Rejection of the null hypothesis leads to the acceptance of the alternative hypothesis. The alternative hypothesis states that there is a difference between the procedures. It is denoted by H1.
Assuming that the upstream BOD and downstream BOD are normally distributed, we test using significance level of 0.05 whether BOD increases at the downstream. The significance level of a statistical hypothesis test is a fixed probability of rejecting the null hypothesis H0, when it is in fact true. It is called a type I error and is set by the investigator in relation to the consequences of such an error. We want to make the significance level as small as possible in order to protect the null hypothesis and to prevent, as far as possible, the investigator from inadvertently making false claims.
Test Statistic: It is the random variable X whose value is tested to arrive at a decision. The Central Limit Theorem states that for large sample sizes (n > 30) drawn randomly from a population, the distribution of the means of those samples will approximate normality, even when the data in the parent population are not distributed normally. A z statistic is usually used for large sample sizes (n > 30), but often large samples are not easy to obtain, in which case the t-distribution can be used. The population standard deviation s is estimated by the sample standard deviation,
The observed t value is calculated from the sample used. Testing of means can be accomplished when the data are in the form of paired observations. We compute for the confidence interval of d - u in the situation with paired observations is based on the random variable
Where and are random variables representing the sample mean and standard deviations of the differences of the observations in the experimental units. This two-sample problem is essentially reduced to a one-sample problem by using the computed difference d1, d2, d3 dn. Thus the hypothesis reduces to Ho: =do
The computed test statistic is then given by:
The Critical Region is the part of the sample space where the null hypothesis H0 is rejected. The size of this region is determined by the probability of the