| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Topic | Measures of Location and Spread |
| Type | Calculate variance from summary statistics |
| Difficulty | Moderate -0.3 Part (a) involves straightforward application of standard mean and standard deviation formulas with given summary statistics, requiring only substitution and calculator work. Part (b) is a combinatorics problem requiring the complementary counting approach (total arrangements minus arrangements with Es together), which is a standard technique but requires careful handling of repeated letters. Both parts are routine A-level exercises with no novel insight required, making this slightly easier than average overall. |
| Spec | 2.02g Calculate mean and standard deviation5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) Boys mean = 14.8 | B1 | Allow better 14.8013 and 14.6996 or rounding to 14.8 and 14.7 |
| Girls mean = 14.7 | ||
| Boys sd = 1.21 | B1 | Allow answers in range [1.21, 1.23] or 2.29 |
| Girls sd = 2.29 | B1 | |
| (ii) Almost the same mean but ages more spread for girls. | B1 | Award only for correct mean and sd. Comment must be made on mean and sd. |
| (b) Permutations of DFATD = \(\frac{5!}{2!} = 60\) | B1 | Sight of 60 or \(\frac{5!}{2!}\) |
| E's can be inserted in 3 of 6 positions | B1 | 20 seen or \(^6C_3\) |
| \(^6C_3 = 20\) | ||
| No of permutations = \(20 \times 60 = 1200\) | B1 | [7] |
(a) (i) Boys mean = 14.8 | B1 | Allow better 14.8013 and 14.6996 or rounding to 14.8 and 14.7
Girls mean = 14.7 |
Boys sd = 1.21 | B1 | Allow answers in range [1.21, 1.23] or 2.29
Girls sd = 2.29 | B1 |
(ii) Almost the same mean but ages more spread for girls. | B1 | Award only for correct mean and sd. Comment must be made on mean and sd.
(b) Permutations of DFATD = $\frac{5!}{2!} = 60$ | B1 | Sight of 60 or $\frac{5!}{2!}$
E's can be inserted in 3 of 6 positions | B1 | 20 seen or $^6C_3$
$^6C_3 = 20$ |
No of permutations = $20 \times 60 = 1200$ | B1 | [7] | Accept 1200 or $20 \times 60$\begin{enumerate}[label=(\alph*)]
\item A random sample of young people in a certain town comprised 312 boys and 253 girls. Denoting a boy's age by $x$ years and a girl's age by $y$ years, the following data were obtained:
$$\sum x = 4618, \quad \sum x^2 = 68812, \quad \sum y = 3719, \quad \sum y^2 = 55998.$$
\begin{enumerate}[label=(\roman*)]
\item Calculate the mean and standard deviation of the ages of the boys in the sample and also of the girls in the sample. [3]
\item Use these results to comment on the distribution of the ages of the boys and girls in the sample. [1]
\end{enumerate}
\item How many arrangements of the letters of the word DEFEATED are there in which the Es are separated from each other? [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2011 Q13 [7]}}