| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2020 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Mixed sum threshold probability |
| Difficulty | Standard +0.8 Part (a) requires understanding sampling distributions and calculating P(X̄ ≥ 200) with variance σ²/n. Part (b) is more demanding: students must form the linear combination 8S - 2M - L, find its mean and variance correctly (handling the subtraction and multiple coefficients), then compute a probability. This goes beyond routine normal distribution questions and requires careful algebraic manipulation of variances, making it moderately challenging for Further Maths students. |
| Spec | 5.04b Linear combinations: of normal distributions |
| Bag size | Mean | Standard deviation |
| Small | 51.5 | 1.1 |
| Medium | 100.7 | 1.6 |
| Large | 201.3 | 1.7 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (a) | Mean of 2 large bags ⁓ N(201.3,1.72/2 ) |
| Answer | Marks |
|---|---|
| P(Mean > 200) = 0.8603 | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.3 |
| Answer | Marks |
|---|---|
| 3.4 | For Normal and mean |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (b) | Need distribution of 8 small – 2 medium – 1 large |
| Answer | Marks |
|---|---|
| P( > 0) = 0.9865 | M1 |
| Answer | Marks |
|---|---|
| [5] | 3.1b |
| Answer | Marks |
|---|---|
| 3.4 | For distribution required. soi eg correct mean or variance |
Question 3:
3 | (a) | Mean of 2 large bags ⁓ N(201.3,1.72/2 )
i.e. N(201.3, 1.445)
P(Mean > 200) = 0.8603 | M1
A1
B1
[3] | 3.3
1.1
3.4 | For Normal and mean
Allow M1 for total N(402.6, 5.78) even if not used
For correct variance
3 | (b) | Need distribution of 8 small – 2 medium – 1 large
mean = 8 × 51.5 – 2 × 100.7 – 1 × 201.3
variance = 8 × 1.12 + 2 × 1.62 + 1 × 1.72
so distribution is N(9.3, 17.69)
P( > 0) = 0.9865 | M1
M1
M1
A1
A1
[5] | 3.1b
3.3
1.1
1.1
3.4 | For distribution required. soi eg correct mean or variance
Method for mean
Method for variance
Correct distribution3 A supermarket sells cashew nuts in three different sizes of bag: small, medium and large. The weights in grams of the nuts in each type of bag are modelled by independent Normal distributions as shown in Table 3.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | l | r | c | }
\hline
Bag size & Mean & Standard deviation \\
\hline
Small & 51.5 & 1.1 \\
\hline
Medium & 100.7 & 1.6 \\
\hline
Large & 201.3 & 1.7 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 3}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the mean weight of two randomly selected large bags is at least 200 g .
\item Find the probability that the total weight of eight randomly selected small bags is greater than the total weight of two randomly selected medium bags and one randomly selected large bag.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2020 Q3 [8]}}