Edexcel S1 2008 June — Question 2 14 marks

Exam BoardEdexcel
ModuleS1 (Statistics 1)
Year2008
SessionJune
Marks14
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMeasures of Location and Spread
TypeInterpret or analyse given back-to-back stem-and-leaf
DifficultyModerate -0.8 This is a straightforward S1 statistics question requiring standard calculations from a stem-and-leaf diagram: mode (reading off), quartiles (counting positions), mean (sum/n), standard deviation (formula application), and skewness (given formula). All techniques are routine recall with no problem-solving insight needed, though the multi-part nature and arithmetic involved make it slightly more substantial than trivial index law questions.
Spec2.02a Interpret single variable data: tables and diagrams2.02f Measures of average and spread2.02g Calculate mean and standard deviation
2. The age in years of the residents of two hotels are shown in the back to back stem and leaf diagram below. Abbey Hotel \(8 | 5 | 0\) means 58 years in Abbey hotel and 50 years in Balmoral hotel Balmoral Hotel
(1)20
(4)97511
(4)983126(1)
(11)999976653323447(3)
(6)9877504005569(6)
\multirow[t]{3}{*}{(1)}85000013667(9)
6233457(6)
7015(3)
For the Balmoral Hotel,
  1. write down the mode of the age of the residents,
  2. find the values of the lower quartile, the median and the upper quartile.
    1. Find the mean, \(\bar { x }\), of the age of the residents.
    2. Given that \(\sum x ^ { 2 } = 81213\) find the standard deviation of the age of the residents. One measure of skewness is found using $$\frac { \text { mean - mode } } { \text { standard deviation } }$$
  4. Evaluate this measure for the Balmoral Hotel. For the Abbey Hotel, the mode is 39 , the mean is 33.2 , the standard deviation is 12.7 and the measure of skewness is - 0.454
  5. Compare the two age distributions of the residents of each hotel.
Question 2:
Part (a)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(50\)B1
Part (b)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(Q_1 = 45\)B1
\(Q_2 = 50.5\)B1 ONLY
\(Q_3 = 63\)B1
Part (c)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\text{Mean} = \dfrac{1469}{28} = 52.464286\ldots\)M1A1 M1 for 1469 between 1300 and 1600 divided by 28; awrt 52.5
\(\text{Sd} = \sqrt{\dfrac{81213}{28} - \left(\dfrac{1469}{28}\right)^2}\)M1 M1 for correct formula including sq root
\(= 12.164\ldots\) or \(12.387216\ldots\) (for divisor \(n-1\))A1 awrt 12.2 or 12.4
Part (d)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\dfrac{52.46\ldots - 50}{sd} =\) awrt \(0.20\) or \(0.21\)M1A1 M1 for values correctly substituted; accept 0.2 as special case of awrt 0.20
Part (e)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
1. mode/median/mean Balmoral \(>\) mode/median/mean AbbeyB1 Technical terms required in correct context
2. Balmoral sd \(<\) Abbey sd, or Balmoral range \(<\) Abbey range, or Balmoral IQR \(>\) Abbey IQRB1
3. Balmoral positive skew or almost symmetrical AND Abbey negative skew, or Balmoral less skew using value from (d)B1 Only one comment of each type; max 3 marks 
## Question 2:

### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $50$ | B1 | |

### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $Q_1 = 45$ | B1 | |
| $Q_2 = 50.5$ | B1 | ONLY |
| $Q_3 = 63$ | B1 | |

### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\text{Mean} = \dfrac{1469}{28} = 52.464286\ldots$ | M1A1 | M1 for 1469 between 1300 and 1600 divided by 28; awrt 52.5 |
| $\text{Sd} = \sqrt{\dfrac{81213}{28} - \left(\dfrac{1469}{28}\right)^2}$ | M1 | M1 for correct formula including sq root |
| $= 12.164\ldots$ or $12.387216\ldots$ (for divisor $n-1$) | A1 | awrt 12.2 or 12.4 |

### Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\dfrac{52.46\ldots - 50}{sd} =$ awrt $0.20$ or $0.21$ | M1A1 | M1 for values correctly substituted; accept 0.2 as special case of awrt 0.20 |

### Part (e)
| Answer/Working | Marks | Guidance |
|---|---|---|
| 1. mode/median/mean Balmoral $>$ mode/median/mean Abbey | B1 | Technical terms required in correct context |
| 2. Balmoral sd $<$ Abbey sd, or Balmoral range $<$ Abbey range, or Balmoral IQR $>$ Abbey IQR | B1 | |
| 3. Balmoral positive skew or almost symmetrical AND Abbey negative skew, or Balmoral less skew using value from (d) | B1 | Only one comment of each type; max 3 marks |

---
2. The age in years of the residents of two hotels are shown in the back to back stem and leaf diagram below.

Abbey Hotel $8 | 5 | 0$ means 58 years in Abbey hotel and 50 years in Balmoral hotel Balmoral Hotel

\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
(1) & 2 & 0 &  &  \\
\hline
(4) & 9751 & 1 &  &  \\
\hline
(4) & 9831 & 2 & 6 & (1) \\
\hline
(11) & 99997665332 & 3 & 447 & (3) \\
\hline
(6) & 987750 & 4 & 005569 & (6) \\
\hline
\multirow[t]{3}{*}{(1)} & 8 & 5 & 000013667 & (9) \\
\hline
 &  & 6 & 233457 & (6) \\
\hline
 &  & 7 & 015 & (3) \\
\hline
\end{tabular}
\end{center}

For the Balmoral Hotel,
\begin{enumerate}[label=(\alph*)]
\item write down the mode of the age of the residents,
\item find the values of the lower quartile, the median and the upper quartile.
\item \begin{enumerate}[label=(\roman*)]
\item Find the mean, $\bar { x }$, of the age of the residents.
\item Given that $\sum x ^ { 2 } = 81213$ find the standard deviation of the age of the residents.

One measure of skewness is found using

$$\frac { \text { mean - mode } } { \text { standard deviation } }$$
\end{enumerate}\item Evaluate this measure for the Balmoral Hotel.

For the Abbey Hotel, the mode is 39 , the mean is 33.2 , the standard deviation is 12.7 and the measure of skewness is - 0.454
\item Compare the two age distributions of the residents of each hotel.
\end{enumerate}

\hfill \mbox{\textit{Edexcel S1 2008 Q2 [14]}}
This paper (7 questions)
View full paper
Q1 9 Q2 14 Q3 11 Q4 15 Q5 10 Q6 5 Q7 12