Probability Distributions: Binomial and Normal — HS Statistics practice questions

X is binomial with n = 4 and p = 0.2. What is P(X = 0)?

1. 0.0016
2. 0
3. 0.4096
4. 0.8

Answer: C — 0.4096

C) Correct — All four trials fail: 0.8⁴. A) Is 0.2⁴, the chance of all four succeeding. B) Assumes at least one success must happen. D) Is the chance of failing just one trial.

Which setting fits a binomial distribution?

1. Measuring a continuous quantity like height
2. A fixed number of independent trials, each a success or failure with the same probability
3. Counting trials until the first success
4. Trials where the probability changes each time

Answer: B — A fixed number of independent trials, each a success or failure with the same probability

B) Correct — Binary outcomes, Independent trials, fixed Number, Same probability. A) Binomial counts successes; it is discrete. C) That is a geometric setting. D) The probability must stay the same.

IQ scores are normal with mean 100 and SD 15. About what percent score between 85 and 115?

1. 95%
2. 68%
3. 99.7%
4. 50%

Answer: B — 68%

B) Correct — 85 and 115 are 1 SD below and above the mean. A) Is the share within 2 SDs (70 to 130). C) Is the share within 3 SDs. D) Is the share above 100.

X is binomial with n = 20 and p = 0.3. What is the variance of X?

1. 6
2. 4.2
3. 2.05
4. 0.21

Answer: B — 4.2

B) Correct — Variance = np(1 − p) = 20(0.3)(0.7). A) Is the mean, np. C) Is the standard deviation. D) Is p(1 − p) without multiplying by n.

For the standard normal distribution, what is the area to the left of z = 0?

1. 0
2. 0.5
3. 1
4. 0.68

Answer: B — 0.5

B) Correct — The curve is symmetric about 0. A) Half the area lies on each side. C) Is the total area. D) Is the area between −1 and 1.

In a normal distribution, about what percent of values lie within 1 SD of the mean?

1. 95%
2. 68%
3. 99.7%
4. 50%

Answer: B — 68%

B) Correct — The 68-95-99.7 rule. A) Is within 2 SDs. C) Is within 3 SDs. D) Is the share above (or below) the mean.

X takes values 0, 1, and 2 with probabilities 0.5, 0.3, and 0.2. What is E(X)?

1. 0.7
2. 1
3. 0.33
4. 3

Answer: A — 0.7

A) Correct — 0(0.5) + 1(0.3) + 2(0.2). B) Uses the middle value instead of weighting by probability. C) Averages the probabilities. D) Adds the values.

X has mean 10 and Y has mean 4. What is the mean of X − Y?

1. 14
2. 40
3. 2.5
4. 6

Answer: D — 6

D) Correct — Means subtract: 10 − 4. A) Adds the means. B) Multiplies the means. C) Divides the means.

Which is a continuous random variable?

1. The time it takes to finish a race
2. The number of goals scored
3. The number of heads in 10 flips
4. The number of students absent

Answer: A — The time it takes to finish a race

A) Correct — Time can take any value in an interval. B) Is a count, so it is discrete. C) Is a count, so it is discrete. D) Is a count, so it is discrete.

About what share of a normal distribution lies more than 2 SDs above the mean?

1. 5%
2. 95%
3. 0.15%
4. 2.5%

Answer: D — 2.5%

D) Correct — 95% is within 2 SDs, leaving 5% split between the two tails. A) Counts both tails. B) Is the share within 2 SDs. C) Is the share beyond 3 SDs on one side.

Binomial distribution?

Fixed n trials, two outcomes (success/failure), constant p, independent trials.

Normal distribution shape?

Bell-shaped. Symmetric around mean.

Geometric distribution?

Number of trials until first success. p constant.

Combining random variables?

Means: E(X ± Y) = E(X) ± E(Y). If X and Y are independent, variances always add: Var(X ± Y) = Var(X) + Var(Y). Standard deviations do not add.