| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Conditional probability with normal |
| Difficulty | Standard +0.3 This is a straightforward S1 normal distribution question requiring standard z-score calculations and basic conditional probability. Parts (a) and (b) are routine applications of tables, while part (c) introduces conditional probability P(X < 24 | X > 22) which requires understanding but is still a standard technique taught in S1. Slightly above average difficulty due to the conditional probability element, but well within expected S1 scope. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(Z < \frac{20-21.6}{1.8}) = P(Z < -0.89) = 0.1867\) | M2 A1 | |
| \(P(Z > \frac{24-21.6}{1.8}) = P(Z > 1.33) = 0.0918\) | M1 A1 | |
| \(\therefore\) in 90 days expect \(0.0918 \times 90 = 8.26\) \(\therefore\) 8 times | M1 A1 | |
| \(P(X < 24 \mid X > 22) = \frac{P(22 < X < 24)}{P(X > 22)}\) | M1 A1 | |
| \(P(X > 22) = P(Z > \frac{22-21.6}{1.8}) = P(Z > 0.22) = 0.4129\) | M1 A1 | |
| \(P(22 < X < 24) = P(X > 22) - P(X > 24) = 0.3211\) | M1 | |
| \(\therefore\) require \(\frac{0.3211}{0.4129} = 0.778\) (3sf) | A1 | (13) |
$P(Z < \frac{20-21.6}{1.8}) = P(Z < -0.89) = 0.1867$ | M2 A1 |
$P(Z > \frac{24-21.6}{1.8}) = P(Z > 1.33) = 0.0918$ | M1 A1 |
$\therefore$ in 90 days expect $0.0918 \times 90 = 8.26$ $\therefore$ 8 times | M1 A1 |
$P(X < 24 \mid X > 22) = \frac{P(22 < X < 24)}{P(X > 22)}$ | M1 A1 |
$P(X > 22) = P(Z > \frac{22-21.6}{1.8}) = P(Z > 0.22) = 0.4129$ | M1 A1 |
$P(22 < X < 24) = P(X > 22) - P(X > 24) = 0.3211$ | M1 |
$\therefore$ require $\frac{0.3211}{0.4129} = 0.778$ (3sf) | A1 | (13)4. Alan runs on a treadmill each day for as long as he can at 7 miles per hour. The length of time for which he runs is normally distributed with a mean of 21.6 minutes and a standard deviation of 1.8 minutes.
\begin{enumerate}[label=(\alph*)]
\item Calculate the probability that on any one day Alan will run for less than 20 minutes.
\item Estimate the number of times in a ninety-day period that Alan will run for more than 24 minutes.
\item On a particular day Alan is still running after 22 minutes. Find the probability that he will stop running in the next 2 minutes.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q4 [13]}}