Get a summary of H2 A Level Math Statistics here. In this post, we’ll go through the key concepts tested in each chapter of H2 A Level Math Statistics.
The topics tested in H2 A Level Math Statistics are:
Click on the chapter below to go to the topic directly.
- permutation and combination
- probability
- discrete random variable
- binomial distribution
- normal distribution
- sampling
- hypothesis testing
- regression and correlation
Permutation and combination
Permutation and combination is the first chapter of statistics. In this chapter, students are asked to find the number of possible ways for a scenario.
Common formulae used in this chapter are:
- Number of ways to arrange n objects = n!
- Number of ways to arrange n objects with r of them being repeated units: n!/r!
- Number of ways to arrange n objects in a circle = (n-1)!
- Number of ways to permutate r objects from a set of n objects = nPr
- Number of ways to choose r objects from a set of n objects without permutation = nCr
Probability
- Probability of event A = number of outcomes for event A / total number of outcomes
- P(AUB) = P(A) + P(B) – P(A∩B)
- P(A|B) = P(A∩B)/P(B)
- If A and B are independent, P(A∩B) = P(A) x P(B), P(A|B) = P(A)
- If A and B are mutually exclusive, P(A∩B) = 0
Discrete random variable
- Expected value of X = E(X) = ΣxP(X=x)
- variance of X = Var(X) = E(X2) -[E(x)]2
- E(aX± b) = aE(X) + b, where a and b are constants
- E(aX± b) = aE(X) + bE(Y), , where a and b are constants, X and Y are independent
- Var(aX± b) = a2Var(X), where a and b are constants
- Var(aX± bY) = a2Var(X)+b2Var(X) , where a and b are constants, X and Y are independent
- standard deviation = √(Var(X))
Binomial Distribution
Conditions for binomial distribution:
- n independent trials
- the probability of success for each trial is a constant, p
- only 2 outcomes
- If X takes binomial distribution, we write it as X~B(n,p)
- Our graphic calculator (Ti-84) is designed to find P(X=x) and P(X≤x)
- To find P(X=x), use binompdf
- The formula booklet also provides information on how to find P(X=x) manually.
- To find P(X≤x), use binomcdf
Normal Distribution
If X takes a normal distribution, we write it as X ~ N( μ, σ2)
μ: mean
σ: standard deviation
σ2: variance
To convert X to Z (Standard normal), we use this formula: (X -μ)/ σ
Standard normal is used when μ and/or σ are unknown
If X and Y are independent, with X ~ N( μ1, σ12) and Y ~ N( μ2, σ22)
aX + bY ~N(μ1 +μ2 , a2σ12 +b2σ22)
If x̄ represents mean of X out of n samples, and X ~ N( μ, σ2),
then x̄ ~ N( μ, σ2/ n)
If x̄ represents mean of X out of n samples, where n is large, if the distribution of X is unknown, or the distribution is NOT a normal distribution, we can still say x̄ ~ N( μ, σ2/ n), by Central Limit Theorem.
You’ll use these functions in your graphic calculator (Ti-84):
To find probability, use normcdf.
If you are given P(X<a) = 0.3, and asked to find a, then use invnorm.
Sampling
unbiased estimate of population mean = μ = Σx/ n
unbiased estimate of population variance = s2 , where the formulae for s2 can be obtained from your data booklet
relationship between sample variance (σ2) and unbiased estimate of population variance (s2): s2 = n/(n-1) x σ2
Hypothesis Testing
For the H2 A Level exams, only hypothesis for mean is tested. (Note: for those of you taking ACT, SAT 2, or AP, you’ll also need to know the hypothesis testing for proportion.)
The important part of this chapter is to be able to conduct a hypothesis testing using either the p- test or test statistics method.
When do you reject Ho:
- when p value is less than the level of significance
- when test statistics lies within the critical region
When using your graphic calculator (Ti-84) to conduct the p- test:
- Press [Stat] button, then go to Test, and select Z-test
- Press in the information you have and you’ll get p- value.
- Compare p- value with level of significance to determine whether you reject or accept Ho.
When using the test statistics method, it’s less direct than the p- value method.
- Find what is the test statistics, Z = (x̄ – μ)/(s/ √n) –> when you use this method (instead of the p value method), one of them is unknown i.e. x̄, μ, s, n
- Find critical region from level of significance using invnorm
- Look at the question, determine whether you want to reject Ho (meaning test statistics lie within critical region) or do not reject Ho (then test statistics lie outside the critical region)
For full details, check out our Statistics Course here.
Regression and Correlation
- In least square regression, we find the product moment correlation, r, and the constants a, and b using the graphic calculator. For Ti-84, go to [Stats], Calc, followed by LinReg(a+bx) to conduct linear regression
- If any of your data points are unknown, remember that (x̄, ȳ) lies on the least square linear regression line.
- For reliable estimate of a value from the regression line, |r| should be close to 1 AND we are doing an interpolation.
- Regression only shows relationship, it does not show causation. If r is positive and close to 1, it shows that there is a strong positive relationship between x and y. However, this does not mean that x causes (or results in ) y.
A Complete Course on H2 A Level Math Statistics
Want a complete course on H2 A Level Math Statistics?
Check out our course here, where we cover all the topics tested in statistics.
The course includes concepts, types of questions, and how to apply them, so that you’ll score for your exam.