| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Compare uniform with other distributions |
| Difficulty | Moderate -0.3 This is a straightforward S2 question requiring sketching two simple pdfs (uniform and quadratic), verbal interpretation, and solving a basic probability equation using integration. All techniques are standard with no novel insight needed, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{64.2 - 63}{\sqrt{12.25/23}} = 1.644\) | M1 dep | Standardise 64.2 with ∀n |
| A1 | \(z = 1.644\) or 1.645, must be + | |
| dep M1 | Find Φ(z), answer < 0.5 | |
| A1 | Answer, a.r.t. 0.05 or 5.0% | 4 |
| (ii) (a) \(63 + 1.645 \times \frac{3.5}{\sqrt{50}}\) | M1 | |
| \(k = 1.645\) (allow 1.64, 1.65) | B1 | |
| A1 | Answer, a.r.t. 63.8, allow >, ≥, ≈, c.w.o. | 3 |
| (b) \(P(< 63.8 \mid \mu = 65)\) | M1 | Use of correct meaning of Type II |
| \(\frac{63.81 - 65}{3.5/\sqrt{50}} = -2.3956\) | M1 | Standardise their \(c\) with ∀50 |
| \(\frac{3.5}{\sqrt{50}}\) | A1 | |
| \(z = (±) 2.40\) [or −2.424 or −2.404 etc] | A1 | Answer, a.r.t. 0.0083 [or 0.00867] |
| A1 | 4 | |
| (iii) B better; Type II error smaller (and same Type I error) | B2 ∀ | This answer: B2. "B because sample bigger": B1. [SR: Partial answer: B1] |
(i) $\frac{64.2 - 63}{\sqrt{12.25/23}} = 1.644$ | M1 dep | Standardise 64.2 with ∀n
| A1 | $z = 1.644$ or 1.645, must be +
| dep M1 | Find Φ(z), answer < 0.5
| A1 | Answer, a.r.t. 0.05 or 5.0% | 4
(ii) (a) $63 + 1.645 \times \frac{3.5}{\sqrt{50}}$ | M1 |
$k = 1.645$ (allow 1.64, 1.65) | B1 |
| A1 | Answer, a.r.t. 63.8, allow >, ≥, ≈, c.w.o. | 3
(b) $P(< 63.8 \mid \mu = 65)$ | M1 | Use of correct meaning of Type II
$\frac{63.81 - 65}{3.5/\sqrt{50}} = -2.3956$ | M1 | Standardise their $c$ with ∀50
$\frac{3.5}{\sqrt{50}}$ | A1 |
$z = (±) 2.40$ [or −2.424 or −2.404 etc] | A1 | Answer, a.r.t. 0.0083 [or 0.00867]
| A1 | 4
(iii) B better; Type II error smaller (and same Type I error) | B2 ∀ | This answer: B2. "B because sample bigger": B1. [SR: Partial answer: B1]7 Two continuous random variables $S$ and $T$ have probability density functions as follows.
$$\begin{array} { l l }
S : & f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\
0 & \text { otherwise } \end{cases} \\
T : & g ( x ) = \begin{cases} \frac { 3 } { 2 } x ^ { 2 } & - 1 \leqslant x \leqslant 1 \\
0 & \text { otherwise } \end{cases}
\end{array}$$
(i) Sketch on the same axes the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { g } ( x )$. [You should not use graph paper or attempt to plot points exactly.]\\
(ii) Explain in everyday terms the difference between the two random variables.\\
(iii) Find the value of $t$ such that $\mathrm { P } ( T > t ) = 0.2$.
\hfill \mbox{\textit{OCR S2 2007 Q7 [10]}}