AP Stats Probability and Distributions 3 — Questions and Answers
Question 1: A binomial random variable has n = 20 and p = 0.4. What is its standard deviation?
- 2.19 (Correct answer)
- 4.80
- 8.00
- 0.48
Correct answer: 2.19
σ = √(np(1−p)) = √(20 × 0.4 × 0.6) = √4.8 ≈ 2.19.
Question 2: Using the standard normal table, approximately what proportion of data falls within one standard deviation of the mean?
- 68% (Correct answer)
- 95%
- 99.7%
- 50%
Correct answer: 68%
The empirical rule states that approximately 68% of data lies within ±1σ of the mean in a normal distribution.
Question 3: A student randomly guesses on a 10-question true/false test. What is the probability of getting exactly 7 correct?
- 0.117 (Correct answer)
- 0.070
- 0.500
- 0.250
Correct answer: 0.117
P(X = 7) = C(10,7)(0.5)^7(0.5)^3 = 120 × (1/1024) ≈ 0.117.
Question 4: If X ~ N(100, 15²), what is P(X > 130) using the empirical rule?
- Approximately 2.5% (Correct answer)
- Approximately 5%
- Approximately 16%
- Approximately 0.15%
Correct answer: Approximately 2.5%
130 is 2σ above the mean; 95% falls within ±2σ, so 2.5% lies above 130.
Question 5: The variance of a discrete random variable X is defined as:
- E[(X − μ)²] (Correct answer)
- E[X²]
- E[X] − μ
- √(E[X²] − μ²)
Correct answer: E[(X − μ)²]
Variance is the expected value of the squared deviation from the mean: σ² = E[(X − μ)²].
Question 6: Two independent random variables X and Y have variances 9 and 16 respectively. What is Var(X + Y)?
- 25 (Correct answer)
- 7
- 144
- 5
Correct answer: 25
For independent variables, Var(X + Y) = Var(X) + Var(Y) = 9 + 16 = 25.
Question 7: A Poisson random variable models the number of emails received per hour with mean λ = 3. What is P(X = 0)?
- e⁻³ ≈ 0.050 (Correct answer)
- 3e⁻³ ≈ 0.149
- 1 − e⁻³ ≈ 0.950
- e³ ≈ 20.09
Correct answer: e⁻³ ≈ 0.050
P(X = 0) = e^(−λ)λ^0/0! = e^(−3) ≈ 0.050.
AP Stats Probability and Distributions 3
A binomial random variable has n = 20 and p = 0.4.
What is its standard deviation?