Statistics

Question 1

notation

Answer 1

Xi - random var with distribution N(a, b^2)

xi - observed value

Question 2

sample mean for observation. x1, x2, .., xn

Answer 2

x’ = 1/n* sum from 1 to n of xi

Question 3

sample var for observations

Answer 3

s^2 = 1/(n-1) * sum (xi - x’)^2 = 1/(n-1) * sum(x1^2) - n*x’^2

Question 4

sample correlation of the observations

Answer 4

r = sum((xi-x’)(yi-y’))/sqrt(sum((Xî-x’))^2sum((yi-y’)^2)) - r linear correlation

Question 5

what is the statistic, estimator and estimate

Answer 5

statistic - random variable giving the formula for the function of the data
λ’ = 1/X’ - estimator
λ’ = 1/x’ - estimate
estimator is random var and has distribution

Question 6

methods of estimation

Answer 6

least squares estimation is useful because it does not make any assumption about the distribution of the data, only about mean
maximum likelihood estimation does require a particular distribution form for the data

Question 7

properties of estimators

Answer 7

bias = E[Φ’ - Φ]
Φ’ - estimator of parameter Φ
mean squared error = E[(Φ’ - Φ)^2]

E[(Φ’-Φ)^2] = Var(Φ’) if Φ’ is unbiased
E[(Φ’ - Φ)^2] = Var(Φ’) + bias^2 - if Φ’ is biased

Question 8

confidence level

Answer 8

expressed as percentage with common values 95%, 99% or 90%, refers to idea that it gives an interval which includes the parameter with the specified probability
α - level associated with confidence level
c - confidence level
α = 1 - c

Question 9

Cl for mean of normal data σ^2 known and X1, X2, …, Xn are random sample from a normal distribution N(a, σ^2)

Answer 9

a 100(1- α)% confidence interval for a is based symmetrically around X’ as:
(X’ - z * σ/sqrt(n), X’ + z * σ/sqrt(n)
where
z’s index is 1 - α/2 and z solves Φ(z) = 1 - α/2 and is evaluated using Matlab quantile function norming(1 - α/2, 0, 1)

Question 10

cl for mean of normal data σ^2 unknown and t distribution with n-1 degrees of freedom

Answer 10

t-distribution similar to normal, you can calculate t quantiles in Matlab using tin(p, v), where p - probability 1 - α/2 and v is degrees of freedom