HS Statistics practice questions and exam guide

A box plot displays:

1. the mean and standard deviation
2. every individual value
3. the counts of each category
4. the five-number summary

Answer: D — the five-number summary

D) Correct — Minimum, Q1, median, Q3, and maximum (with outliers marked). A) A box plot is built from quartiles, not the mean and SD. B) That is a dot plot or stem plot. C) That is a bar chart.

A distribution is skewed right (a long tail of high values). Typically:

1. the mean is less than the median
2. the mean is greater than the median
3. the mean equals the median
4. the median is greater than the mean

Answer: B — the mean is greater than the median

B) Correct — The high values pull the mean toward the tail. A) Describes a left-skewed distribution. C) Describes a symmetric distribution. D) Also describes left skew.

A simple random sample of size n is one in which:

1. the researcher picks people who seem typical
2. the first n people who respond are used
3. each subgroup is sampled separately
4. every group of n individuals has the same chance of being chosen

Answer: D — every group of n individuals has the same chance of being chosen

D) Correct — That is what makes it simple and random. A) Personal choice introduces bias. B) Is a convenience or voluntary sample. C) Describes a stratified sample.

Researchers compare the health of people who already choose to exercise with people who do not. This is:

1. a randomized experiment
2. a matched-pairs experiment
3. an observational study
4. a census

Answer: C — an observational study

C) Correct — The researchers do not assign the exercise. A) Subjects chose their own exercise habits. B) No treatment is assigned at all. D) Not everyone in the population is measured.

What is the expected value of one roll of a fair six-sided die?

1. 3
2. 6
3. 3.5
4. 21

Answer: C — 3.5

C) Correct — (1 + 2 + 3 + 4 + 5 + 6)/6 = 21/6. A) Rounds down; the expected value need not be a possible roll. B) Is the largest outcome. D) Adds the outcomes without dividing by 6.

A fair coin is flipped 3 times. What is P(at least one head)?

1. 3/8
2. 7/8
3. 1/2
4. 3/2

Answer: B — 7/8

B) Correct — 1 − P(no heads) = 1 − 1/8. A) Is P(exactly one head). C) Uses the probability for one flip. D) Adds 1/2 three times; probabilities cannot exceed 1.

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.

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.

A 95% confidence interval for a proportion is (0.42, 0.58). What are the point estimate and margin of error?

1. p̂ = 0.50, margin of error 0.16
2. p̂ = 0.50, margin of error 0.08
3. p̂ = 0.42, margin of error 0.58
4. p̂ = 0.16, margin of error 0.50

Answer: B — p̂ = 0.50, margin of error 0.08

B) Correct — The center is the estimate, and half the width is the margin of error. A) Uses the full width as the margin of error. C) Reads the endpoints as the estimate and margin. D) Swaps the width and the center.

A Type II error is:

1. rejecting H₀ when it is true
2. rejecting H₀ when it is false
3. choosing α too small
4. failing to reject H₀ when it is actually false

Answer: D — failing to reject H₀ when it is actually false

D) Correct — A missed detection; its probability is β. A) Is a Type I error. B) Is a correct decision. C) Can make Type II errors more likely, but is not one.

A category has observed count 15 and expected count 10. What is its contribution to χ²?

1. 5
2. 2.5
3. 25
4. 1.67

Answer: B — 2.5

B) Correct — (15 − 10)²/10 = 25/10. A) Uses O − E without squaring or dividing. C) Forgets to divide by E. D) Divides by O instead of E.

A student's actual score is 70, and the line predicts 74. What is the residual?

1. −4
2. 4
3. 70
4. 16

Answer: A — −4

A) Correct — Residual = actual − predicted = 70 − 74. B) Subtracts in the wrong order. C) Is the actual value. D) Squares the residual.

Choose 3 of 7: ?

7C3 = 35.

What should you check before trusting a statistical claim?

Sample size, methodology, source bias, confounders, statistical vs practical significance.

Finding a value from a percentile (normal)?

Find the z with that area to its left (table or invNorm), then x = μ + zσ. Example: the 90th percentile of N(100, 15) ≈ 100 + 1.28(15) ≈ 119.

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.

Placebo?

Inert treatment. Subjects can show effects from belief alone.

Tree diagram?

Branches show each stage's outcomes and probabilities. Multiply along a path for P(A and B); add the paths that lead to the same event.