Statistics Mastery

📝 Explanation The correct answer is D — Skewed Left. Most data (high scores 85–100) is on the right side, but the tail stretches to the left where a few very low scores exist. The mean is pulled left by those outliers: Mean < Median.

📝 Explanation A — Skewed Right. Most people cluster at 0 (the left), and the tail extends far to the right for the heavy exercisers. This is the classic skewed-right pattern: Mean > Median > Mode.

📝 Explanation B — Bimodal. Two separate peaks exist — one around age 20, one around age 50 — with a valley in between. Bimodal distributions signal two distinct sub-groups in the data.

📝 Explanation C — Mean > Median. In a right-skewed distribution, the extreme high values in the right tail pull the mean upward more than the median (which is resistant to outliers). Key rule: Skewed Right → Mean > Median | Skewed Left → Mean < Median.

📝 Explanation D — Uniform. Each value (1 through 6) has exactly 1/6 probability. The histogram is perfectly flat with no peaks or tails. A uniform distribution has no mode.

📝 Explanation B — Negative Correlation. As x (absences) goes up, y (grade) goes down. The pattern slopes downward. This is negative (inverse) correlation. The r-value would be negative (e.g., r ≈ −0.85).

📝 Explanation D — r = −0.95. Strength is measured by |r|. Comparing: |0.75| = 0.75, |−0.50| = 0.50, |0.10| = 0.10, |−0.95| = 0.95 — the largest. The sign only indicates direction, not strength.

📝 Explanation C — Lurking variable. Hot weather increases both ice cream sales AND swimming (thus drowning risk). This is the classic example: Correlation does not imply causation. A lurking (confounding) variable drives both variables simultaneously.

📝 Explanation B — Outlier. A point that does not follow the general pattern of the rest of the data is an outlier. It falls far from the regression line and can significantly affect the correlation coefficient r and the line of best fit.

📝 Explanation C — 96.04%. The coefficient of determination is r² = (0.98)² = 0.9604 = 96.04%. This means 96.04% of the variability in quiz scores can be explained by sleep hours. The remaining 3.96% is due to other factors.

📝 Explanation B. The slope always means: "for each 1-unit increase in x, y changes by [slope] units." Template: "For each additional [x-unit], the predicted [y] increases/decreases by [|slope|]."

📝 Explanation C — 78.41. Substitute x = 6:
The actual data value at x = 6 is 79, so our prediction is very close!

📝 Explanation A. The y-intercept is always the predicted y when x = 0. Here: a student sleeping 0 hours is predicted to score 47.21. Note: this may not be realistic (extrapolation below data range), but that is the mathematical interpretation.

📝 Explanation C — −1.01. Residual = Actual − Predicted: A negative residual means the actual value is below the regression line — the model slightly overestimated.

📝 Explanation B — Extrapolation. x = 20 is far outside the observed range of 1–8. Using the regression line here is unreliable because we don't know if the linear relationship continues. Predicted score would be 5.20(20) + 47.21 = 151.21 — impossible on a 100-point quiz!

📝 Explanation C — 70.625.

📝 Explanation B — r = 0.05. A random cloud with no pattern means no linear correlation → r ≈ 0. r = 0.05 is closest to zero. (Note: in reality, temperature and hot cocoa sales have a strong negative correlation — but the question describes a random cloud specifically.)

📝 Explanation B — $54,500. x = 5 (thousands). Substitute: Don't forget: x = 5 because the units are already in $thousands.

📝 Explanation C — Skewed Left. Rule: Mean < Median < Mode → Skewed Left. Here 45 < 62 < 70 confirms the distribution is skewed left (the mean is dragged down by low outliers in the left tail).

📝 Explanation B. |r| = 0.87 → strong (≥ 0.8). Negative sign → as stress increases, sleep quality decreases (negative direction). We say "tends to predict" — not "causes" — because correlation ≠ causation. Option D is wrong because it reverses the cause-effect claim incorrectly.