Topic 6: Confidence intervals

Question 1

What are confidence intervals (3)

Answer 1

-Confidence intervals give a range of values for the true population parameter, together with a measure of confidence/likelihood that the range contains the true value
-For unknown population parameter θ, based on sample data we find two values a and b such that P(a<θ<b) = 1 - α
-We then say with 100(1-α)% confidence that θ lies in the range a to b

Question 2

How can we make a 95% confidence interval for the standard normal distribution (8,1)

Answer 2

-Take X~N(0, 1²)
-We know P(-1.96<Z<1.96) = 0.95
-X̄ = Σx/n ~ N(μ, σ²/n)
-Z = (X̄ - μ)/(√σ²/n) ~ N(0, 1²)
-P(-1.96 < (X̄ - μ)/(√σ²/n) < 1.96) = 0.95
-P(-1.96(√σ²/n) < X̄ - μ <1.96(√σ²/n)) = 0.95
-P(-1.96(√σ²/n) < X̄ - μ <1.96(√σ²/n)) = 0.95
-P(X̄+1.96(√σ²/n) > μ > X̄ - 1.96(√σ²/n) = 0.95

-μ ∈ [X̄ +- 1.96(√σ²/n)] with 95% confidence

Question 3

What is the formula for the confidence interval of a single mean with a normal distribution, unknown mean but known variance (2)

Answer 3

-μ ∈ [sample mean +- critical value x standard error of sample mean]
-μ ∈ [X̄ +- (Z^a/2)((√σ²/n))]

Question 4

What is the width of the confidence interval, and what does it depend on (1, 3)

Answer 4

-The width = the critical value x standard error of sample mean

Depends on:
-The amount of confidence (90, 95 or 99), a higher interval being wider
-Uncertainty in underlying distribution (σ), higher σ = wider
-Number of observations, higher n = less wide

Question 5

What is the formula for the confidence interval of a mean with a not normal distribution, unknown mean, known population variance, and n > 25 (1)

Answer 5

-μ ∈ [X̄ +- (Z^a/2)((√σ²/n))

Question 6

What is the formula for the confidence interval of a mean with a not normal distribution, unknown mean, unknown population variance, and n > 25 (1)

Answer 6

-μ ∈ [X̄ +- (Z^a/2)((√S²/n))

Question 7

What is the formula for the confidence interval of a mean with a normal distribution, unknown mean and unknown population variance (1)

Answer 7

-μ ∈ [X̄ +- (t^a/2_n-1)((√S²/n))

Question 8

What is the formula for the confidence interval of a mean with a bernoulli distribution, unknown mean and unknown population variance (1)

Answer 8

-μ ∈ [p +- (Z^a/2)(√(p(1-p)/n))]

Question 9

What is the formula for a difference in means confidence interval with normal distributions, unknown means and known population variance (1)

Answer 9

-μ₁ - μ₂ ∈ [(x̄₁ - x̄₂ ± (z_α/2)(√((σ²₁/n₁) + (σ²₂/n₂))]

Question 10

What is the formula for a difference in means confidence interval with not normal distributions, unknown means and known population variance, with n>25 (1)

Answer 10

-μ₁ - μ₂ ∈ [(x̄₁ - x̄₂ ± (z_α/2)(√((σ²₁/n₁) + (σ²₂/n₂))]

Question 11

What is the formula for a difference in means confidence interval with not normal distributions, unknown means and unknown population variance, with n>25 (1)

Answer 11

-μ₁ - μ₂ ∈ [(x̄₁ - x̄₂ ± (z_α/2)(√((S²₁/n₁) + (S²₂/n₂))]

Question 12

What is the formula for a difference in means confidence interval with bernoulli distributions with n>25 (1)

Answer 12

-μ₁ - μ₂ ∈ [(p₁ - p₂ ± (z_α/2)(√((p₁(1 - p₁)/n₁) + (p₂(1 - p₂)/n₂))]

Question 13

What is the formula for a difference in means confidence interval with normal distributions, unknown means and unknown but equal population variance (2)

Answer 13

-μ₁ - μ₂ ∈ [(x̄₁ - x̄₂ ± (t_{n1 + n2 - 2, α/2})(√((S²₀/n₁) + (S²₀/n₂))]
-s²₀ = ((n₁ - 1)s²₁ + (n₂ - 1)s²₂)/(n₁ + n₂ - 2)

Question 14

What is the formula for a difference in means confidence interval with normal distributions, unknown means and unknown but unequal population variance (2)

Answer 14

-μ₁ - μ₂ ∈ [(x̄₁ - x̄₂ ± (t_DoF, α/2</sub>)(√((S²₁/n₁) + (S²₂/n₂))]
- DoF = [(s²₁/n₁) + (s²₂/n₂)]²/(((s²₁/n₁)²/(n₁-1)) + ((s²₂/n₂)²/(n₂-1)))

Question 15

What is the expected value and variance of D = X̄₁ - X̄₂ for matched pairs (2)

Answer 15

-E(D) = µ₁ - µ₂ = µ_d
-V(D) = σ²₁ + σ²₂ - 2σ₁₂ = σ²_d

Question 16

How do we work out the mean difference and sample variance estimator for matched pairs (3)

Answer 16

-Define the difference as d₁ = x¹₁ - x¹₂, …, d_n = xⁿ₁ - xⁿ₂
-Calculate đ = ∑d_i/n
-Calculate s²_d = (∑d_i - đ)²/(n-1)

Question 17

What is the formula for a matched pairs difference in means confidence interval (1)

Answer 17

µ_d ∈ [đ ± t^α/2_n-1(√S²/n)

Question 18

What condition do we need to create a confidence interval for the variance of a distribution (1)

Answer 18

-X must be normally distributed

Question 19

How can we set up a confidence interval ()

Answer 19

S²