How To Ace Your A Level H1 Mathematics Examinations (Section B)
Probability and Statistics is often lauded as most relevant to daily life among all the sub-disciplines of Mathematics. Indeed, newspapers, magazines, company reports, population white papers, and even advertisements feature statistics extensively. At Zenith, we believe strongly that the study of Mathematics is highly relevant to daily life. Our tutors always strive to draw connections between Mathematical concepts and real life. After all, Section B of the A Level H1 Mathematics Examination requires the application of concepts in various situations.
Many students at Zenith go on to pursue tertiary education and careers in fields such as Biology, Psychology, or even Data Science. These disciplines might require you to conduct field research, where the concepts that you learn when preparing for Section B of the A Level H1 Mathematics Examination will act as an important foundation for data collection. Otherwise, jobs in Marketing and Sales might also need you to, for instance, track the popularity of certain products, ad performance, and the demographics of consumers, which similarly require your knowledge of Statistics. As seen in Fig. 1 below, the A Level H1 Mathematics Syllabus does require you to apply the concepts you learn. This is why, in the larger scheme of things, it will benefit you to master Section B of the A Level H1 Mathematics syllabus! In this article, Zenith, Singapore’s top A Level Mathematics tuition center, shares with you the important things to take note of for Section A (Statistics and Probability) of the A Level H1 Mathematics Examinations.
Fig 1. Integration and Application in the A Level H1 Mathematics syllabus as provided by SEAB
First, let’s take a look at how the A Level H1 Mathematics syllabus is structured. The A Level H1 Mathematics syllabus consists of only one 3-hour exam paper. It is graded out of 100 marks, but it is nonetheless split into two different sections, just like the A Level H2 Mathematics paper. As shown in Fig 2., Section A (40 marks) covers questions on Pure Math while Section B (60 marks) covers questions on Probability and Statistics. This mark allocation means that a larger percentage of your grade at the A Level H1 Mathematics examinations is dependent on your grasp of concepts under Section B. It hence goes without saying that being confident in the topics covered under Statistics and Probability is imperative to your securing of that “A” grade! Check out our article on Section A: Pure Math here.
Fig 2. Breakdown of the A Level H1 Mathematics Examination paper
Here are the topics that are tested during the A Level H1 Mathematics examinations under Probability and Statistics in Section B:
Correlation and Linear Regression
Present your answers in the correct format, with the correct language.
The A Level H1 Mathematics Examination is not a test of your language flair. It requires the use of specific Mathematical language, presented in a universally-accepted Mathematical format. Let’s take the topic of Hypothesis Testing as an example. Many students might not present their answers correctly for questions on Hypothesis Testing during their A Level H1 Mathematics Examination if they are not careful.
Here are some key terms used in Hypothesis Testing:
The null hypothesis is denoted as H0 (pronounced as “H-naught”). The null hypothesis is assumed true until proven otherwise.
The alternative hypothesis is denoted as H1 (pronounced as “H-one”). The alternative hypothesis is not typically assumed to be true. However, when we conduct a hypothesis test, we are trying to find evidence to see if we can support this alternative hypothesis.
We test these definitions at a specific level of significance, which refers to the probability that we (wrongly) reject the null hypothesis when it is in fact true. It is, in other words, the risk of getting a wrong conclusion when rejecting the null hypothesis.
For the following example, let’s assume that the test is being done at the 5% level of significance and that the hypotheses are as follows:
Null Hypothesis (H0): All children grow to a height of 1.64m.
Alternate Hypothesis (H1): Children who eat chocolate can grow to a height taller than 1.64m
Now, let’s assume that you have completed testing your hypothesis, and you want to present your answer. This is how you should do it:
1. When you do not have sufficient evidence to prove that H1 is true:
Therefore, we do not reject H0 as there is insufficient evidence at the 5% level of significance to conclude that children who eat chocolate can grow to a height taller than 1.64m.
2. When you have sufficient evidence to prove that H1 is true:
Therefore, we reject H0 as there is sufficient evidence at the 5% level of significance to conclude that children who eat chocolate can grow to a height taller than 1.64m.
