Final

Question 1

Probability

Answer 1

The measure of the likelihood that an event will occur

Question 2

Sample Space

Answer 2

The set of all possible outcomes of an experiment

Question 3

Event

Answer 3

A subset of the sample space

Question 4

Compliment

Answer 4

Denoted as A’ or A^c; all outcomes not in A.

Question 5

Addition Rule

Answer 5

P(A∪B)=P(A)+P(B)−P(A∩B)

Question 6

Multiplication Rule (for Independent Events)

Answer 6

P(A∩B)=P(A)⋅P(B)

Question 7

Conditional Probability

Answer 7

P(A|B)= P(A∩B) / P(B)

Question 8

Probability Mass Function (PMF)

Answer 8

Gives the probability of each possible value in a discrete random variable.

Question 9

Probability Density Function (PDF)

Answer 9

Gives the probability density of a continuous random variable.

Question 10

Expected Value (μ)

Answer 10

μ=∑(i=1, n) xi * P(X=xi)

Question 11

Mean (X^-)

Answer 11

(X^-) = [∑(i=1, n) xi] / n

Question 12

Variance (Var(x))

Answer 12

Var(x) = [∑(i=1, n) (xi-[X^-])^2] / n

Question 13

Standard Deviation (σ)

Answer 13

σ = SQRT (Var(x))

Question 14

Summation Notation

Answer 14

∑(i=1, n) xi
“The sum of xi from i=1 to n.”

Question 15

Factorial Notation

Answer 15

n! = the product of all positive integers up to n
EX: 3! = 321 = 6

Question 16

Central Limit Theorem (CLT)

Answer 16

States that the distribution of the sum (or average) of a large number of independent, identically distributed random variables (typically n > 30) approaches a normal distribution, regardless of the original distribution

Question 17

Conditions for CLT

Answer 17

The random variables must be independent.
The sample size should be sufficiently large.
The original distribution’s shape doesn’t matter.

Question 18

CLT Formula for Sample Means

Answer 18

If X is a random variable with mean μ and standard deviation
σ, then the distribution of the sample mean (X^-) approaches a normal distribution with a mean, μ, and standard deviation, σ/SQRT(n)

Question 19

Maximum Likelihood Estimation (MLE)

Answer 19

A method for estimating the parameters of a statistical model that maximizes the likelihood function

Question 20

Likelihood Function

Answer 20

L(θ|X)=P(X|L) where:
L is the likelihood function
θ is the parameter
X is the data.

Question 21

Log-Likelihood Function

Answer 21

ℓ(θ)=ln(L(θ|X)); often used for easier calculations
Set dℓ/dθ =0 and solve for the parameter θ.

Question 22

Confidence Interval (CI)

Answer 22

A range of values constructed from sample data so that the population parameter is likely to occur within that range at a certain level of confidence

Question 23

Confidence Level

Answer 23

The probability that the interval contains the true parameter
Common choices are 90%, 95%, and 99%

Question 24

CI Formula for a Mean

Answer 24

(X^ˉ) ± Z⋅σ/SQRT(n) where:
Z is the Z-score corresponding to the desired confidence level.

Question 25

Hypothesis Testing Steps

Answer 25

State the null hypothesis (H0) and alternative hypotheses (H1).
Choose the significance level (α).
Calculate the test statistic.
Make a decision: reject or fail to reject the null hypothesis.

Question 26

Type I Error

Answer 26

Rejecting a true null hypothesis (false positive)

Question 27

Type II Error

Answer 27

Failing to reject a false null hypothesis (false negative)

Question 28

P-Value for Z TEST

Answer 28

1 Sided: P(Z > z) or P(Z < z) as appropriate.
2 Sided: P(|Z| > |z|) where z is the calculated Z-value.

Question 29

P-Value for T TEST

Answer 29

1 Sided: P(t > t_observed) or P(t < t_observed) as appropriate.
2 Sided: P(|t| > |t_observed|) where t_observed is the calculated t-value.

Question 30

CLT for Sample Proportions

Answer 30

If X is the number of successes in a sample of size, n, from a population with proportion, p, the distribution of
(^p) (sample proportion) approaches a normal distribution with mean, p and standard deviation SQRT [(p*(1-p)) / n].

EX: Z = (^p)-p / (SQRT [(p*(1-p)) / n])

Question 31

MLE for Normal Distribution (Known Variance)

Answer 31

If X1, X2,…, Xn are independent and identically distributed (i.i.d.) random variables from a normal distribution with mean, μ, and known variance, σ^2, the MLE for μ is the sample mean, (X̄)

Question 32

CI for Population Proportion

Answer 32

(^p) ± Z * SQRT [(^p * (1 - ^p)) / n

Question 33

Test Statistic for Population Mean (Known Variance) (Z TEST)

Answer 33

For testing a hypothesis about the population mean (μ) with known variance (σ^2), the test statistic is: Z = (X̄ - μ₀) / (σ / √n), where:
X̄ is the sample mean,
μ₀ is the hypothesized population mean.

Question 34

Z-Score (Z)

Answer 34

The Z-score is determined by the desired level of confidence. You can find this value in a Z-table or use statistical software.

EX: If you’re constructing a 95% confidence interval, use a Z-score corresponding to the critical value for a 95% confidence interval.

Question 35

Test Statistic for Population Mean (Unknown Variance) (T TEST)

Answer 35

t = (X̄ - μ₀) / (s / √n)

Question 36

Degrees of Freedom (df)

Answer 36

df = n - 1 where:
n is the sample size