Standard +0.3 This is a straightforward application of linear combinations of normal distributions requiring students to set up a profit equation, form a linear combination (200 - 13.50X - 0.90Y - 55), find its mean and variance using standard formulas, then calculate a single probability. While it involves multiple steps, each is routine and the question clearly signposts the approach, making it slightly easier than average.
6 A factory makes loaves of bread in batches. One batch of loaves contains \(X\) kilograms of dried yeast and \(Y\) kilograms of flour, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 0.7,0.02 ^ { 2 } \right)\) and \(\mathrm { N } \left( 100.0,3.0 ^ { 2 } \right)\) respectively.
Dried yeast costs \(\\) 13.50\( per kilogram and flour costs \)\\( 0.90\) per kilogram. For making one batch of bread the total of all other costs is \(\\) 55\(. The factory sells each batch of bread for \)\\( 200\).
Find the probability that the profit made on one randomly chosen batch of bread is greater than \(\\) 40$. [7]
\)\frac{160 - 154.45}{\sqrt{7.3629}}\( or \)\frac{40 - 45.55}{\sqrt{7.3629}}$. No mixed methods.
\([P(T < 105) = P(z < 2.045) =]\ \Phi(2.045)\)
M1
For area consistent with their working.
\(= 0.9795\) or \(0.9796\) or \(0.98(0)\) or \(0.979\) (3 sf)
A1
## Question 6:
Cost of dried yeast and flour: $\$D$ and $\$F$; $E(D) = 13.5 \times 0.7 = 9.45$, $E(F) = 0.9 \times 100 = 90$ | B1 | One of these soi – can be given at early stage.
$\text{Var}(D) = 0.02^2 \times 13.50^2 = 0.0729$; $\text{Var}(F) = 3.0^2 \times 0.90^2 = 7.29$ | B1 | One of these soi – can be given at early stage.
Total cost: $T \sim N(99.45,\ (0.02^2 \times 13.50^2 + 3.0^2 \times 0.90^2))$ | M1 | Attempt to combine $D$ and $F$ with or without 55 and 200 (but variance must not include 55 or 200).
$N(99.45,\ 7.3629)$ accept 99.4 or 99.5 | A1 | Or $N(154.45, 7.3629)$ or $N(45.55, 7.3629)$. Accept 3sf.
$[P(\text{profit} > \$40) = P(T < 105)]$; $\dfrac{105 - 99.45}{\sqrt{7.3629}} [= 2.045]$ | M1 | $\frac{160 - 154.45}{\sqrt{7.3629}}$ or $\frac{40 - 45.55}{\sqrt{7.3629}}$. No mixed methods.
$[P(T < 105) = P(z < 2.045) =]\ \Phi(2.045)$ | M1 | For area consistent with their working.
$= 0.9795$ or $0.9796$ or $0.98(0)$ or $0.979$ (3 sf) | A1 |
6 A factory makes loaves of bread in batches. One batch of loaves contains $X$ kilograms of dried yeast and $Y$ kilograms of flour, where $X$ and $Y$ have the independent distributions $\mathrm { N } \left( 0.7,0.02 ^ { 2 } \right)$ and $\mathrm { N } \left( 100.0,3.0 ^ { 2 } \right)$ respectively.
Dried yeast costs $\$ 13.50$ per kilogram and flour costs $\$ 0.90$ per kilogram. For making one batch of bread the total of all other costs is $\$ 55$. The factory sells each batch of bread for $\$ 200$.
Find the probability that the profit made on one randomly chosen batch of bread is greater than $\$ 40$. [7]\\
\hfill \mbox{\textit{CAIE S2 2023 Q6 [7]}}