RI S5 Hypothesis Testing_Lecture Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 __________________________ Chapter S5: Hypothesis Testing Page 1 of 19 Chapter S5: Hypothesis Testing SYLLABUS INCLUDES  concepts of null hypothesis (0H ) and alternative hypotheses (1H), test statistic, critical region, critical value, level of significance and -valuep  formulation of hypotheses and testing for a population mean based on: - a sample from a normal population of known variance - a large sample from any population  1-tail and 2-tail tests  Interpretation of the results of a hypothesis test in the context of the problem PRE-REQUISITES  S3 Normal Distribution  S4 Sampling CONTENT 1 Terminology 1.1 Null and Alternative Hypotheses 1.2 Level of Significance 1.3 Test Statistic, Critical Value, Critical Region, p-value 1.4 Performing a Hypothesis Test 2 Hypothesis Tests on Population Mean 2.1 Using the z-test Appendix 1 Type I and Type II Errors Appendix 2 Performing the z-test using GC

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ __________________________ Chapter S5: Hypothesis Testing Page 2 of 19 INTRODUCTION Let’s look at the following scenario: A manufacturer claims that the light bulbs he produces have a mean lifespan of 600 hours, and a standard deviation of 60 hours. A retailer, having received numerous complaints from his customers, suspects that they do not last as long. He contacts the manufacturer and they decide to test a random sample of 50 bulbs. It turns out that the average lifespan of this sample is 586 hours. Does this constitute proof that the average lifespan of light bulbs is below 600 hours? Well, the manufacturer might argue that this is simply a chance occurrence; after all, we cannot expect the average lifespan of a sample to be exactly the same as the mean lifespan. How might the retailer present his case if he is to argue that this represents objective evidence that the mean lifespan is below 600 hours? This type of scenario occurs sufficiently commonly in various fields of study that statisticians have devised a procedure, called hypothesis testing, to deal with it. In this chapter we will describe how this procedure is carried out in the context of the testing of the population mean. 1 TERMINOLOGY 1.1 Null and Alternative Hypotheses The manufacturer’s stand, that the observed variation from his stated lifespan on the light bulbs is purely due to chance, is what we will call the “null hypothesis”. The retailer might postulate that the mean is actually lower, and his claim is known as the “alternative hypothesis”. For the “A” Levels, we perform hypothesis tests on the population mean  and compare the measured sample mean against a sta