Essential Knowledge — All Topics
Statistics
\(\bar{x}=\dfrac{\sum x}{n}\)
\(\sigma^2=\dfrac{\sum(x-\bar{x})^2}{n}\)
\(r=\dfrac{S_{xy}}{S_xS_y}\)
\(\sigma^2=\dfrac{\sum(x-\bar{x})^2}{n}\)
\(r=\dfrac{S_{xy}}{S_xS_y}\)
Regression
\(y=b x+a\)
\(b=r\cdot\dfrac{S_y}{S_x}\)
\(a=\bar{y}-b\bar{x}\)
\(b=r\cdot\dfrac{S_y}{S_x}\)
\(a=\bar{y}-b\bar{x}\)
Probability
\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
Bayes: \(P(A|B)=\dfrac{P(B|A)P(A)}{P(B)}\)
Bayes: \(P(A|B)=\dfrac{P(B|A)P(A)}{P(B)}\)
Distributions
\(X\sim N(\mu,\sigma^2)\)
\(X\sim B(n,p)\)
\(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\)
\(X\sim B(n,p)\)
\(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\)
Calculus
\(\dfrac{d}{dx}[x^n]=nx^{n-1}\)
\(\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\)
\(\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\)
Growth Models
Exp: \(N=N_0 e^{kt}\)
Logistic: \(P=\dfrac{L}{1+Ae^{-kt}}\)
Logistic: \(P=\dfrac{L}{1+Ae^{-kt}}\)
Finance
\(FV=PV(1+r)^n\)
Annuity: \(R=\dfrac{PVr(1+r)^n}{(1+r)^n-1}\)
Annuity: \(R=\dfrac{PVr(1+r)^n}{(1+r)^n-1}\)
Geometry
Arc: \(l=r\theta\)
Sector: \(A=\tfrac{1}{2}r^2\theta\)
(\(\theta\) in radians)
Sector: \(A=\tfrac{1}{2}r^2\theta\)
(\(\theta\) in radians)
🔑 Chi-squared: degrees of freedom for an \(m\times n\) contingency table \(= (m-1)(n-1)\). Reject \(H_0\) if \(\chi^2_{\text{calc}}>\chi^2_{\text{crit}}\).
EGWorked Example — Normal Distribution
The masses of apples are normally distributed with mean 180 g and standard deviation 12 g. Find the probability that a randomly chosen apple has mass less than 195 g.
Solution:
Standardise: \(z=\dfrac{195-180}{12}=1.25\)
\(P(X<195)=\Phi(1.25)=0.8944\)
Answer: 0.8944 (4 s.f.)
Standardise: \(z=\dfrac{195-180}{12}=1.25\)
\(P(X<195)=\Phi(1.25)=0.8944\)
Answer: 0.8944 (4 s.f.)
Practice Questions
Questions 1 – 20Your Score
—
—
Correct
—
Incorrect
—
Time Used
Full Solutions & Explanations
Answers and worked solutions for all 20 questions