Probability Distributions: Binomial and Normal — HS Statistics practice questions

Median of {3, 7, 8, 12, 15}:

1. 8 (middle value when ordered)
2. 7
3. 9
4. 12
5. 15

Answer: A — 8 (middle value when ordered)

Median = middle value (odd count) or average of two middle (even count). Already ordered, middle of 5 values is the 3rd: 8. Median resistant to outliers, unlike mean.

What is the Central Limit Theorem (CLT)?

1. Random
2. For large enough sample sizes (n ≥ 30 rule of thumb), the distribution of sample means is approximately NORMAL regardless of population shape
3. Means equal medians
4. All samples are normal
5. Doesn't apply

Answer: B — For large enough sample sizes (n ≥ 30 rule of thumb), the distribution of sample means is approximately NORMAL regardless of population shape

CLT: sampling distribution of x̄ → normal for large n, regardless of population distribution. Mean: μ. SE = σ/√n. Foundation of inference (z-tests, t-tests). One of most important results in statistics.

The probability of flipping heads then tails on two fair coins is:

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

Answer: C — 1/4

C) Independent events multiply: (1/2)(1/2) = 1/4. A) Used probability of a single coin. B) Computed the complement of "both tails." D) Used three coins.

The LEAST SQUARES regression line is chosen to:

1. Connect points exactly
2. Minimize the sum of squared vertical distances (residuals) from data points to the line; finds best linear fit
3. Pass through origin
4. Pass through mean only
5. Be random

Answer: B — Minimize the sum of squared vertical distances (residuals) from data points to the line; finds best linear fit

Least squares: minimizes Σ(residuals²) where residual = observed - predicted. Best linear fit. Line passes through (x̄, ȳ). Coefficient of determination r²: proportion of variance explained. Foundation of regression.

What is the SIGNIFICANCE LEVEL (α) commonly used in hypothesis testing?

1. 0.05 (5%; threshold for rejecting null hypothesis; lower α = stricter standard)
2. 0.5
3. 0.01 (also common, stricter)
4. 0.95
5. 0.5

Answer: A — 0.05 (5%; threshold for rejecting null hypothesis; lower α = stricter standard)

α = significance level = probability of Type I error (false positive). Conventional: 0.05 (5%). Stricter: 0.01 or 0.001. Reject H₀ if p < α. Trade-off with Type II error (β).

Range of {12, 7, 15, 9, 20}:

1. 15
2. 20
3. 13 (max - min = 20 - 7 = 13)
4. 11
5. 14

Answer: C — 13 (max - min = 20 - 7 = 13)

Range = maximum - minimum. Identifies max (20) and min (7). 20 - 7 = 13. Simple but limited spread measure; only uses two values, ignores middle of distribution.

WORKED EXAMPLE: A free-throw shooter makes 70% of shots. What is the probability of making exactly 4 out of 6 attempts? Use the binomial formula.

Step 1 — IDENTIFY binomial setup (BINS): Binary (make/miss), Independent shots, fixed N = 6, Same p = 0.7. Step 2 — FORMULA: P(X = k) = C(n, k) · p^k · (1 − p)^(n − k). Step 3 — SUBSTITUTE: n = 6, k = 4, p = 0.7, 1 − p = 0.3. Step 4 — COMBINATIONS: C(6, 4) = 6! / (4!·2!) = (6·5)/(2·1) = 15. Step 5 — POWERS: 0.7^4 = 0.2401. 0.3^2 = 0.09. Step 6 — MULTIPLY: P(X = 4) = 15 · 0.2401 · 0.09 = 15 · 0.021609 ≈ 0.324. Step 7 — INTERPRET: about a 32.4% chance of making exactly 4 of 6. Step 8 — CHECK with mean: E(X) = np = 6(0.7) = 4.2, so getting 4 is near the expected value — high probability makes sense.

Scatter plot?

Plots two quantitative variables. Reveals relationship (direction, form, strength).

Coefficient of determination r²?

Proportion of variation in y explained by linear regression with x. 0 to 1.

Linear regression?

Fits a straight line to data: ŷ = a + bx. Slope = change in y per unit x. Intercept = y when x=0.