S3 June 2016 Q4
4. The weights of eggs are normally distributed with mean 60 g and standard deviation 5 g
Sairah chooses 2 eggs at random.
Sairah is packing eggs into cartons. The weight of an empty egg carton is normally distributed with mean 40 g and standard deviation 1.5 g
| Scheme | Marks |
|---|---|
| \(X_i\) be rv ‘weight of \(i\)th randomly chosen egg’ | |
| \(\mathrm{E}(X_1 - X_2) = 0\) | B1 |
| \(\mathrm{Var}(X_1 - X_2) = 2 \times 5^2 = 50\) | B1 |
| \(\mathrm{P}(|X_1 - X_2| \gt 2) = 2\mathrm{P}(X_1 - X_2 \gt 2)\) | M1 |
| \(= 2\mathrm{P}\left(Z \gt \dfrac{2}{\sqrt{50}}\right)\) \(= 2\mathrm{P}(Z \gt 0.2828\ldots)\) | dM1 |
| \(= 2(1 - 0.6103) = 0.7794\) awrt 0.777-0.779 | A1 |
| (5) |
Notes
B1 for 0
B1 for 50
M1 for \(|X_1 - X_2| \gt 2\) seen. Accept \(X_1 - X_2 \gt 2\) provided a subsequent doubling of the probability is seen. i.e. 0.3897 x 2.
dM1 standardise with their 0 and their \(\sqrt{50}\) dependent on previous M.
A1 awrt 0.777-0.779
| Scheme | Marks |
|---|---|
| \(W = C + X_1 + X_2 + \ldots + X_{12}\) | |
| \(\mathrm{E}(W) = 40 + 12 \times 60 = 760\) | B1 |
| \(\mathrm{Var}(W) = 1.5^2 + 12 \times 5^2\) | M1 |
| \(= 302.25\) | A1 |
| Distribution is \(\mathrm{N}(760, 302.25)\) | |
| (3) |
Notes
B1 for 760
M1 requires squares
A1 cao
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(W \gt 800) = \mathrm{P}\left(Z \gt \dfrac{800 - 760}{\sqrt{302.25}}\right)\) \(= 1 - \mathrm{P}(Z \lt 2.3007\ldots)\) | M1 |
| \(= 0.0107\) awrt 0.0107 | A1 |
| (2) | |
| (10 marks) |
Notes
Must be finding correct probability (ie \(\mathrm{P}(W \gt 800)\) or \(\mathrm{P}(Z \gt 2.3007\ldots)\) etc) and standardise with 800 and their 760 and their \(\sqrt{302.25}\)
A1 awrt 0.0107 from correct working.