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 stated value, usually denoted as 0. In our introductory scenario, 0 600 and the null hypothesis, denoted as 0H , is stated formally as 0 .H 0: 60 The alternative hypothesis denoted as 1H , representing the retailer’s claim that the mean is less than 600 hours, is stated formally as 1 .H 0: 60 Thus we are testing 0H : 600 vs 1 .H 0: 60 In general, we can also test for an increase in the mean, or simply a difference of the mean from 0 , that is 0 0H :  vs 1 0H :  (test for a decrease), or 1 0H :  (test for an increase), or 1 0H :  (test for a difference) The tests for a decrease or an increase are known as 1-tail tests, while the test for a difference is a 2-tail test.

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ __________________________ Chapter S5: Hypothesis Testing Page 3 of 19 Example 1 For each of the following cases, formulate appropriate null and alternative hypotheses. (a) Flour is sold, on average, at 500 g per bag. Test whether the average mass of the contents of a bag is being understated. To test 0H : 500 vs 1H : 500 (b) The company Kookakola produces cans of aerated water. The company claims that the mean content of aerated water is 505 ml. Test whether it is overstating the mean content. To test 0H : 505 vs 1H : 505 (c) A machine produces components of mean length 10 cm if it is correctly set up. A random sample of components produced by the machine is taken, and the length of each component is measured. Test whether the machine is correctly set up. To test 0H : 10 vs 1H : 10 When carrying out a hypothesis test, we are looking for significant evidence to reject 0H in favour of 1H. But what constitutes significant evidence? 1.2 Level of Significance When testing hypotheses there is always a possibility of making a wrong decision or error. The threshold under which we consider 0H to be sufficiently unlikely to be correct is called the level of significance (or significance level) of the hypothesis test. The level of significance (or significance level) of a hypothesis test, denoted by , is defined as the probability of rejecting 0H when 0H is true. i.e. Level of significance = P(reject 0H when 0H is in fact true) = 0 0P(reject H H is true) In the context of our introductory scenario,  P(concluding that the mean lifespan of a light bulb is shorter than 600 hrs when it actually is 600 hrs). As such, it is tempting to choose  to be as small as possible, since the conclusion is a wrong one; so why not pick 0? Refer to Appendix 1 (Type I and Type II Errors) for more information. Appropriate values for  depends on which area of study we are engaged in and are usually chosen by consensus. For the social sciences, it might be chosen as high as 0.3, the biological and medical fields mostly use 0.05 (5% level of significance) or 0.01 (1% level of significance), while the physical sciences might use a value as small as 6.10  Note that the lower the level of significance (i.e. the smaller  is), the stronger the evidence needed to reject 0H .

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ __________________________ Chapter S5: Hypothesis Testing Page 4 of 19 1.3 Test Statistic, Critical Value, Critical Region, p-value Test Statistic To carry out the test, the focus moves from X, the lifespan of lightbulbs to the distribution of ,X the mean lifespan from a sample of lightbulbs. X is the test statistic for the population mean . A test statistic is a random variable used to make the decision “do not reject 0H ” or “reject 0H ”. The test statistic measures the degree of agreement between a sample of data and the null hypothesis. Its observed value changes randomly from one random sample to another. We will discuss in Section 2, how to make use of the test statistic, X , and its distribution, when performing a hypothesis test on the population mean, . In our example, 0H : 600 vs 1H : 600. Under the assumption that the null hypothesis is true, the population mean and standard deviation are 600 and 60 (these values are given by the manufacturer), so, by the Central Limit Theorem, we have 260~ N 600,50X     approximately, since 50n is large. Distribution of X (under 0H ) Critical Value, Critical Region The result of the test depends on the whereabouts of the test value in the sampling distribution, 586 hours (observed mean value of lifespan of a random sample of 50 bulbs). If the value is close to 600 hours then it is likely to have come from a distribution with mean 600 hours. On the other hand, if