| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Conditional probability with normal |
| Difficulty | Standard +0.3 This is a standard S1 normal distribution question with routine calculations: (a) basic z-score lookup, (b) straightforward conditional probability P(X>50|X>39), and (c) simultaneous equations from two z-score conditions. All techniques are textbook exercises requiring no novel insight, though part (c) involves slightly more algebraic manipulation than average. |
| Spec | 2.03d Calculate conditional probability: from first principles2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(H < 18) = P\left(Z < \frac{18-22}{10}\right) = P(Z < -0.4)\) | M1 | For standardising with 18, 22 and 10. Allow \(\pm\frac{18-22}{10}\) |
| \(= 1 - 0.6554\) | dM1 | Dependent on 1st M1, for \(1-p\) where \(0.6 < p < 0.7\) |
| \(= 0.3446\) or awrt \(0.345\) | A1 (3) | For 0.3446 or better or awrt 0.345. Calculator gives 0.3445783. Ans only 3/3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(H > 50) = P(Z > 2.8) = 1 - 0.9974 = 0.0026\) | M1, A1 | 1st M1 for correct standardisation and \(1-q\) for one of these probs. 1st A1 for 0.0026 or better |
| \(P(H > 39) = P(Z > 1.7) = 1 - 0.9554 = 0.0446\) | A1 | 2nd A1 for 0.0446 or better |
| \(P(H > 50 \mid H > 39) = \frac{P(H>50)}{P(H>39)} = \frac{0.0026}{0.0446} = 0.057 \sim 0.0585\) | M1, A1 (5) | 2nd M1 for correct ratio. A1 for answer in range 0.057–0.0585. No fractions but \(\frac{13}{223}\) can score full marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{18 - \mu}{\sigma} = -0.8416\) and \(\frac{28 - \mu}{\sigma} = 1\) | M1, B1, A1 | M1 for attempt to standardise with \(\mu\), \(\sigma\) and 18 or 28 set equal to \(z\) value (\(\pm\)). B1 for using \(z = 0.8416\) or better. A1 for both equations with \(\pm 1\) and \(\pm 0.84\) or better |
| \(10 = 1.8416\sigma\) | M1 | For solving their linear equations in \(\mu\) and \(\sigma\) |
| \(\sigma =\) awrt \(5.43\) | A1 | |
| \(\mu =\) awrt \(22.57\) | A1 (6) |
## Question 5:
### Part (a)
| $P(H < 18) = P\left(Z < \frac{18-22}{10}\right) = P(Z < -0.4)$ | M1 | For standardising with 18, 22 and 10. Allow $\pm\frac{18-22}{10}$ |
| $= 1 - 0.6554$ | dM1 | Dependent on 1st M1, for $1-p$ where $0.6 < p < 0.7$ |
| $= 0.3446$ or awrt $0.345$ | A1 (3) | For 0.3446 or better or awrt 0.345. Calculator gives 0.3445783. Ans only 3/3 |
### Part (b)
| $P(H > 50) = P(Z > 2.8) = 1 - 0.9974 = 0.0026$ | M1, A1 | 1st M1 for correct standardisation and $1-q$ for one of these probs. 1st A1 for 0.0026 or better |
| $P(H > 39) = P(Z > 1.7) = 1 - 0.9554 = 0.0446$ | A1 | 2nd A1 for 0.0446 or better |
| $P(H > 50 \mid H > 39) = \frac{P(H>50)}{P(H>39)} = \frac{0.0026}{0.0446} = 0.057 \sim 0.0585$ | M1, A1 (5) | 2nd M1 for correct ratio. A1 for answer in range 0.057–0.0585. No fractions but $\frac{13}{223}$ can score full marks |
### Part (c)
| $\frac{18 - \mu}{\sigma} = -0.8416$ and $\frac{28 - \mu}{\sigma} = 1$ | M1, B1, A1 | M1 for attempt to standardise with $\mu$, $\sigma$ and 18 or 28 set equal to $z$ value ($\pm$). B1 for using $z = 0.8416$ or better. A1 for both equations with $\pm 1$ and $\pm 0.84$ or better |
| $10 = 1.8416\sigma$ | M1 | For solving their linear equations in $\mu$ and $\sigma$ |
| $\sigma =$ awrt $5.43$ | A1 | |
| $\mu =$ awrt $22.57$ | A1 (6) | |
---5. Rosie keeps bees. The amount of honey, in kg, produced by a hive of Rosie's bees in a season, is modelled by a normal distribution with a mean of 22 kg and a standard deviation of 10 kg .
\begin{enumerate}[label=(\alph*)]
\item Find the probability that a hive of Rosie's bees produces less than 18 kg of honey in a season.
The local bee keepers' club awards a certificate to every hive that produces more than 39 kg of honey in a season, and a medal to every hive that produces more than 50 kg in a season. Given that one of Rosie's bee hives is awarded a certificate
\item find the probability that this hive is also awarded a medal.\\
(5)
Sam also keeps bees. The amount of honey, in kg, produced by a hive of Sam's bees in a season, is modelled by a normal distribution with mean $\mu \mathrm { kg }$ and standard deviation $\sigma \mathrm { kg }$. The probability that a hive of Sam's bees produces less than 28 kg of honey in a season is 0.8413
Only 20\% of Sam's bee hives produce less than 18 kg of honey in a season.
\item Find the value of $\mu$ and the value of $\sigma$. Give your answers to 2 decimal places.\\
(6)\\
\begin{center}
\end{center}
\includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-11_2261_47_313_37}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2016 Q5 [14]}}