Source: ACE Exam Paper 1 ADV
Tags: normal distribution, z-scores, standard deviation, Pearson's correlation coefficient, scatterplot, line of best fit, continuous random variable, probability density function, combined mean, 68-95-99.7 rule, empirical rule
Statistics in this paper covers the normal distribution (z-scores, the empirical rule), correlation (scatterplots, Pearson's r), continuous random variables (integrating piecewise functions), and probability density function properties. The maths is usually straightforward once you identify which formula to use, but reading the question carefully is half the battle.
Normal distribution
A symmetric, bell-shaped distribution defined by its mean (μ) and standard deviation (σ). Roughly 68% of data falls within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.
z-score
z = (x - μ) / σ. Tells you how many standard deviations a value is from the mean. A z-score of -1 means one standard deviation below the mean.
Empirical rule (68-95-99.7 rule)
Approximately 68% of data lies between z = -1 and z = 1, 95% between z = -2 and z = 2, and 99.7% between z = -3 and z = 3.
Pearson's correlation coefficient (r)
A number between -1 and 1 measuring the strength and direction of a linear relationship. Close to 1 means strong positive linear correlation. Close to 0 means little to no linear relationship.
Scatterplot
A graph of paired data points on an x-y plane, used to visualise the relationship between two variables.
Line of best fit
A straight line drawn through a scatterplot that best represents the trend in the data. Used for interpolation (predicting within the data range).
Continuous random variable
A variable that can take any value in an interval. Its probability is found by integrating its probability density function (PDF).
Probability density function (PDF)
A function f(x) ≥ 0 where the total area under the curve equals 1. The probability of X falling between a and b is ∫ from a to b of f(x) dx.
Combined mean (weighted mean)
When combining groups, the overall mean = (n₁ × mean₁ + n₂ × mean₂) / (n₁ + n₂).
A factory produces bags of cashews: μ = 900 g, σ = 50 g. What percentage weigh more than 1000 g?
z = (1000 - 900)/50 = 2
About 95% of bags lie within 2 standard deviations (between 800 g and 1000 g)
5% lies outside, split equally between both tails
The percentage above 1000 g is approximately 2.5%
Answer: (B).
Mean weight = 12.5 kg, σ = 0.5 kg.
(a) Minimum weight accepted (z-score > -1):
z = -1: x = μ + zσ = 12.5 + (-1)(0.5) = 12.0 kg
(b) Maximum weight accepted (z-score < 2):
z = 2: x = 12.5 + (2)(0.5) = 13.5 kg
Mean = 60, σ = 10.
(a) Score with z = -2:
x = 60 + (-2)(10) = 40
(b) z-score of x = 65:
z = (65 - 60)/10 = 0.5
Mean = 167 cm, σ = 12 cm. What percentage are taller than 179 cm?
z = (179 - 167)/12 = 1
68% of the group is between 155 cm and 179 cm
32% is outside this range, split evenly
Percentage above 179 cm: 16%
When data points trend upward from left to right and are loosely clustered (not tightly around a line), the correlation is moderate positive. A tight cluster would be strong positive, and a very scattered cloud with a slight upward trend would be weak positive.
The exam scatterplot shows a clear upward trend with moderate scatter. Answer: (D) Moderate positive.
A Pearson's correlation coefficient of 0.12 between assessment results and height indicates a very weak positive linear relationship. In practical terms, there is almost no meaningful linear association between how tall a student is and how they perform on the assessment. Height is effectively not a useful predictor of results.
Data pairs: forearm (cm) vs hand (cm), ranging from 25.0 to 29.2 and 17.2 to 20.6 respectively. The relationship is strongly positive and roughly linear.
For a forearm length of 27.8 cm, read from the line of best fit. The expected hand length is approximately 19.6 cm (depending on the line drawn, answers in the range 19.4 to 19.8 are reasonable).
Pearson's r calculation:
Using the formula r = [nΣxy - ΣxΣy] / √{[nΣx² - (Σx)²][nΣy² - (Σy)²]}, with n = 10 data points, the calculation yields r ≈ 0.9919. This confirms a very strong positive linear correlation.
f(x) = |3 - x| for 2 ≤ x ≤ 4, and 0 otherwise.
