A2 October 2021 Q3
3. A courier delivers parcels. The random variable \(X\) represents the number of parcels delivered successfully each day by the courier where \(X \sim \mathrm{B}(400, 0.64)\)
A random sample \(X_1, X_2, \ldots X_{100}\) is taken.
Estimate the probability that the mean number of parcels delivered each day by the courier is greater than 257 (4)
| Scheme | Marks | AO |
|---|---|---|
| \(\overline{X} \approx \mathrm{N}(256, \ldots)\) oe | M1 | 3.1a |
| \(\overline{X} \approx \mathrm{N}(256, 0.9216)\) | A1 | 1.1b |
| \(\mathrm{P}(\overline{X} \gt 257) = \mathrm{P}\left(Z \gt \dfrac{257 - 256}{\sqrt{\text{“}0.9216\text{”}}}\right)\ [= \text{awrt } 1.04]\) | dM1 | 3.4 |
| \(p = 0.1492\ldots\) | A1 | 1.1b |
| (4) | ||
| (4 marks) |
Notes
M1: For realising the need to use the CLT with correct mean
A1: For a correct normal stated
dM1: Dep on previous Method mark. Use of the normal model to find \(\mathrm{P}(\overline{X} \gt 257)\) If final answer is incorrect then we need to see the standardisation using their \(\sigma\).
A1: awrt 0.149 (0.14878… from calculator)
NB Allow awrt 0.148 if a continuity correction is used.