Normal Distribution: Linear combinations — ASRJC 2025 H2 Math Prelim Paper 2
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 \(T = 0.95(X_1 + X_2) + 0.97(Y_1 + Y_2 + Y_3) + S_1 + \cdots + S_7 + W\)
\[\begin{aligned} \mathrm{E}(T) &= (0.95 \times 2 \times 280) + (0.97 \times 3 \times 220) + (7 \times 8) + 540 \\ &= 532 + 426.8 + 56 + 540 \\ &= 1768.2 \end{aligned}\] \[\begin{aligned} \mathrm{Var}(T) &= (0.95^2 \times 2 \times 6^2) + (0.97^2 \times 3 \times 5^2) + (7 \times 1.2^2) + 64 \\ &= 64.98 + 70.5675 + 10.08 + 64 \\ &= 209.6275 \end{aligned}\] \(\mathrm{P}(T > 1785) = 0.123\)