Note that the answers are always phrased in terms of the null hypothesis––we either “reject” or “do not reject” H0. We do not “accept” either of the hypotheses. This is as there is a degree of uncertainty that is associated with Hypothesis Testing. We can only safely and surely conclude that a hypothesis is wrong, we cannot confirm that it is definitely correct. Even in the case where we do not reject H0, we are only confirming that H1 cannot be true. As mentioned above, it is usually assumed that the status quo is true. However, in reality, further testing is typically done to confirm that the status quo is also definitely correct. Hypothesis Testing is generally adopted to check if an alternative hypothesis has a likeliness of being true.
Students have to be clear about the above, as they will be penalised and lose marks even if their calculations are right if the wrong terminology is used. This is as there are nuances within the Mathematical language, as explained above. Making mistakes suggests that you are unclear about the difference between rejecting H0 and accepting H1. The rejection of H0 does not translate automatically into the acceptance of H1.
As such, it is key that you use the appropriate Mathematical language when attempting questions at the A Level H1 Mathematics examinations. In particular, Hypothesis Testing, in comparison to all the topics under Probability and Statistics, requires more textual explanation and use of accurate Mathematical jargon.
Always remember the correlation coefficient.
The correlation coefficient indicates the strength of the linear relationship between two variables, for instance, x and y. Denoted by r, it is used for Binomial Theorem, Normal Distribution, Sampling as well as Correlation and Linear Regression. It is important because it is a measure of the interdependence between two variables. The range of r is between -1.0 and 1.0––for accuracy purposes, you have to present r to one decimal place (i.e. 1.0 instead of 1).
When r = 1.0, there is a perfectly positive correlation between the two variables. This means that, for example, for every unit increase in the value of variable x, there will be a corresponding unit increase in the value of variable y.
When r is between 0.5 and 0.9, there is a strong positive correlation between the two variables. This means that while the positive relationship between the two variables is not directly proportional, they have a significant impact on each other.
When r is between 0.1 and 0.4, there is a weak positive correlation between the two variables. This means that the positive relationship between the two variables is not directly proportional and they do not have a significant impact on each other.
When r = -1.0, there is a perfectly negative correlation between the two variables. This means that, for every unit increase in the value of variable x, there will be a corresponding unit decrease in the value of variable y.
When r is between -0.5 and -0.9, there is a strong negative correlation between the two variables. This means that while the negative relationship between the two variables is not directly proportional, they have a significant impact on each other.
When r is between -0.1 and -0.4, there is a weak negative correlation between the two variables. This means that the negative relationship between the two variables is not directly proportional and they do not have a significant impact on each other.
When r = 0, there is no relation between the two variables.
Questions might ask you to gauge the correlative relationship of two variables. It might also ask you to plot a graph based on the value of r. Therefore, it is important that you understand how the value of r works. It is commonly overlooked as Standard Deviation (SD) and the sample population or mean are looked upon as more major concepts in Statistics, however, questions on r might still appear for the A Level H1 Mathematics Examinations.
Take note of the Central Limit Theorem.
The Central Limit Theorem (CLT) accounts for how independent random variables, when summed up, typically tend toward a Normal Distribution. Theoretically, this happens despite how the variables are not normally distributed. CLT is extremely important because it allows us to approximate the distribution of a set of independent random variables to the trends of Normal Distribution. Thereafter, we can apply the rules and concepts of Normal Distribution to the analysis of that set, which enables us to draw more productive conclusions than if we had to factor in many different types of distributions. In other words, we are conducting data analysis on the “assumption” that a set is normally distributed.
This is how you should present your answer during the A Level H1 Mathematics Examination when “assuming” that a set of independent random variables are normally distributed:
Since n = 55 is large, by Central Limit Theorem, Y is approximately normally distributed.
Remember that any sample size (n) equivalent to or bigger than 30 is generally considered large enough for approximation to a normal distribution. You are not supposed to introduce the Central Limit Theorem for numbers smaller than 30 as the sample size is not large enough for approximation. In such cases, the questions typically specify that the set is already in a normal distribution.