| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Mixed calculations with boundaries |
| Difficulty | Moderate -0.3 This is a straightforward application of normal distribution with standard z-score calculations and expected value. Part (a) requires two z-score lookups for boundaries, part (b) is simple arithmetic using probabilities, and part (c) is inverse normal lookup. All techniques are routine for S1 level with no problem-solving insight required, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| \([P(142 < X < 205)] = P\left(\frac{142-170}{25} < z < \frac{205-170}{25}\right)\) | M1 | Use of \(\pm\) standardisation formula once substituting 170, 25 and either 142 or 205 appropriately. Condone \(25^2\) and continuity correction \(\pm 0.5\) |
| \(P(-1.12 < z < 1.4)\) | A1 | Both correct. Accept unsimplified. |
| \(\Phi(1.4) - (1 - \Phi(1.12)) = 0.9192 + 0.8686 - 1\) | M1 | Calculating the appropriate area from stated phis of \(z\)-values. |
| \(0.788\) | A1 | AWRT, not from wrong working |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X > 205) = 1 - 0.9192 = 0.0808\) | B1 FT | Correct or FT from part 5(a) |
| \((0.0808 \times 0.30 + \text{their } 0.788 \times 0.24) \times 20000\) | M1 | Correct or their \(0.0808 \times 0.30 \times k +\) their \(0.788 \times 0.24 \times k\), \(k\) positive integer |
| \([\\)]4266.24\( | A1 | \)4265 <\( income \)\leqslant 4270$, not from wrong working |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left[P\left(Z > \frac{w-182}{20}\right) = 0.72\right]\) \(\frac{w-182}{20} = -0.583\) | B1 | \(0.5828 \leqslant z \leqslant 0.583\) or \(-0.583 \leqslant z \leqslant -0.5828\) seen |
| M1 | 182 and 20 substituted in \(\pm\) standardisation formula, no continuity correction, not \(\sigma^2\), \(\sqrt{\sigma}\), equated to a \(z\)-value | |
| \(w = 170\) | A1 | \(170 \leqslant w < 170.35\) |
## Question 5:
**Part 5(a):**
$[P(142 < X < 205)] = P\left(\frac{142-170}{25} < z < \frac{205-170}{25}\right)$ | M1 | Use of $\pm$ standardisation formula once substituting 170, 25 and either 142 or 205 appropriately. Condone $25^2$ and continuity correction $\pm 0.5$
$P(-1.12 < z < 1.4)$ | A1 | Both correct. Accept unsimplified.
$\Phi(1.4) - (1 - \Phi(1.12)) = 0.9192 + 0.8686 - 1$ | M1 | Calculating the appropriate area from stated phis of $z$-values.
$0.788$ | A1 | AWRT, not from wrong working
**Part 5(b):**
$P(X > 205) = 1 - 0.9192 = 0.0808$ | B1 FT | Correct or FT from part 5(a)
$(0.0808 \times 0.30 + \text{their } 0.788 \times 0.24) \times 20000$ | M1 | Correct or their $0.0808 \times 0.30 \times k +$ their $0.788 \times 0.24 \times k$, $k$ positive integer
$[\$]4266.24$ | A1 | $4265 <$ income $\leqslant 4270$, not from wrong working
**Part 5(c):**
$\left[P\left(Z > \frac{w-182}{20}\right) = 0.72\right]$ $\frac{w-182}{20} = -0.583$ | B1 | $0.5828 \leqslant z \leqslant 0.583$ or $-0.583 \leqslant z \leqslant -0.5828$ seen
| M1 | 182 and 20 substituted in $\pm$ standardisation formula, no continuity correction, not $\sigma^2$, $\sqrt{\sigma}$, equated to a $z$-value
$w = 170$ | A1 | $170 \leqslant w < 170.35$
---5 Farmer Jones grows apples. The weights, in grams, of the apples grown this year are normally distributed with mean 170 and standard deviation 25. Apples that weigh between 142 grams and 205 grams are sold to a supermarket.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that a randomly chosen apple grown by Farmer Jones this year is sold to the supermarket.\\
Farmer Jones sells the apples to the supermarket at $\$ 0.24$ each. He sells apples that weigh more than 205 grams to a local shop at $\$ 0.30$ each. He does not sell apples that weigh less than 142 grams.
The total number of apples grown by Farmer Jones this year is 20000.
\item Calculate an estimate for his total income from this year's apples.\\
Farmer Tan also grows apples. The weights, in grams, of the apples grown this year follow the distribution $\mathrm { N } \left( 182,20 ^ { 2 } \right) .72 \%$ of these apples have a weight more than $w$ grams.
\item Find the value of $w$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2022 Q5 [10]}}