S1 January 2006 Q7
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
(a) is taller than 188 cm, (3)
(b) weighs less than 97 kg. (2)
(c) 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. (3)
(d) Comment on the assumption that height and weight are independent. (1)
| Scheme | Marks |
|---|---|
| Let \(H\) be rv height of athletes, so \(H \sim \mathrm{N}(180, 5.2^2)\) \(\mathrm{P}(H \gt 188) = \mathrm{P}\left(Z \gt \dfrac{188 - 180}{5.2}\right) = \mathrm{P}(Z \gt 1.54) = 0.0618\) | M1A1A1 |
| (3) |
Notes
M1A1A1 \(\pm\) stand. \(\sqrt{}\), sq, awrt 0.062
| Scheme | Marks |
|---|---|
| Let \(W\) be rv weight of athletes, so \(W \sim \mathrm{N}(85, 7.1^2)\) \(\mathrm{P}(W \lt 97) = \mathrm{P}(Z \lt 1.69) = 0.9545\) | M1A1 |
| (2) |
Notes
M1A1 standardise, awrt 0.9545
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(H \gt 188\ \&\ W \gt 97) = 0.0618(1 - 0.9545)\) | M1A1ft |
| \(= 0.00281\) | A1 |
| (3) |
Notes
M1 allow (a)×(b) for M
A1 awrt 0.0028
(corrected from the printed mark scheme: it prints \(\mathrm{P}(H \gt 188\ \&\ W \lt 97)\); the question asks for weighs more than 97 kg, which is what \(1 - 0.9545\) gives)
| Scheme | Marks |
|---|---|
| Evidence suggests height and weight are positively correlated / linked Assumption of independence is not sensible | B1 |
| (1) | |
| (9 marks) |