervals in a previous table, we find that a value of 2.98 here is way bigger than the value we would expect for a 95% reliability, or confidence level, which would have been 1.96. The magnitude is much bigger than that. So therefore, the conclusion is that there is at least a 95% probability that the plant is not operating properly. In other words, the mean value is more than two standard deviations away from what we would have expected in this case if it was operating properly. Finally, a couple of other comments, these are further tables which are given in the Reference handbook, tests of other means which are other statistical hypotheses which can be used, table B and table C. In the first case where the variance is unknown, and in the second case, where the mean value is unknown. And, these can be employed in different types of problems. So this concludes my discussion of hypothesis testing.