AP Stats Probability and Distributions 3

Question 1

A binomial random variable has n = 20 and p = 0.4. What is its standard deviation?

Accepted Answer

2.19

Answer

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?

Accepted Answer

68%

Answer

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?

Accepted Answer

0.117

Answer

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?

Accepted Answer

Approximately 2.5%

Answer

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:

Accepted Answer

E[(X − μ)²]

Answer

Variance is the expected value of the squared deviation from the mean: σ² = E[(X − μ)²].

AP Practice Test