| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2019 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Mixed sum threshold probability |
| Difficulty | Standard +0.3 This is a straightforward application of linear combinations of normal random variables with clear parameters. Part (a) requires summing 5 independent normals, part (b) applies a simple linear transformation (scaling), and part (c) combines multiple normals with given means and variances. All parts use standard techniques with no conceptual challenges beyond routine variance calculations and normal table lookups, making it slightly easier than average for Further Maths Statistics. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (a) | Total weight : N(5 × 205, 5 × 112) |
| P(Total ≥ 1000 g) = 0.8453 | M1 |
| Answer | Marks |
|---|---|
| [2] | For distribution |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (b) | Peeled weight : N(0.65 × 205, 0.652 × 112) |
| P(weight ≤ 150 g) = 0.9904 | B1 |
| Answer | Marks |
|---|---|
| [3] | For N and mean |
| Answer | Marks |
|---|---|
| BC | (mean = 133.25) |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (c) | Weight of smoothie |
| Answer | Marks |
|---|---|
| P(weight < 700 g) = 0.1474 | M1 |
| Answer | Marks |
|---|---|
| [4] | Method for mean FT their part (b) |
Question 3:
3 | (a) | Total weight : N(5 × 205, 5 × 112)
P(Total ≥ 1000 g) = 0.8453 | M1
A1
[2] | For distribution
BC
3 | (b) | Peeled weight : N(0.65 × 205, 0.652 × 112)
P(weight ≤ 150 g) = 0.9904 | B1
M1
A1
[3] | For N and mean
For variance
BC | (mean = 133.25)
(variance =
51.1225)
3 | (c) | Weight of smoothie
: N(2×133.25 + 20×22.5, 2×51.1225 + 20×2.72)
N(716.5, 248.045)
P(weight < 700 g) = 0.1474 | M1
M1
A1
A1
[4] | Method for mean FT their part (b)
Method for variance FT their part (b)
For both correct
BC3 The weights of bananas sold by a supermarket are modelled by a Normal distribution with mean 205 g and standard deviation 11 g .
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the total weight of 5 randomly selected bananas is at least 1 kg .
When a banana is peeled the change in its weight is modelled as being a reduction of $35 \%$.
\item Find the probability that the weight of a randomly selected peeled banana is at most 150 g
Andy makes smoothies. Each smoothie is made using 2 peeled bananas and 20 strawberries from the supermarket, all the items being randomly chosen. The weight of a strawberry is modelled by a Normal distribution with mean 22.5 g and standard deviation 2.7 g .
\item Find the probability that the total weight of a smoothie is less than 700 g .
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2019 Q3 [9]}}