ABSTRACT
At the end of this chapter, you should be able to:
• understand hypotheses • appreciate type I and type II errors • calculate type I and type II errors using binomial and Poisson approximations • appreciate significance tests for population means • determine hypotheses using significance testing • compare two sample means given a level of significance
Industrial applications of statistics are often concerned with making decisions about populations and population parameters. For example, decisions about which is the better of two processes or decisions aboutwhether to discontinue production on a particular machine because it is producing an economically unacceptable number of defective components are often based on deciding the mean or standard deviation of a population, calculated using sample data drawn from the population. In reaching these decisions, certain assumptions are made, which may or may not be true. A statistical hypothesis is an assumption about the distribution of a random variable, and is usually concernedwith statements about probability distributions of populations. For example, in order to decidewhether a dice is fair, that is, unbiased, a hypothesis can be made that a particular
number, say 5, should occur with a probability of one in six, since there are six numbers on a dice. Such a hypothesis is called a null hypothesis and is an initial statement. The symbol H0 is used to indicate a null hypothesis. Thus, if p is the probability of throwing a 5, then H0: p= 16 means, ‘the null hypothesis that the probability of throwing a 5 is 16 ’. Any hypothesis which differs from a given hypothesis is called an alternative hypothesis, and is indicated by the symbol H1. Thus, if after many trials, it is found that the dice is biased and that a 5 only occurs, on average, one in every seven throws, then several alternative hypotheses may be formulated. For example: H1: p= 17 or H1: p< 16 or H1: p> 18 or H1: p = 16 are all possible alternative hypotheses to the null hypothesis that p= 16 Hypotheses may also be used when comparisons are being made. If we wish to compare, say, the strength of two metals, a null hypothesis may be formulated that
there no of the two canwithstand are F1 and F2, then the null hypothesis is H0: F1=F2. If it is found that the null hypothesis has to be rejected, that is, that the strengths of the twometals are not the same, then the alternative hypotheses could be of several forms. For example, H1: F1>F2 or H1: F2>F1 or H1: F1 =F2. These are all alternative hypotheses to the original null hypothesis.
