7. Survey Sampling

Question 1

Confidence Interval

Definition

Answer 1

-a 100(1-α)% confidence interval for an unknown parameter θ is defined as the random interval (θ1^,θ2^) where θ1^=g1(X_) and θ2^=g2(X_) are statistics (random variables) such that:
P(θ1^ < θ < θ2^) = 1-α

Question 2

Confidence Interval

Notes

Answer 2

• confidence intervals are not unique since there are infinitely many choices for these random variables
• θ is the true parameter value so is not random whereas θ1^ and θ2^ are random variables
• the usual value taken for α is 0.05, this corresponds to a 95% confidence interval

Question 3

Confidence Interval

Interpretation

Answer 3

-if we have α=0.05, we have a 95% confidence interval for parameter θ -
if we do many samplings for each observation, a random sample x_, we construct (g1(x_),g2(x_)) we should expect to have the true value θ in this interval 95% of the time

Question 4

Z-Statistic

Definition

Answer 4

-a statistic with standard normal distribution:
√(n) (X_-μ) / σ ~ N(0,1)
-this result is given by the central limit theorem

Question 5

Z-Statistic

Purpose

Answer 5

-used to calculate the range of plausible values for μ assuming σ² is KNOWN

Question 6

When can the z-statistic be used?

Answer 6

-when the variance is known

Question 7

How do you find a confidence interval for the mean when the variance is known?

Answer 7

-the 100(1-α)% confidence interval is:

x_ - σ/√(n) * z_α/2 , x_ + σ/√(n) * z_α/2

Question 8

T-Statistic

Definition

Answer 8

-we know that if X1,X2,…,Xn are i.i.d. with N(μ,σ²) then:

T = √(n) (X_-μ) / S ~ t(n-1)

Question 9

T-Statistic

Purpose

Answer 9

-used to calculate the range of plausible values for μ assuming σ² is UNKNOWN

Question 10

How do you find a confidence interval for the mean when the variance is unknown?

Answer 10

-the 100(1-α)% confidence interval is:

X_ - t_α/2/√(n) * S, X_ + t_α/2/√(n) *S

Question 11

Two Sample Problems

Description

Answer 11

-consider two populations N(μ1,σ1²) and N(μ2,σ2²) with two independent random samples, thus:
Var|X1-X2| = σ1²/n1 + σ2²/n2
-we are interested in comparing μ1 and μ2

Question 12

Two Sample Problems

Known Variances

Answer 12

-if σ1² and σ2² are known then a 100(1-α)% confidence interval for μ1-μ2 is:
(X1-X2) z_(1-α/2) *√[σ1²/n1 + σ2²/n2]
-where P(Z

Question 13

Two Sample Problems

Unknown Variance

Answer 13

-if σ1² and σ2² are unknown then a 100(1-α)% confidence interval for μ1-μ2 is:
(X1-X2) t_(1-α/2,n1+n1-2) *Sp√[1/n1 + 1/n2]
-where Sp²= [(n1-1)S1² + (n2-1)S2²] / [n1+n2-2] is the pooled variance