Topic 6 Confidence Intervals Flashcards
What does the Law of Large Numbers (LLN) state?
The average of a large number of i.i.d. random variables converges to the expected value.
True/False
The probability that the sample mean deviates significantly from the population mean increases with sample size.
False
What does the Central Limit Theorem state?
The normalized sum of i.i.d. variables converges in distribution to a standard normal distribution.
What adjustment should be made for discrete variables in CLT approximation?
a) Add standard deviation
b) Use z²
c) Apply a continuity correction
d) None
c) Apply a continuity correction
Confidence intervals allow us to quantify the ________ in our estimation.
uncertainty
What does a 95% confidence interval mean?
That 95% of such intervals will contain the true value of the parameter.
What is the two-sided CI formula?
P(C_l ≤ X ≤ C_u) = 1 − α
What is the one-side CI formula?
P(X ≤ C_u) = 1 − α or P(C_l ≤ X) = 1 − α
What is a sampling distribution?
The distribution of a sample statistic across repeated samples.
True/False
The sampling distribution must be known to construct a confidence interval.
True
What is the formula for the CI for the mean for sigma known?
P(X̄ − z_{α/2} * (σ / √n) < μ < X̄ + z_{α/2} * (σ / √n)) = 1 − α
Why can’t we use Z when σ is unknown?
Because replacing σ with sample standard deviation introduces extra variability.
What is the formula for uniform normal distribution?
Zₙ = (∑Xᵢ − nμ) / √(nσ²) = (X̄ − μ) / (σ / √n)
What is the formula for
E(X̄)
Var(X̄)
E(X), Var(X) / n
Formula using student to find the mean?
P(X̄ − t_{n−1, α/2} * (Sₓ / √n) < μ < X̄ + t_{n−1, α/2} * (Sₓ / √n)) = 1 − α
What is the chi-squared distribution?
The sum of squares of n standard normal variables, used to model variance.
What is the mean and variance using Chi-squared?
E[Y] = n, Var(Y) = 2n
How is the Chi-squared distribution denoted?
Y = ∑ from i=1 to n of Xᵢ² ∼ χ²(n)
What is the mean and variance of the t-distribution?
E[X] = 0, Var(X) = n / (n − 2) for n > 2
CI formula for variant using Chi-squared?
V = ((n − 1) * S²) / σ² ∼ χ²(n − 1)
P(((n − 1) * S²) / χ²{n−1, α/2} < σ² < ((n − 1) * S²) / χ²{n−1, 1−α/2}) = 1 − α
When σ is unknown, we use the ________ distribution to estimate the confidence interval for the mean.
Student t
