| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Find minimum/maximum n for probability condition |
| Difficulty | Standard +0.3 This is a straightforward S2 question requiring standard application of normal approximation to binomial (with continuity correction) and finding a critical value from tables. Both parts are routine textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(X\sim\text{B}(80,0.6)\approx\text{N}(48,19.2)\) | M1 A1 | 1st M1 writing/using normal approximation; 1st A1 correct mean and variance |
| \(P\!\left(Z>\frac{(n-0.5)-48}{\sqrt{19.2}}\right)<0.05\) | M1 | 2nd M1 continuity correction \((n\pm 0.5)\) or \(((n-1)\pm 0.5)\) |
| \(\frac{(n-0.5)-48}{\sqrt{19.2}}>1.6449\) | M1 B1 | 3rd M1 standardising; B1 for 1.6449 or better |
| \(n>55.7\), \(n=56\) | A1cao | A1 56 cao; dependent on all M marks. NB: use of binomial scores 0/6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0:\lambda=9\), \(H_1:\lambda>9\); \([B\sim\text{Po}(9)]\) | B1 | |
| \(P(B\leq 14)=0.9585\) / \(P(B\geq 15)=0.0415\ (<0.05)\) | M1 A1 | M1 for either probability (may be implied by correct CR); A1 allow any letter but must be CR not probability statement |
| \(B\geq 15\) |
## Question 6:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $X\sim\text{B}(80,0.6)\approx\text{N}(48,19.2)$ | M1 A1 | 1st M1 writing/using normal approximation; 1st A1 correct mean and variance |
| $P\!\left(Z>\frac{(n-0.5)-48}{\sqrt{19.2}}\right)<0.05$ | M1 | 2nd M1 continuity correction $(n\pm 0.5)$ or $((n-1)\pm 0.5)$ |
| $\frac{(n-0.5)-48}{\sqrt{19.2}}>1.6449$ | M1 B1 | 3rd M1 standardising; B1 for 1.6449 or better |
| $n>55.7$, $n=56$ | A1cao | A1 56 cao; dependent on all M marks. NB: use of binomial scores 0/6 |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0:\lambda=9$, $H_1:\lambda>9$; $[B\sim\text{Po}(9)]$ | B1 | |
| $P(B\leq 14)=0.9585$ / $P(B\geq 15)=0.0415\ (<0.05)$ | M1 A1 | M1 for either probability (may be implied by correct CR); A1 allow any letter but must be CR not probability statement |
| $B\geq 15$ | | |\begin{enumerate}
\item In a local council, $60 \%$ of households recycle at least half of their waste. A random sample of 80 households is taken.
\end{enumerate}
The random variable $X$ represents the number of households in the sample that recycle at least half of their waste.\\
(a) Using a suitable approximation, find the smallest number of households, $n$, such that
$$\mathrm { P } ( X \geqslant n ) < 0.05$$
The number of bags recycled per family per week was known to follow a Poisson distribution with mean 1.5
Following a recycling campaign, the council believes the mean number of bags recycled per family per week has increased. To test this belief, 6 families are selected at random and the total number of bags they recycle the following week is recorded.
The council wishes to test, at the 5\% level of significance, whether or not there is evidence that the mean number of bags recycled per family per week has increased.\\
(b) Find the critical region for the total number of bags recycled by the 6 families.
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\hfill \mbox{\textit{Edexcel S2 2018 Q6 [8]}}