Probability and Statistics practice questions

Answer: C) Median, quartiles and range

• AIncorrect. That's a regression plot.
• BIncorrect. Box plots show quartiles.
• CCorrect. Outliers are often shown as points.
• DIncorrect. That's a time-series plot.

Answer: C) 10%

• AIncorrect. That's compliance.
• BIncorrect. Divide by 30 samples.
• CCorrect. 3 exceedances ÷ 30 samples.
• DIncorrect. Recheck: 3/30.

Answer: A) No linear relationship

• ACorrect. A nonlinear relationship could still exist.
• BIncorrect. That's r = 1.
• CIncorrect. That's r = −1.
• DIncorrect. That's r = 1.

Answer: D) 1.645 standard deviations

• AIncorrect. That's about the 99.9th percentile.
• BIncorrect. That's the 97.5th percentile.
• CIncorrect. That's about the 84th percentile.
• DCorrect. One-sided 95%.

Answer: A) 15%

The coefficient of variation is the standard deviation divided by the mean, usually given as a percentage: 12/80 = 0.15 = 15%. Because it has no units, it measures relative variability.

• ACorrect. CV = (s / mean) × 100% = (12 / 80) × 100% = 0.15 × 100% = 15%.
• BIncorrect. This flips the ratio: (80 / 12) × 100% ≈ 667%. The CV puts the standard deviation in the numerator.
• CIncorrect. This uses the variance (12² = 144) in place of the standard deviation: 144 / 80 = 1.8 = 180%.
• DIncorrect. This misplaces the decimal point. The ratio 0.15 equals 15%, not 1.5%.

Answer: B) 2.58

Sample standard deviation = √[Σ(x − x̄)²/(n − 1)] = √(20/3) ≈ 2.58. Dividing by n gives the population value instead, and stopping before the square root gives the variance.

• AIncorrect. This is the population standard deviation: √(20/4) = √5 ≈ 2.24. The sample standard deviation divides by n − 1 = 3.
• BCorrect. The mean is 5. The squared deviations are 9 + 1 + 1 + 9 = 20, and 20/3 = 6.67. √6.67 ≈ 2.58.
• CIncorrect. 20/3 = 6.67 is the sample variance. You still need to take its square root to get the standard deviation.
• DIncorrect. 5 is the arithmetic mean of the data, which measures central tendency, not spread.

Answer: A) 90% of measured values are at or below it

• ACorrect. It's often used for compliance.
• BIncorrect. That's the 10th percentile.
• CIncorrect. That isn't a percentile.
• DIncorrect. That isn't the definition.

Answer: C) 26.0 mg/L

Flow-weighted mean = Σ(Qi·Ci)/ΣQi = 130/5 = 26 mg/L. Each concentration is weighted by its flow before dividing by the total flow.

• AIncorrect. This comes from weighting by flow but dividing by the wrong total, or averaging the two higher-flow samples with only part of the first. The correct value is Σ(Q·C)/ΣQ = 130/5 = 26 mg/L.
• BIncorrect. This is the simple average (10 + 20 + 40)/3. It ignores the different flows.
• CCorrect. Σ(Q·C) = 1·10 + 2·20 + 2·40 = 130, and ΣQ = 5, so 130/5 = 26 mg/L.
• DIncorrect. 20 mg/L is the median concentration. It does not account for the flow weights.

Answer: C) 3

• AIncorrect. Order the data first.
• BIncorrect. That's the mean.
• CCorrect. Ordered: 1, 2, 3, 5, 8.
• DIncorrect. Order the data first.

Answer: D) ±3.92

When σ is known, the 95% confidence half-width for a mean is 1.96σ/√n. Here that is 1.96 × 10/5 = 3.92.

• AIncorrect. This is the standard error σ/√n = 10/5 = 2.00 with no multiplier. A 95% interval needs z = 1.96 times the standard error.
• BIncorrect. This is 1.96 × σ, which leaves out the √n. The half-width uses the standard error of the mean, σ/√n, not σ.
• CIncorrect. This is 1.96 × 10/25, which divides σ by n instead of √n.
• DCorrect. Half-width = z(0.025) × σ/√n = 1.96 × 10/√25 = 1.96 × 2 = 3.92.