Statistics - Confidence Intervals

Question 1

what is the idea of a confidence interval

Answer 1

an interval of error around the approximation

Question 2

what is ε

Answer 2

the error

Question 3

what would be the confidence interval for the population mean

Answer 3

xbar - ε < μ < xbar + ε

Question 4

what is the general expression for the 100(1- α)% confidence interval

Answer 4

P(Xbar - ε< μ <Xbar + ε) = 1-α

Question 5

what is considered a small sample

Answer 5

n<=30

Question 6

what is considered a large sample

Answer 6

n>=30

Question 7

for small samples with sample mean X and sample standard deviation s what is the 100(1-α)% confidence interval for u

Answer 7

(xbar - tn-1;α/2s/sqrt(n), xbar + tn-1;α/2s/sqrt(n)

Question 8

if the sample standard deviation increases what happens to the confidence interval

Answer 8

increases

Question 9

if the sample size increases what happens to the confidence interval

Answer 9

decreases

Question 10

what is tn-1;α/2

Answer 10

the critical value for the student’s t distribution with n-1 degrees of freedom T~tn-1

Question 11

what is the standard variable for the student’s t distribution

Answer 11

T=Xbar-μ/s/sqrt(n)

Question 12

how do you express P(μ - ε < Xbar < μ + ε) = 1 – α using the standard variable T

Answer 12

P(− 𝜖sqrt(n)/s < T < 𝜖sqrt(n)/s)

Question 13

why isn’t T normally distributed

Answer 13

because s/sqrt(n) is based on the sample standard deviation s and not σ

Question 14

what does T~tn-1 mean

Answer 14

T obeys the student t distribution with v=n-1 degrees of freedom

Question 15

what happens to the student t distribution when the degrees of freedom is very large

Answer 15

tv tends to the standard normal N(0,1)

Question 16

what are the critical values of the standard normal distribution

Answer 16

P(Z>zα)=α

Question 17

what are the critical values of the student t distribution with v degrees of freedom

Answer 17

P(T>tv;α)=α

Question 18

when can we use the critical value of the standard normal

Answer 18

for large degrees of freedom

Question 19

what happens to the critical value tv;α above v=31

Answer 19

it becomes almost constant

Question 20

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

Answer 20

(xbar - zα/2s/sqrt(n), xbar + zα/2s/sqrt(n))