Statistics - Confidence Intervals Flashcards
what is the idea of a confidence interval
an interval of error around the approximation
what is ε
the error
what would be the confidence interval for the population mean
xbar - ε < μ < xbar + ε
what is the general expression for the 100(1- α)% confidence interval
P(Xbar - ε< μ <Xbar + ε) = 1-α
what is considered a small sample
n<=30
what is considered a large sample
n>=30
for small samples with sample mean X and sample standard deviation s what is the 100(1-α)% confidence interval for u
(xbar - tn-1;α/2s/sqrt(n), xbar + tn-1;α/2s/sqrt(n)
if the sample standard deviation increases what happens to the confidence interval
increases
if the sample size increases what happens to the confidence interval
decreases
what is tn-1;α/2
the critical value for the student’s t distribution with n-1 degrees of freedom T~tn-1
what is the standard variable for the student’s t distribution
T=Xbar-μ/s/sqrt(n)
how do you express P(μ - ε < Xbar < μ + ε) = 1 – α using the standard variable T
P(− 𝜖sqrt(n)/s < T < 𝜖sqrt(n)/s)
why isn’t T normally distributed
because s/sqrt(n) is based on the sample standard deviation s and not σ
what does T~tn-1 mean
T obeys the student t distribution with v=n-1 degrees of freedom
what happens to the student t distribution when the degrees of freedom is very large
tv tends to the standard normal N(0,1)
what are the critical values of the standard normal distribution
P(Z>zα)=α
what are the critical values of the student t distribution with v degrees of freedom
P(T>tv;α)=α
when can we use the critical value of the standard normal
for large degrees of freedom
what happens to the critical value tv;α above v=31
it becomes almost constant
if there are n independent r.v’s distributed X~N(μ, σ^2) with unknown μ and σ^2 and the sample is large what is the 100(1 - α)% confidence interval for population mean μ
(xbar - zα/2s/sqrt(n), xbar + zα/2s/sqrt(n))
