AS Statistics

which distribution can be approximated as another?

binomial as Poisson

when does this approximation work, and what is λ?

when p gets smaller, and n gets bigger and λ= E(x)/ the mean

what characterises a Poisson distribution?

the mean and variance are equal

how to calculate variance

mean of the squares-square of the mean

E(aX+b)=

aE(X)+b

Var(aX+b)=

a² Var(X)

4 conditions that must be met for binomial distribution to work

• 2 mutually exclusive outcomes

• fixed number of trials

• each event is independent of the one before

• probability of success is constant

4 conditions that must be met for Poisson distribution to work

• event occurs randomly in space/time

• events cannot occur simultaneously

• each event is independent of the one before

• the mean remains constant

if X and Y are Poisson distributions, how to calculate X+Y?

X+Y∼Po(sum of means)

how to calculate: P(x≥n) must be

P(x≤n-1) must be >(1-p)

two hypotheses needed for hypothesis testing

H₀, the null hypothesis and H₁, the alternative hypothesis

5 steps in a hypothesis testing question

• let variable X= ?

• state H₀ and H₁

• state distribution and find probability

• compare probability to significance level

• state conclusion in context

formula for Χ²

∑(Oᵢ²/Eᵢ) -N (where N is number of trials)

how to generally calculate degrees of freedom

number of cells (after combining) - number of constraints

what counts as a constraint

estimating a parameter or number of trials

what does ν represent

number of degrees of freedom

what does χ²₃ represent

the family of χ² distributions with 3 degrees of freedom

how to combine the table whilst using a Χ² test

no __**expected**__ value is allowed to be

how to estimate p for a binomial distribution

p= (number of successes)/(number of attempts) or

p= Σ(r x fᵣ)/(N x n)

how to estimate λ for a Poisson distribution

calculate it as if it were the mean of a frequency table

how to do a Χ² test on observed data

calculate p or λ, then use this to generate expected frequencies, then conduct the test as normal

how to calculate expected values for each box in a contingency table

Eᵢ= (row total x column total)/ grand total

how to calculate ν for a contingency table

ν = (no of rows-1)(no of columns-1)