SAJC 2021 H1 Chapter 6 Math Hypothesis Testing Notes Teacher V2

2021 SAJC JC1 H1 Maths Chapter 6 : Hypothesis Testing 1 Chapter 6 (Statistics) : Hypothesis Testing tatistic Pre-requisite Knowledge  Sampling Distribution of Means;  Point Estimates Content 6.1 Introduction to Hypothesis Testing 6.2 Terms and definitions 6.2.1 Null Hypothesis and Alternative Hypothesis 6.2.2 Test Statistic 6.2.3 Comparing z-score and critical value; or p-value and level of significance 6.2.4 Formal definitions of p-value and level of significance 6.3 Carrying out a Hypothesis Test (Examples) 6.3.1 Sample size large/small, Population normal, 2 known 6.3.2 Sample size large, Population non-normal, 2 known 6.3.3 Sample size large, Population non-normal, 2 unknown 6.4 Further Examples 6.5 Summary and Checklist Objectives: At the end of the chapter, you should have learnt about the following:  Concepts of null hypothesis (H 0) and alternative hypothesis (H1), test statistic, critical region, critical value, level of significance, and p-value  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-tailed and 2-tailed tests  Interpretation of the results of a hypothesis test in the context of the question

2021 SAJC JC1 H1 Maths Chapter 6 : Hypothesis Testing 2 6.1 Introduction to Hypothesis Testing Are Coke cans really 330 ml? Are th is type of dumbbells at least 10 kg? Will that slimming centre help customers lose up to 16 pounds in just 4 weeks as promised? Claims abound in advertising and merchandise, as well as elsewhere e.g. social sciences. The most objective way to test whether a claim is valid is to take a (random) sample and measure it. To be specific: In H1 Maths, we measure its sample mean ( x ) and calculate how much it deviates from an assumed population mean ( 0 , which could be based on known facts or some unverified claim). This deviation is expressed in terms of a relative score or probability, which we compare with a predetermined standard to reach a conclusion about whether the claim is valid. This whole procedure is known as hypothesis testing. A statistical hypothesis is a statement, which may or may not be true, concerning one or more population parameters (e.g. mean, variance, proportion). Hypothesis testing is the procedure used to test the validity of this statement based on evidence from random samples drawn from the population. Let us look at some concrete examples: Coke cans are 330ml. Dumbbells are at least 10kg each. We will help you lose up to 16 lbs.

2021 SAJC JC1 H1 Maths Chapter 6 : Hypothesis Testing 3 Example 1a A fizzy drink brand claims that the mean volume of its canned drinks (the volume is normally distributed with standard deviation 10ml) is 330ml. A consumer weighs a random carton of 24 drinks and found that it has a mean volume of 325ml. He suspects that the mean volume is less than what is claimed. Test the manufacturer’s claim at the 5% level of significance. Solution: Let X be the volume (in ml) of a canned drink with population mean volume  . 2~ N( ,10 )X  210~ N(330, )24X Test 0H : 330  vs. 1H : 330 at the 5% level of significance. Under 0H , XZ n    330 ~ N(0,1) 10 / 24 X  Option 1: Critical Value Method Option 2: p-value Method Using a one-tailed z-test, reject 0H if 1.64485z . Using GC, 325x  gives z-cal = –2.4495 < –1.64485  We reject 0H and conclude that at the 5% level of significance, there is sufficient evidence that the mean volume of a canned drink is less than 330ml. Using a one-tailed z-test, reject 0H if 0.05p . Using GC, 325x  gives p-value = 0.00715 < 0.05  We reject 0H and conclude that at the 5% level of significance, there is sufficient evidence that the mean volume of a canned drink is less than 330ml. Note:  P 325pX  P2 .4495Z  Press [Stats], choose [Z-Test] Choose [Stats], key in stats, select [Calculate] The output: Note to teachers: These examples are meant to reduce the abstractness. Teachers can use Section 6.2 to elucidate the theory and demonstrate the calculations involved.

2021 SAJC JC1 H1 Maths Chapter 6 : Hypothesis Testing 4 Example 1b (two-tailed test) A fizzy drink brand claims that the mean volume of its canned drinks (the volume is normally distributed with standard deviation 10ml) is 330ml. A random carton of 24 drinks has a mean volume of 325ml. Test the manufacturer’s claim at 5% level of significance. Solution: Let X be the volume (in ml) of a canned drink with population mean volume  . 2~ N( ,10 )X  210~ N(330, )24X Test 0H : 330  vs. 1H : 330  at the 5% level of significance. Under 0H , XZ n    330 ~ N(0,1) 10 / 24 X  Option 1: Critical Value Method Option 2: p-value Method Using a two-tailed z-test, reject 0H if 1.95996z or 1.95996z . Using GC, 325x  gives z-cal = –2.4495 < –1.95996  We reject 0H and conclude that at the 5% level of significance, there is sufficient evidence that the mean volume of a canned drink is not 330ml. Using a two-tailed z-test, reject 0H if 0.05p . Using GC, 325x  gives p-value = 0.0143 < 0.05  We reject 0H and conclude that at the 5% level of significance, there is sufficient evidence that the mean volume of a canned drink is not 330ml. Note: The implied message is that since the p-value is really low (  0.05), it is unlikely that the mean volume is indeed 330ml. Press [Stats], choose [Z-Test] Choose [Stats], key in stats, select [Calculate] The output:

2021 SAJC JC1 H1 Maths Chapter 6 : Hypothesis Testing 5 6.2 Terms & definitions 6.2.1 Null Hypothesis and Alternative Hypothesis Are we testing for (a) any change/difference from the (assumed) population mean 0 , or (b)/(c) a definite increase/decrease in the (assumed) population mean? Null hypothesis (denoted by 0H ) 0H is a statement that says that a population parameter takes a specific value. Alternative hypothesis (denoted by 1H ) 1H is a statement that says that the population parameter takes a value different, in some way, from the value given in 0H . Test 00H:  (Assumption: population mean is 0 ) vs. (a) 10H:  (b) 10H:  (c) 10H:  Note: 1. The rejection of the null hypothesis will generally imply that the alternative hypothesis is accepted. In this case, we conclude that the null hypothesis is false. 2. Failure to reject the null hypothesis is not the same as accepting the null hypothesis. It only means there is insufficient statistical evidence to reject 0H in favour of 1H (i.e. 0H may or may not be true). In other words, we cannot conclude that we accept H0. 6.2.2 Test Statistic The sample statistic used to test the hypotheses is called the test statistic. In the H1 Maths syllabus, we take the sample mean X and standardise it to obtain the standard normal distribution Z. In this way, any particular value x will have a corresponding standard score (also known as z-score), which is a measure of its number of standard deviations above or below  . Recall that if we use a lar ge sample size n, then X approximately follows a normal distributio n by Central Limit Theorem (CLT), so a bell curve appears! If we now assume our null hypothesis is true, then we get something familiar: Under 0H , 2 ~ N ,X n   and so ~ N(0,1) / XZ n    . Note: 1. If the population X is already normal, then Central Limit Theorem is not required. 2. If 2 is unknown, we use the unbiased estimate s2 2( in GC)xS instead. = sample variance1 n n  In a two-tailed test, 1H