S1 June 2006 Q5
5. From experience a high-jumper knows that he can clear a height of at least 1.78 m once in 5 attempts. He also knows that he can clear a height of at least 1.65 m on 7 out of 10 attempts.
Assuming that the heights the high-jumper can reach follow a Normal distribution,
(a) draw a sketch to illustrate the above information, (3)
(b) find, to 3 decimal places, the mean and the standard deviation of the heights the high-jumper can reach, (6)
(c) calculate the probability that he can jump at least 1.74 m. (3)
| Scheme | Marks |
|---|---|
| Bell Shape | B1 |
| 1.78 & 0.2 | B1 |
| 1.65 & 0.3 | B1 |
| (3) |
Notes
2 separate sketches OK.
Accept clear alternatives to 0.3: 0.7/0.5/0.2
| Scheme | Marks |
|---|---|
| \(\dfrac{1.78 - \mu}{\sigma} = 0.8416 \Rightarrow 1.78 - \mu = 0.8416\sigma\) | M1 B1 |
| \(\dfrac{1.65 - \mu}{\sigma} = -0.5244 \Rightarrow 1.65 - \mu = -0.5244\sigma\) | B1 |
| Solving gives \(\mu = 1.70, \sigma = 0.095\) | M1A1A1 |
| (6) |
Notes
M1 either for method
B1 0.8416
B1 (−)0.5244
N.B. awrt 0.84, 0.52 B1B0
M1A1A1 awrt 1.7, 0.095 cao
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(\text{height} \geqslant 1.74) = 1 - \mathrm{P}(\text{height} \lt 1.74)\) | M1 |
| \(= 1 - \mathrm{P}\left(Z \lt \dfrac{1.74 - 1.70}{0.095}\right)\) | M1 |
| \(= 1 - \mathrm{P}(Z \lt 0.42) = 0.3372\) | A1 |
| (3) | |
| (12 marks) |
Notes
M1 ‘one minus’
M1 standardise with their mu and sigma
A1 awrt 0.337