This is a V-shaped function centred at x = 3:
For 2 ≤ x ≤ 3: f(x) = 3 - x
For 3 < x ≤ 4: f(x) = x - 3
(a) P(2 ≤ X ≤ 3.5):
Split at x = 3:
∫₂³ (3 - x) dx = [3x - x²/2] from 2 to 3 = (9 - 9/2) - (6 - 2) = 9/2 - 4 = 1/2
∫₃^{3.5} (x - 3) dx = [x²/2 - 3x] from 3 to 3.5 = (12.25/2 - 10.5) - (9/2 - 9) = (6.125 - 10.5) - (4.5 - 9) = -4.375 + 4.5 = 1/8
P(2 ≤ X ≤ 3.5) = 1/2 + 1/8 = 5/8
(b) P(2 ≤ X ≤ 2.5):
∫₂^{2.5} (3 - x) dx = [3x - x²/2] from 2 to 2.5 = (7.5 - 3.125) - (6 - 2) = 4.375 - 4 = 3/8
A valid PDF must satisfy two conditions:
f(x) ≥ 0 for all x (the graph never goes below the x-axis)
The total area under the curve equals 1
Any graph that dips below the x-axis cannot be a PDF. Among the exam options, the graph that takes negative values is the one that could not represent a PDF. Answer: (A), the graph with a portion below the x-axis.
Class A: 24 students, mean 75.5%. Class B: 28 students, mean 80.5%.
Combined mean = (24 × 75.5 + 28 × 80.5) / (24 + 28) = (1812 + 2254) / 52 = 4066 / 52 = 78.2%
z = (x - μ) / σ, rearranged: x = μ + zσ
Empirical rule: 68% within 1σ, 95% within 2σ, 99.7% within 3σ
Combined mean: x̄ = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)
Pearson's r = [nΣxy - ΣxΣy] / √{[nΣx² - (Σx)²][nΣy² - (Σy)²]}
For a continuous random variable: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx
⚠️ The empirical rule is approximate. "Best approximation" in a multiple-choice question means pick the answer closest to the rule's prediction, not an exact calculation.
⚠️ z-score questions are essentially substitution into z = (x - μ)/σ. The most common error is subtracting the wrong way round.
⚠️ Pearson's r only measures linear association. A value near zero does not mean "no relationship" in general, only "no linear relationship."
⚠️ For PDFs, area under the curve = probability. A function that goes negative somewhere fails the f(x) ≥ 0 requirement and is immediately disqualified.
⚠️ Combined mean is a weighted average, not the average of the two means. Using (75.5 + 80.5)/2 = 78 is wrong here because the class sizes differ.
⚠️ When splitting |3 - x| into a piecewise function, be precise about which piece applies on which side of x = 3.
Q: Bags of cashews have mean 900 g and standard deviation 50 g (normal distribution). What percentage weigh more than 1000 g?
A: z = 2. By the empirical rule, about 2.5% are above 2 standard deviations.
Q: Mean = 12.5 kg, σ = 0.5 kg. What weight has a z-score of -1?
A: x = 12.5 + (-1)(0.5) = 12.0 kg.
Q: Class A (24 students, mean 75.5%) and Class B (28 students, mean 80.5%). What is the combined mean?
A: (24 × 75.5 + 28 × 80.5) / 52 = 78.2%.
Q: Pearson's r between height and test score is 0.12. What does this mean?
A: There is a very weak positive linear relationship. Height is essentially not a useful predictor of test performance.
Q: f(x) = |3 - x| for 2 ≤ x ≤ 4. Find P(2 ≤ X ≤ 2.5).
A: ∫₂^{2.5} (3 - x) dx = 3/8.
normal distribution, bell curve, z-score, standard score, empirical rule, 68-95-99.7, standard deviation, Pearson's correlation coefficient, r value, scatterplot, line of best fit, interpolation, continuous random variable, probability density function, PDF, combined mean, weighted average, HSC maths advanced, Year 12 statistics