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MN1025 – Business Statistics 19 Lecture 5—Friday 8/2/2008 TESTING HYPOTHESES: TESTING THE MEAN Reference: Lind et al. , Chapters 9, 10. 5.1 Example: less than 99 defects? Last week we used the T statistic to derive con- fidence intervals for the unknown mean of a nor- mally distributed population. In this lecture we will use the same formalism to answer yes/no questions about the population mean. We are again concerned with a sample drawn from an (approximately) nor- mally distributed population. We start with an ex- ample. A manufacturer has a production line which has had 99 defects per day, on average. A new system is introduced and the manufacturer wants to know if there has been a significant decrease in the average number of defects. Hypothesis testing proceeds in five steps. 5.2 Step 1: Setting up the hypotheses In our example, the previous system had an average fault rate of 99 defects per day. The average fault rate of the new system, denoted by μ , is unknown. Our test will consider two hypotheses: The Null Hypothesis , H 0 , is that nothing has changed, i.e. that μ = 99 as previously. The Alternative Hypothesis , H 1 , is that there has been a decrease, i.e. that μ < 99 or, in words, the new population mean is less than the previous one. In some books, the alternative hypothesis is written as H A or H a . We have to choose between H 0 and H 1 . We will choose, or accept , H 0 (no change) unless there is sig- nificant evidence in favour of H 1 , in which case we say that we reject H 0 . We will see below what ex- actly is meant by this. Notice that we never test an hypothesis in isolation; we always need to know the alternative. 5.3 Step 2: Setting a level of significance The level of significance is the probability of reject- ing the null hypothesis when it is actually true (this is called a type I error ). Commonly used levels of significance are 10%, 5% and 1%. The level of sig- nificance must be set before the sample is taken. To choose, e.g., a 5% level of significance in Minitab’s 1-sample t test, click on Options and en- ter a 95% confidence level. 5.4 Step 3: Choosing a test statistic In this chapter, we use T as the test statistic (see lec- ture 4). We will encounter other useful test statistics later in the course. 5.5 Step 4: Setting a decision rule Assume we take a sample of size n from the produc- tion line and compute the sample mean, ¯ x , and the sample standard deviation, s . If the average number of defects for the new system, μ , is less than 99, we expect the sample mean to be also less than 99, i.e., ¯ x < 99. But of course, the sample mean could turn out to be less than 99 even if μ is equal to 99, i.e. if the population mean μ has not decreased. We should therefore reject the null hypothesis μ = 99 only if ¯ x is significantly smaller than 99. To make this idea precise, we assume that μ = 99 and look at the T statistic, which is in this case given by T = ¯ x μ s/ n = ¯ x 99 s/ n .

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