Free practice / JC2 H2 Maths Prelim
Normal Distribution: Linear combinations — ASRJC 2025 H2 Math Prelim Paper 2
Attempt the question on paper, then open the worked solution to check your method.
What this question tests
Question
In this question, state clearly all distributions used, together with the values of the appropriate parameters.
A toy designer creates weighted wooden blocks. The mass (in grams) of each block type follows a normal distribution:
| Block type | Mean | Standard Deviation |
|---|---|---|
| Cylindrical | 280 | 6 |
| Cuboidal | 220 | 5 |
- Find the probability that the mass of a randomly selected cylindrical block exceeds the mass of a randomly selected cuboidal block by more than 65 grams.
- The probability that the total of twice the mass of a cylindrical block and thrice the mass of a cuboidal block is less than \(M\,\)g is \(0.355\). Find \(M\) to the nearest integer.
To build a deluxe seesaw model, the designer uses: 2 cylindrical blocks, 3 cuboidal blocks, 7 screws, 1 wooden plank. Before assembly, each cylindrical block loses 5% of its mass after drilling, and each cuboidal block loses 3% of its mass after drilling. The remaining components follow:
| Component | Mean | Standard Deviation |
|---|---|---|
| Screw | 8 | 1.2 |
| Wooden plank | 540 | 8 |
- Find the probability that the total mass of all modified components in a deluxe seesaw model is more than 1785 g.
Show full worked solution▾
Let \(X \sim N(280, 36)\) be the mass of a cylindrical block and \(Y \sim N(220, 25)\) be the mass of a cuboidal block.
(a)
\(X \sim \mathrm{N}(280,\, 6^2)\) and \(Y \sim \mathrm{N}(220,\, 5^2)\)
\[\begin{aligned} X - Y &\sim \mathrm{N}(280 - 220,\; 5^2 + 6^2) \\ X - Y &\sim \mathrm{N}(60,\; 61) \\ \mathrm{P}(X - Y > 65) &= 0.261 \end{aligned}\]\(P(X-Y > 65) = 0.0261\) (3 s.f.)
(b) Let \(C = 2X + 3Y\). \[\begin{aligned} \mathrm{E}(C) &= 2(280) + 3(220) = 560 + 660 = 1220,\\ \mathrm{Var}(C) &= 4(36) + 9(25) = 144 + 225 = 369. \end{aligned}\] \(C \sim N(1220, 369)\).
Given \(P(C < M) = 0.355\).
From GC: \(M = 1212.86\), so \(M = 1213\) (nearest integer).
\(M = 1213\) g.
(c)
Let \(S\) and \(W\) be the mass of a randomly selected screw and wooden plank (in g) respectively.
\(S \sim \mathrm{N}(8,\, 1.2^2)\) and \(W \sim \mathrm{N}(540,\, 8^2)\)
Let \(T = 0.95(X_1 + X_2) + 0.97(Y_1 + Y_2 + Y_3) + S_1 + \cdots + S_7 + W\)
\[\begin{aligned} \mathrm{E}(T) &= (0.95 \times 2 \times 280) + (0.97 \times 3 \times 220) + (7 \times 8) + 540 \\ &= 532 + 426.8 + 56 + 540 \\ &= 1768.2 \end{aligned}\] \[\begin{aligned} \mathrm{Var}(T) &= (0.95^2 \times 2 \times 6^2) + (0.97^2 \times 3 \times 5^2) + (7 \times 1.2^2) + 64 \\ &= 64.98 + 70.5675 + 10.08 + 64 \\ &= 209.6275 \end{aligned}\] \(\mathrm{P}(T > 1785) = 0.123\)