ACJC 2026 Hypothesis Testing Lecture Notes

1 17 HYPOTHESIS TESTING SYLLABUS • Concepts of null and alternative hypotheses, test statistic , critical region, critical value, level of significance and p-value • Understand the nature of a statistical test of hypothesis, the concepts and formulation of a null hypothesis ( 0H ) and an alternative hypothesis ( 1H ), the difference between 1-tail and 2-tail tests, and the concepts of significance level and test statistic; • Understand that the p-value is the smallest level of significance at which the null hypothesis can be rejected, calculate the p- value using a graphic calculator, and determine whether there is sufficient evidence to reject the null hypothesis • Test for a population mean based on • A sample from a normal population of known variance; • A large sample from any population • Formulate hypotheses and carry out a hypothesis test for a population mean in the following cases: • A sample from a normal population of known variance; • A large sample drawn from any population, using the Central Limit Theorem and an unbiased estimate of the population variance • 1-tail and 2-tail tests • Interpret the results of a hypothesis test in the context of the problem

ACJC 2025/26 H2 Mathematics (9758) 2 CONTENTS 1 Hypothesis Tests ............................................................................... 3 1.1 The Null and Alternative Hypotheses ..................................... 6 1.2 Level of Significance of a Test ............................................... 7 1.3 Test Statistic ............................................................................ 8 1.4 The p-value ........................................................................... 10 2 Z-Test .............................................................................................. 13 3 Critical Region and Critical Values ................................................ 17 4 Using the Result of Hypothesis Testing to Find Unknowns .......... 19

17 Hypothesis Testing 3 LECTURE 1 Lesson Outline • Introduction to hypothesis testing and how it is done • Understand the meaning of null and alternative hypotheses, 1 -tail and 2-tail tests, and significance level 1 HYPOTHESIS TESTS In real life, we are often required to make decisions and draw conclusions about an entire population based on data from a random sample. For example, we may be asked whether eating a supplement decreases the risk of cancer based on a study of 80 randomly selected people. To arrive at a conclusion, assumptions typically need to be made about the population. Such assumptions are called statistical hypotheses. They may or may not be true, and can be tested. Here are two other scenarios: Scenario 1: The Fair Coin We may hypothesise that a coin is fair . If we tossed th at coin 100 times and obtained 52 heads, is the hypothesis that the coin is fair still valid? What if 70 heads were obtained instead? More generally, u nder what conditions would we reject the hypothesis that the coin is fair? Scenario 2: Lifespan of Batteries A manufacturer may claim that the mean lifespan of their batteries for pocket calculations is 600 hours. An independent party test s the manufacturer’s claim using a sample of 50 batteries. If the mean lifespan of the 50 batteries tested is 585 hours, how confident is the third party to conclude that the manufacturer overstated their battery lifespan?

ACJC 2025/26 H2 Mathematics (9758) 4 Example 1 We use a fictional (but familiar) story to illustrate hypothesis testing. We state th e two possibilities as t wo statistical hypothese s, the null hypothesis against the alternative hypothesis. Denote the population mean prelim score by  . Now, we set up a model on the assumption that H0 is true to calculate the probability that a sample mean of 62 or more could arise by chance. We let X be the random variable denoting the “H2 math prelim score of a randomly chosen student”. Under the assumption that H0 is true, the mean prelim score is normally distributed with mean 60 and variance 100. That is, ( ) 2N 60,10X  . For a random sample of 50 students, the sampling distribution is 210N 60, 50X    . Using the GC, ( )P 62 0.0786X = (to 3 s.f.). The null hypothesis, denoted by H0 , is that the mean prelim score of students in 2024 is still 60, i.e. the special programme may not have an effect on prelim scores. The alternative hypothesis, denoted by H1 , is that the mean prelim score of students in 202 4 is more than 60 , i.e. t he programme may have an effect on improving prelim scores. We write 0H :  = 60 The past decade showed that H2 math prelim scores are normally distributed with mean 60 and variance 100. In 2024, JC2s underwent a special programme in the hope of improving their scores For 2024’s prelim, the mean score of a sample of 50 students is found to be 62. While the sample mean prelim score in 2024 is higher at 62, did this arise by chance due to random sampling or… …is there sufficient evidence that students in 2024, who underwent the special programme, had a higher mean prelim score? We write 1H :  > 60 60 62

17 Hypothesis Testing 5 So, under the assumption that H0 is true , we have a probability of 0.0786 of seeing a sample mean score of at least 62, and we could either: • Not reject 0H as 0.0786 is large enough that we feel a sample mean of at least 62 could arise by chance (insufficient evidence to re ject 0H that the mean prelim score in 2024 is still 60); or • Reject 0H in favour of 1H as 0.0786 is too small that we feel that the sample mean of 62 is unlikely to arise by chance ( sufficient evidence to favour 1H that the mean prelim score in 2024 is more than 60). An equivalent way of interpreting the probability of 0.0786 is: • If 0H were actually true, but we decided to reject 0H whenever we see a sample mean score of at least 62, then we have a 7.86% chance of incorrectly rejecting 0H . So, we must set a threshold for when to reject 0H , i.e. when it is considered too improbable for the sample data to arise by chance. But, • The threshold cannot be too high, as we may incorrectly reject 0H too often; and • The threshold cannot be too low, as we may not reject 0H even if the sample data were improbable and unlikely to arise by chance. In practice, this threshold is decided before data is collected, and is called the level of significance of the test. The significance level is typically taken as 5% (or 0.05), i.e. 5% chance of incorrectly rejecting 0H given that 0H is true. In Example 1, we do not reject 0H at a 5% level of significance, as the probability of 0.0786 is more than 0.05 . S o, it is considered probable enough for a sample mean score of 62 to arise by chance. • Fields like medicine may use a lower level of significance at 1% (or 0.01) for it is vital to ensure that the effect seen is not due to chance; • Some fields like psychology use a higher level of significance of 10% (or 0.1) . In Example 1, we reject 0H at a 10% level of significance. This is because the probability of 0.0786 is less than 0.1, which is now considered sufficiently improbable to arise by chance. Determining what to choose as the level of significance of a hypothesis test depends on the problem context and subject discipline. In Example 1, would we reject H0 at a 1% level of significance?

ACJC 2025/26 H2 Mathematics (9758) 6 1.1 The Null and Alternative Hypotheses In general, when conducting a hypothesis test, we first state two statistical hypotheses to decide between, namely: • The null hypothesis 0H , which states that the population mean is of the claimed value; and • The alternative hypothesis 1H , which is used to contradict 0H . In