Sampling, Estimation, Numerical Solutions to equations and Differential Equations CT

Question 1

Target population and sampling frame

Definition of each

Answer 1

Population being investigated and for what
Who is being investigated

Question 2

Rows and columns

Answer 2

Rows (horizontal)
Columns (veritcal)

Rows are horizontal like how rowers hold their paddles horizontally

Question 3

Central limit theorem

Distributions of the total

Answer 3

T = X1 + X2 + … + Xn
E(T) = nµ
Var(T) = nσ²
Sd(T) = √n σ

X1 + X2 ≠ 2X , sum of 2 random variables which can have different values, 2X is the outcome of one trial multiplied by 2

Question 4

Distribution of the sample mean x̄

Answer 4

E(x̄) = µ
Var(x̄) = σ²/n

if n>30

Question 5

Types of sampling (5)

Answer 5

1. Small sample from a homogenous population
2. Simple random sampling (whole population available for survey)
3. Systematic sampling (pattern of choice to choose population that is sampled)
4. Cluster sampling (break down a large population into geographical or other groups)
5. Stratified sampling (seperating a population into mutually exclusive sub-groups [weighting] eg. yr 9, yr 10, yr 11)

Question 6

Unbiassed estimated of mean

Answer 6

given in formula sheet
1/n-1 Σ(x - x̄)²

further simplification of formula not provided in booklet

Question 7

Confidence interval

example

Answer 7

If the mean from a sample size n from a normally distributed population is x̄, it is fairly simple to show that in 95% of all cases, the population mean µ lies in the interval

( x̄ - 1.96(σ/√n) < µ < x̄ + 1.96(σ/√n) )

Question 8

Different levels of confidence

Answer 8

100(1 - α)%

φ(z) = 1 - 1/2(α)

Question 9

Confidence intervals for proportions

Answer 9

E(p̄) = p
Var(p̄) = pq/n
Sd(p̄) = √pq/n

for binomial np>5 and nq>5

Question 10

Sign change rule

Answer 10

the real root or solution for an equation lies between a sign change

Question 11

Finding roots by iteration

Answer 11

1. Rearrange equation to have smallest modulus of the gradient
2. If gradient is negative the terms will be alternately above and below the root
3. If the gradient is positive the terms will approach the root steadily from one side

Question 12

Separating variables for differential equations

Answer 12

∫1/x dx = ln|x|+ c or ln|kx|

multiply dx on one side and divide by the y term on the other side

Question 13

Newton’s law of cooling

Answer 13

dT/dt = -k(T - T₀)

T is the temperature
t is the time
T₀ is the surrounding temperature