AP®︎ Statistics: Topic 2.11 Flashcards

A variable that can take any value in an interval; probabilities are assigned to intervals, not single points

A curve representing a continuous probability distribution; probability is area under the curve and total area is 1

A continuous probability distribution with a unimodal, bell-shaped, symmetric density curve, determined by mean μ and standard deviation σ.

μ gives the center and σ gives the spread; larger σ makes the curve wider and shorter, smaller σ makes it narrower and taller

The normal distribution with mean 0 and standard deviation 1; written Z ~ N(0,1)

A standardized value showing how many standard deviations a raw value is above or below the mean: z = (x - μ)/σ.

Converting x to z with z = (x - μ)/σ; if X ~ N(μ, σ), then Z ~ N(0, 1)

Converting from z back to the original scale with x = μ + zσ.

For a normal random variable, the probability of an interval equals the area under the curve over that interval; total area is 1

A table of cumulative left-tail probabilities P(Z ≤ z) for the standard normal distribution

Technology command that gives the probability or area for an interval under a normal distribution: normalcdf(lower, upper, μ, σ)

Technology command that gives the cutoff value for a specified left-tail area in a normal distribution: invNorm(left-tail area, μ, σ)

The pth percentile is a value with about p% of the distribution at or below it

Percentile gives the proportion at or below a value; z-score gives signed distance from the mean in standard deviation units

A boundary x_a with P(X ≤ x_a) = p% ; the lowest p% of values lie at or below x_a

A boundary x_b with P(X ≥ x_b) = p% ; equivalently, P(X ≤ x_b) = 1 - p% in proportion form

An interval with P(x_a ≤ X ≤ x_b) = p%; the remaining area, 100% - p%, is split equally between the two tails

The combined p% in both tails of a normal distribution, split equally so P(X ≤ x_a) = (p/2)% and P(X ≥ x_b) = (p/2)%.

For a normal distribution, about 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3

P(X = a) = 0; probabilities for continuous variables come from intervals, not single points

For a continuous random variable, including or excluding an endpoint does not change the probability: P(X < a) = P(X ≤ a)