| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Geometric/graphical PDF with k |
| Difficulty | Standard +0.3 This is a straightforward S2 question covering standard PDF concepts: finding k using area=1, calculating mean by symmetry and variance by integration, applying independence for repeated observations, and stating the Central Limit Theorem result. All techniques 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 integration5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((11 - 3)k = 1\) | M1 | Use area = 1 [e.g. \(\int kdx = 1\) with limits 3, 11] |
| \(k = 1/8\) | A1 2 | Answer 1/8 or 0.125 only |
| (ii) \(\mu = \frac{1}{2}(3 + 11) = 7\) | B1 | Mean 7, cwd |
| \(\int_3^{11} x \cdot f(x)dx = [\frac{x^4}{24}]_{= 54 \frac{1}{4}}\) | M1 | Attempt \(\int x^2f(x)dx\), correct limits |
| A1 | Indefinite integral \(\frac{x^3}{3k}\), their k | |
| \(\sigma^2 = 54\frac{1}{4} - 7^2 = 5\frac{1}{4}\) | M1 | Subtract their μ² |
| A1 5 | Correct answer, \(5\frac{1}{4}\) or a.r.t. 5.33 | |
| (iii) \(P(X < 9) = 6k \quad [= \frac{3}{4}]\) | B1V | Correct p for their k |
| \((\frac{3}{4})^3 = \frac{27}{64}\) or 0.421875 | M1 | Work out their \(p^3\), \(0 < p < 1\) |
| A1 3 | Answer \(\frac{27}{64}\) or a.r.t. 0.422 | |
| (iv) Normal | B1 | "Normal" distribution stated |
| Mean is 7 | B1V | Mean same as in (ii) √ |
| Variance is \(5\frac{1}{4} + 32 (= \frac{k}{3})\) | B1V 3 | Variance is (iii) + 32 √ [not V errors] |
**(i)** $(11 - 3)k = 1$ | M1 | Use area = 1 [e.g. $\int kdx = 1$ with limits 3, 11]
$k = 1/8$ | A1 2 | Answer 1/8 or 0.125 only
**(ii)** $\mu = \frac{1}{2}(3 + 11) = 7$ | B1 | Mean 7, cwd
$\int_3^{11} x \cdot f(x)dx = [\frac{x^4}{24}]_{= 54 \frac{1}{4}}$ | M1 | Attempt $\int x^2f(x)dx$, correct limits
| A1 | Indefinite integral $\frac{x^3}{3k}$, their k
$\sigma^2 = 54\frac{1}{4} - 7^2 = 5\frac{1}{4}$ | M1 | Subtract their μ²
| A1 5 | Correct answer, $5\frac{1}{4}$ or a.r.t. 5.33
**(iii)** $P(X < 9) = 6k \quad [= \frac{3}{4}]$ | B1V | Correct p for their k
$(\frac{3}{4})^3 = \frac{27}{64}$ or 0.421875 | M1 | Work out their $p^3$, $0 < p < 1$
| A1 3 | Answer $\frac{27}{64}$ or a.r.t. 0.422
**(iv)** Normal | B1 | "Normal" distribution stated
Mean is 7 | B1V | Mean same as in (ii) √
Variance is $5\frac{1}{4} + 32 (= \frac{k}{3})$ | B1V 3 | Variance is (iii) + 32 √ [not V errors]7 The continuous random variable $X$ has the probability density function shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{b69b1fe8-790d-4727-a892-8ab2ade08962-3_364_766_1229_699}\\
(i) Find the value of the constant $k$.\\
(ii) Write down the mean of $X$, and use integration to find the variance of $X$.\\
(iii) Three observations of $X$ are made. Find the probability that $X < 9$ for all three observations.\\
(iv) The mean of 32 observations of $X$ is denoted by $\bar { X }$. State the approximate distribution of $\bar { X }$, giving its mean and variance.
\section*{[Question 8 is printed overleaf.]}
\hfill \mbox{\textit{OCR S2 2005 Q7 [13]}}