Normal Computations Flashcards
What is a random sample?
A sequence of independent and identically distributed (iid) random variables.
What does iid mean?
Independent and identically distributed.
How is a normal distribution denoted?
N(μ, σ²) where μ is the mean and σ² is the variance.
What is the expected value of a random variable X?
The mean of its probability distribution, denoted E[X] = μ.
What is variance?
The expected squared deviation from the mean, denoted Var(X) = σ².
What is standard deviation?
The square root of the variance.
What does the shape of the normal distribution depend on?
Its mean (center) and variance (spread).
What happens to the normal curve if variance increases?
The curve stretches and flattens to keep area under the curve equal to 1.
What is the sample mean X̄ and its distribution?
A linear combination of iid normal variables with mean μ and variance σ²/n.
What is the standard normal distribution?
A normal distribution with mean 0 and variance 1.
What is the Z-score transformation?
(X - μ) / σ, standardizes a normal variable to standard normal.
How do you transform a standard normal Z to a normal variable X?
X = σZ + μ.
What is the CDF of the standard normal distribution?
The function Φ(z) = P(Z ≤ z).
How do you compute P(Z ≤ z) in R?
Use the command pnorm(z).
How do you compute a critical value zα in R?
Use the command qnorm(1 - α) to find the value that cuts off α area to the right.
What is the variance of the sample mean X̄?
σ² / n.
Why is the sample mean less variable than individual observations?
Because variability reduces when averaging over more data.
What is a critical value in hypothesis testing?
A number that cuts off a specified tail probability under the standard normal curve.
What is Zα?
The critical value such that P(Z > Zα) = α.
How is the standard deviation of X̄ written?
σ / √n.
