| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Combined events across distributions |
| Difficulty | Moderate -0.8 This is a straightforward S1 normal distribution question requiring basic standardization and table lookup for parts (a) and (b), simple probability multiplication for independence in (c), and a common-sense comment for (d). All techniques are routine with no problem-solving insight needed, making it easier than average. |
| Spec | 2.03a Mutually exclusive and independent events2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation2.04g Normal distribution properties: empirical rule (68-95-99.7), points of inflection |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(H\) be rv height, \(H \sim N(180, 5.2^2)\); \(P(H>188) = P\!\left(Z > \frac{188-180}{5.2}\right) = P(Z>1.54) = 0.0618\) | M1A1A1 | \(\pm\) stand, \(\sqrt{}\), sq; awrt 0.062 (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(W\) be rv weight, \(W \sim N(85, 7.1^2)\); \(P(W<97) = P(Z<1.69) = 0.9545\) | M1A1 | standardise; awrt 0.9545 (2) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(H>188 \text{ \& } W<97) = 0.0618(1-0.9545) = 0.00281\) | M1A1ft, A1 | allow (a)\(\times\)(b) for M; awrt 0.0028 (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Evidence suggests height and weight are positively correlated/linked; assumption of independence is not sensible | B1 | (1) |
# Question 7:
## Part (a)
| Let $H$ be rv height, $H \sim N(180, 5.2^2)$; $P(H>188) = P\!\left(Z > \frac{188-180}{5.2}\right) = P(Z>1.54) = 0.0618$ | M1A1A1 | $\pm$ stand, $\sqrt{}$, sq; awrt 0.062 (3) |
## Part (b)
| Let $W$ be rv weight, $W \sim N(85, 7.1^2)$; $P(W<97) = P(Z<1.69) = 0.9545$ | M1A1 | standardise; awrt 0.9545 (2) |
## Part (c)
| $P(H>188 \text{ \& } W<97) = 0.0618(1-0.9545) = 0.00281$ | M1A1ft, A1 | allow (a)$\times$(b) for M; awrt 0.0028 (3) |
## Part (d)
| Evidence suggests height and weight are positively correlated/linked; assumption of independence is not sensible | B1 | (1) |7. The heights of a group of athletes are modelled by a normal distribution with mean 180 cm and a standard deviation 5.2 cm . The weights of this group of athletes are modelled by a normal distribution with mean 85 kg and standard deviation 7.1 kg .
Find the probability that a randomly chosen athlete
\begin{enumerate}[label=(\alph*)]
\item is taller than 188 cm ,
\item weighs less than 97 kg .\\
(2)
\item Assuming that for these athletes height and weight are independent, find the probability that a randomly chosen athlete is taller than 188 cm and weighs more than 97 kg .
\item Comment on the assumption that height and weight are independent.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2006 Q7 [9]}}