Author: haroonkhan

In the manufacturing example of the previous section we found such a large value for the test statistic that we are quite confident that the null hypothesis should be rejected, whatever significance level we choose. In other cases, finding a suitable value of α can be tricky. Recall that the larger the value of α, the easier it…

When the null hypothesis is of the form H0 : μ = μ0, we consider a two-tail rejection region. In many problems, the null hypotheses has the form H0 : μ ≥ μ0 or H0 : μ ≤ μ0 are more appropriate. As one could expect, this leads to a rejection region consisting of one tail. Before illustrating the technicalities involved, it is useful to consider a practical example. Example 9.14 A firm…

The need for testing a hypothesis about an unknown parameter arises from many problems related to inferential statistics. There are general and powerful ways to build appropriate procedures for testing hypotheses, which we outline in Section 9.10. Since they do require some level of mathematical sophistication, we offer here an elementary treatment that is strongly linked…

From a qualitative perspective, the form of the confidence interval (9.10) suggests the following observations: The last statement is quite relevant, and is related to an important issue. So far, we have considered a given sample and we have built a confidence interval. However, sometimes we have to go the other way around: Given a…

A further point concerns the correct interpretation of the confidence level. Consider the 95% confidence interval we calculated in Example 9.8. We cannot say that the confidence interval (34.0893, 84.5107) contains the unknown expected value with probability 95%. What we can say is that if we repeat the sampling procedure many times, and we compute a confidence interval for each…

Consider the queueing system in Fig. 9.2. Customers arrive according to a random process. If there is a free server, service is started immediately; otherwise, the customer joins the queue and waits. Service time is random as well, and whenever a server completes a service, the first customer in the queue (if any) starts her service.…

The sample mean is a point estimator for the expected value, in the sense that it results in an estimate that is a single number. Since this estimator it is subject to some variance, it would be nice to have some measure of how much we can trust that single number. In other words, we would like to…

The typical estimator of variance is sample variance: This formula can be understood as a sample counterpart of the definition of variance: It is basically an average squared deviation with respect to sample mean. When doing calculations by hand, the following rearrangement can be useful: The sample standard deviation is just S, the square root of sample variance. We…

The sample mean is a well-known concept from descriptive statistics: If data come from a legitimate random sample, sample mean is a statistic. A natural use of sample mean is to estimate the expected value of the underlying random variable, that is unknown in practice. It is important to understand what we are doing: We are using a…

Inferential statistics relies on random samples. There are many ways to take a sample: The last point should be stressed: Quite often we assume that observed data have been generated by a process that we represent as a statistical model. If we want to use tools from probability and statistics, this is a necessary step. However, we…