5. Confidence Intervals

Question 1

Confidence Intervals

Description

Answer 1

• confidence intervals provide parameter estimates where a range of values is provided instead of a single number
• they serve a similar purpose to estimators but instead of just one ‘plausible’ value of the parameter they determine a range of possible values
• this way the true parameter value falls inside the range with high probability

Question 2

Confidence Interval

Definition

Answer 2

-let X1,…,Xn be generated from a model with an unknown parameter θ∈R
-consider an interval [U,V] where U=U(X1,…,Xn) and V=V(X1,…,Xn) are statistics
-the interval [U,V] is a confidence interval with confidence level α if:
P(θ∈[U,V]) ≥ 1-α
-for every possible value of θ

Question 3

Confidence Interval Observations

Significance Level

Answer 3

• a confidence interval with confidence level 1-α is sometimes called a (1-α)-confidence interval
• as for tests, a typical value of α is 5% corresponding to 95%-confidence intervals

Question 4

Confidence Interval Observations

θ

Answer 4

• the symbol θ in the definition of the confidence interval stands for a generic parameter
• this could be the mean μ, a proportion p, etc.

Question 5

Confidence Interval Observations

Randomness

Answer 5

-in the equation:
P(θ∈[U,V]) ≥ 1-α
-the interval [U,V] is random since it depends on the random sample X1,…,Xn
-but the value θ is not

Question 6

Confidence Interval Observations

Usefullness

Answer 6

-the usefulness of confidence intervals lies in the fact that the condition:
P(θ∈[U,V]) ≥ 1-α
-holds true for all values of θ simultaneously
-thus without even knowing the true value of θ we can be certain that the condition holds
-and for given data x1,…,xn, we can use [U(x1,..,xn),V(x1,..,xn)] as an interval estimate for θ

Question 7

Confidence Intervals for the Mean

Normally Distributed Data, Known Variance Lemma

Answer 7

-let X1,…,Xn~N(μ,σ²) with known variance σ²
-define the interval:
[U,V] = [X~-(q_α/2σ/√n) , X~+(q_α/2σ/√n)]
-where α∈(0,1) and q_α/2 is the (1-α/2)-quantile of the standard normal distribution
-then [U,V] is the (1-α)-confidence interval for the mean μ
-X~ indicates X bar, the sample mean

Question 8

Confidence Intervals for the Mean

Normally Distributed Data, Known Variance Lemma Proof

Answer 8

-prove that:
P(μ<u>V) = α/2
-then the probability that μ falls outside of the interval [U,V] is α
-thus:
P(μ∈[U.V]) = 1 - α</u>

Question 9

Confidence Intervals for the Mean

Normally Distributed Data, Unknown Variance Lemma

Answer 9

-let X1,…,Xn~N(μ,σ²) where the variance σ² is unknown
-define:
[U,V] = [X~-t_n-1(α/2)σx^/√n ,X~+t_n-1(α/2)σx^/√n]
-where α∈(0,1), t_n-1(α/2) is the (1-α/2)-quantile of the t(n-1)-distribution and σx^ is the sample standard deviation of x1,…,xn
-then [U,V] is a (1-α)-confidence interval for μ

Question 10

Confidence Intervals for the Mean

Normally Distributed Data, Unknown Variance Lemma Proof

Answer 10

-similar to the proof for known variance, consider the two cases where the confidence interval can fail to cover the mean μ
-we end up with:
P(U≤μ≤V) = 1 - α

Question 11

Confidence Intervals for the Mean Observations

Width

Answer 11

• width of the interval is proportional to 1/√n

- i.e. if we use more observations to construct a confidence interval, the resulting interval will be narrower

Question 12

Confidence Intervals for the Mean Observations

Confidence Level

Answer 12

• the confidence level 1-α affects the width of the confidence interval
• the smaller α, i.e. the higher the confidence level, the wider the confidence interval gets
• while we can adjust α, there is a trade-off to be made here
• increasing α has the advantage of reducing the width of the confidence interval but it also increases the probability that the interval doesn’t contain the true value

Question 13

Confidence Intervals for the Mean Observations

Standard Deviation

Answer 13

• if the data is spread out, σ and σx^ will be large

- the resulting confidence interval has width proportional to σ or σx^ respectively

Question 14

Confidence Intervals for the Mean Observations

Variance

Answer 14

-all other things being equal, confidence intervals for unknown variance (in particular at small sample size) are wider than confidence intervals for known variance

Question 15

Confidence Intervals for a Proportion

Description

Answer 15

• assume that we have observed attribute data x1,…,xn∈{1,…,K} and we want to obtain a confidence interval for the proportion p of individuals in the population which have class x=1
• to formalise this estimation problem, we introduce a statistical model:
• consider random variables X1,…,Xn∈{1,…,K} i.i.d. with P(Xi=k)=pk for all i∈(1,…,n} and all k∈{1,…,K}
• we know that we can use the proportion of observed values in a class k as a point estimate for pk

Question 16

Confidence Intervals for a Proportion

Lemma

Answer 16

-let X1,…,Xn be i.i.d. with P(Xi=1)=p
-define:
p^ = |{i=1,…,n|Xi=1}| / n
-and:
[U,V] = [p^-q_α/2√(p^(1-p^))/√n , p^+q_α/2√(p^(1-p^))/√n]
-then, the limit as n->∞:
lim P(p∈[U,V]) = 1-α
-i.e. for large n, the interval [U,V] is a (1-α)-confidence interval for p

Question 17

Confidence Intervals for a Proportion

Lemma Proof

Answer 17

-let:
Y = 1/n Σ1{1}Xi
-be the number of samples with class 1
-then Y~B(n,p)
-normal approximation to the binomial:
Y~N(np,np(1-p))
-divide by n, showing:
p^ = 1/nY ~ N(p , p(1-p)/n)
-for large n
-the value for the variance in this case is unknown, but for large n, we can use the point estimator:
 σ^² = p^(1-p^)/n
-we can now sub into the formula for the confidence interval [U,V]