| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Measurement error modeling |
| Difficulty | Moderate -0.3 This is a straightforward application of continuous uniform distribution properties with standard formulas. Parts (a)-(d) require only direct recall of uniform distribution pdf, variance formula, and probability calculations. Part (e) adds a binomial component but remains routine. The question is slightly easier than average due to its formulaic nature and clear structure, though it does test multiple connected concepts across probability distributions. |
| Spec | 2.04c Calculate binomial probabilities2.04d Normal approximation to binomial5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{f}(d) = \frac{1}{5}\), \(-2.5\leq d \leq 2.5\) | B1 | 1st B1 for \(\frac{1}{5}\) (ignore range) |
| \(\text{f}(d) = 0\) otherwise | B1 | 2nd B1 fully correct distribution including ranges; allow \(<\) or \(\leq\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sqrt{\frac{(2.5-(-2.5))^2}{12}} = 1.4433...\) awrt 1.44 | M1 A1 | M1 for correct expression with square root; A1 awrt 1.44, allow \(\frac{5\sqrt{3}}{6}\). For integration allow \(\sqrt{\int_{-2.5}^{2.5}\frac{1}{5}x^2\,dx}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left[\frac{1-(-1)}{5}\right] = \frac{2}{5}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(X\sim\text{B}(10,\ 0.4)\) | M1 | 1st M1 for writing/using binomial with 10 and 'their (d)' |
| \(P(X\geq 6) = 1-P(X\leq 5) = 1-0.8338 = 0.1662\) awrt 0.166 | M1 A1 | 2nd M1 for writing/using \(1-P(X\leq 5)\); A1 awrt 0.166. Alt if 'their(d)'>0.5: use \(Y\sim B(10,1-\text{their d})\), 2nd M1 for \(P(Y\leq 4)\) |
## Question 3:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{f}(d) = \frac{1}{5}$, $-2.5\leq d \leq 2.5$ | B1 | 1st B1 for $\frac{1}{5}$ (ignore range) |
| $\text{f}(d) = 0$ otherwise | B1 | 2nd B1 fully correct distribution including ranges; allow $<$ or $\leq$ |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sqrt{\frac{(2.5-(-2.5))^2}{12}} = 1.4433...$ **awrt 1.44** | M1 A1 | M1 for correct expression with square root; A1 awrt 1.44, allow $\frac{5\sqrt{3}}{6}$. For integration allow $\sqrt{\int_{-2.5}^{2.5}\frac{1}{5}x^2\,dx}$ |
### Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0$ | B1 | |
### Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left[\frac{1-(-1)}{5}\right] = \frac{2}{5}$ | B1 | |
### Part (e)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $X\sim\text{B}(10,\ 0.4)$ | M1 | 1st M1 for writing/using binomial with 10 and 'their (d)' |
| $P(X\geq 6) = 1-P(X\leq 5) = 1-0.8338 = 0.1662$ **awrt 0.166** | M1 A1 | 2nd M1 for writing/using $1-P(X\leq 5)$; A1 awrt 0.166. Alt if 'their(d)'>0.5: use $Y\sim B(10,1-\text{their d})$, 2nd M1 for $P(Y\leq 4)$ |
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\item Albert uses scales in his kitchen to weigh some fruit.
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The random variable $D$ represents, in grams, the weight of the fruit given by the scales minus the true weight of the fruit. The random variable $D$ is uniformly distributed over the interval $[ - 2.5,2.5 ]$\\
(a) Specify the probability density function of $D$\\
(b) Find the standard deviation of $D$
Albert weighs a banana on the scales.\\
(c) Write down the probability that the weight given by the scales equals the true weight of the banana.\\
(d) Find the probability that the weight given by the scales is within 1 gram of the banana's true weight.
Albert weighs 10 bananas on the scales, one at a time.\\
(e) Find the probability that the weight given by the scales is within 1 gram of the true weight for at least 6 of the bananas.
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\hfill \mbox{\textit{Edexcel S2 2018 Q3 [9]}}