Statistics and Probability Review – Notes Flashcards (BSHS Sigma Club)

A set of practice flashcards covering key concepts from the lecture notes: random variables (discrete/continuous), probability distributions, mean/variance/standard deviation, normal distribution and z-scores, CLT, sampling distributions, confidence intervals, t-distribution, and sample size calculations.

What is a random variable?

A function that assigns a real number to each element in the sample space.

What are the two main types of random variables?

Discrete random variables and continuous random variables.

How do you determine the random variable for an experiment?

1) Determine the sample space; 2) Assign letters to outcomes; 3) Count the number of values of the random variable.

In a two-coin toss, if X represents the number of heads, what are the possible values of X?

0, 1, and 2.

What characterizes a discrete random variable?

A set of outcomes that are countable (digital).

What characterizes a continuous random variable?

Values on a continuous scale with infinitely many possible values.

What is a probability distribution for a discrete random variable?

Assign probabilities to each value of the variable; the sum of all probabilities is 1.

How do you construct the probability distribution for a discrete X?

Determine the sample space, list all values of X, and assign P(X) to each value so the total is 1.

What is the mean (expected value) of a discrete random variable X?

μ = Σ xi P(xi) over all values x_i.

How do you compute the mean from a probability distribution?

Multiply each value by its probability and sum the products.

What is the formula for the variance of a discrete random variable?

Var(X) = Σ xi^2 P(xi) − μ^2 (equivalently E[X^2] − μ^2).

What is the standard deviation?

σ = sqrt(Var(X)).

What is the Normal distribution?

A bell-shaped, symmetric distribution characterized by mean μ and standard deviation σ; total area under the curve is 1.

What is the Standard Normal distribution?

A normal distribution with mean μ = 0 and standard deviation σ = 1.

What does the Empirical Rule (68-95-99.7) state for a normal distribution?

About 68.26% within 1 SD, 95.44% within 2 SD, and 99.74% within 3 SD of the mean.

What is a z-score?

z = (X − μ)/σ; the number of standard deviations a value is from the mean.

How do you use the z-table?

Find P(Z ≤ z) for a given z, where Z is standard normal (μ=0, σ=1).

What does the Central Limit Theorem (CLT) say about the sampling distribution of the mean?

As sample size n increases, the distribution of the sample mean approaches a normal distribution; its mean equals the population mean, and its standard error is σ/√n.

What is the finite population correction factor (fpc)?

√(1 − n/N); used when sampling without replacement from a finite population.

What is the t-distribution and when is it used?

A bell-shaped distribution like the normal but with heavier tails, used when the sample size is small and the population standard deviation is unknown (df = n − 1).

When should you use z vs. t for confidence intervals?

Use z if σ is known and n > 30; use t if σ is unknown and n < 30.

What is the confidence interval formula using the z-distribution?

X̄ ± z_{α/2} * (σ/√n).

What is the confidence interval formula using the t-distribution?

X̄ ± t_{α/2, df} * (s/√n).

How do you determine the minimum sample size for estimating a population mean with a desired error E?

n = (Z_{1−α/2} * σ / E)^2.

How do you determine the minimum sample size for estimating a population proportion?

n = p q (Z_{1−α/2} / E)^2, with p the estimated proportion and q = 1 − p.

What is a sampling distribution of the sample mean (infinite population) key property?

The mean of the sampling distribution equals the population mean μ, and Var(X̄) = σ^2 / n (no finite-population correction for infinite populations).

What is sampling distribution of the mean with finite populations?

Var(X̄) = (σ^2 / n) * (N − n)/(N − 1); includes finite population correction.

Why is the Central Limit Theorem important in practice?

It justifies using normal-curve methods and simplifies probability calculations for sample means, even when the underlying population is not normal.