| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Session | Specimen |
| Marks | 14 |
| Topic | Normal Distribution |
| Type | Mixed calculations with boundaries |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question covering standard normal distribution and binomial probability calculations. Parts (i)-(iii) require routine z-score calculations and table lookups, (iv) is a standard binomial probability, and (v) is a simple expectation calculation. While it has multiple parts, each component is a textbook exercise requiring only direct application of formulas without problem-solving or novel insight. |
| Spec | 2.04c Calculate binomial probabilities2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
**(i)** $z = 0.667$ B1
$P(t > 160) = 1 - \Phi(\text{their } z)$ M1
$= 1 - 0.7477 = 0.252$ A1 **[3]**
**(ii)** $\Phi^{-1}(0.95) = 1.645$ B1
their $1.645 = \frac{x - 150}{15}$ M1
$175$ A1 **[3]**
**(iii)** $Z$ values of $-1.333$ and $0.667$ B1
Obtain $1 - \Phi(1.333) = 0.0913$ B1
Attempt $\Phi(0.667)$ and subtraction $(= 0.7477 - 0.0913)$ M1
$(0.6564)$ $65.6\%$ **cao** A1 **[4]**
**(iv)** Use of Binomial distribution with $p = 0.4$, $q = 0.6$, $n = 18$ M1
$P(X > 8) = 1 - P(X \leq 8)$ M1
$= 1 - 0.7368 = 0.263$ A1 **[3]**
**(v)** $180$ B1 **[1]**14 The maximum pressure exerted by the blood on the arteries in a population of elderly male patients may be modelled by a random variable having a normal distribution with a mean of 150 and standard deviation 15, measured in suitable units.\\
(i) Find the probability that the maximum pressure for a randomly chosen patient is more than 160.\\
(ii) If the maximum pressure is found to be $t$ or more, the patient must be referred to a consultant. If $5 \%$ of the patients are referred to a consultant, find the value of $t$.\\
(iii) Find the percentage of patients whose maximum pressure is between 130 and 160 .
The probability that a randomly chosen patient attending a doctor's surgery has their blood pressure measured is 0.4 .\\
(iv) Find the probability that of 18 people attending a doctor's surgery more than 8 have their blood pressure measured, assuming that each measurement is random and independent of any other.\\
(v) If 450 patients visited the surgery in a week, find the expected number of patients whose blood pressure would be measured.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 Q14 [14]}}