| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Forward transformation: find new statistics |
| Difficulty | Moderate -0.8 This is a straightforward application of standard formulas for mean and standard deviation with given data. Part (i) requires basic understanding that zero standard deviation means all values are equal to the mean. Part (ii) is a direct calculation using the standard deviation formula with known frequencies and mean. Both parts are routine exercises requiring recall and basic computation rather than problem-solving or insight. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
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| Fei | 4 | 2 | 0 | ||||||
| Graeme | 1 | 3 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| sd = 0, so all rides must cost the same i.e. the mean | B1*, B1 dep | Must see this and some relevant comment, e.g. no change o.e. |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 \times 2.5 + 3 \times 2.5 + 6 \times x = 3.76 \times 10\) | M1 | attempt to find cost of revolving drum ride |
| \(6x = 37.6 - 10\) and \(x = 4.6\) for revolving drum | A1, A1 | correct equation; correct \(x\) |
| \(\sigma^2 = (2.5^2 \times 1 + 2.5^2 \times 3 + 4.6^2 \times 6)/10 - 3.76^2\) and \(\sigma = 1.03\) | M1, A1 | substituting in correct variance formula; correct answer |
**(i) Standard deviation:**
sd = 0, so all rides must cost the same i.e. the mean | B1*, B1 dep | Must see this and some relevant comment, e.g. no change o.e. | **[2]**
**(ii) Variance calculation:**
$1 \times 2.5 + 3 \times 2.5 + 6 \times x = 3.76 \times 10$ | M1 | attempt to find cost of revolving drum ride |
$6x = 37.6 - 10$ and $x = 4.6$ for revolving drum | A1, A1 | correct equation; correct $x$ |
$\sigma^2 = (2.5^2 \times 1 + 2.5^2 \times 3 + 4.6^2 \times 6)/10 - 3.76^2$ and $\sigma = 1.03$ | M1, A1 | substituting in correct variance formula; correct answer | **[5]**
---4 The numbers of rides taken by two students, Fei and Graeme, at a fairground are shown in the following table.
\begin{center}
\begin{tabular}{ | l | c c c | }
\hline
& \begin{tabular}{ c }
Roller \\
coaster \\
\end{tabular} & \begin{tabular}{ c }
Water \\
slide \\
\end{tabular} & \begin{tabular}{ c }
Revolving \\
drum \\
\end{tabular} \\
\hline
Fei & 4 & 2 & 0 \\
Graeme & 1 & 3 & 6 \\
\hline
\end{tabular}
\end{center}
(i) The mean cost of Fei's rides is $\$ 2.50$ and the standard deviation of the costs of Fei's rides is $\$ 0$. Explain how you can tell that the roller coaster and the water slide each cost $\$ 2.50$ per ride. [2]\\
(ii) The mean cost of Graeme's rides is $\$ 3.76$. Find the standard deviation of the costs of Graeme's rides.
\hfill \mbox{\textit{CAIE S1 2010 Q4 [7]}}