Easy -1.3 This is a straightforward S1 question requiring only direct reading from a stem-and-leaf diagram (mode, quartiles), application of standard formulas (mean, standard deviation), and basic interpretation of skewness. All techniques are routine recall with no problem-solving or novel insight required.
Over a period of time, the number of people \(x\) leaving a hotel each morning was recorded. These data are summarised in the stem and leaf diagram below.
Number leaving
3
2 means 32
Totals
2
7
9
9
(3)
3
2
2
3
5
6
(5)
4
0
1
4
8
9
(5)
5
2
3
3
6
6
6
(7)
6
0
1
4
5
(4)
7
2
3
(2)
8
1
(1)
For these data,
write down the mode,
find the values of the three quartiles.
Given that \(\Sigma x = 1335\) and \(\Sigma x ^ { 2 } = 71801\), find
the mean and the standard deviation of these data.
One measure of skewness is found using
$$\frac { \text { mean - mode } } { \text { standard deviation } } \text {. }$$
Evaluate this measure to show that these data are negatively skewed.
Give two other reasons why these data are negatively skewed.
\begin{enumerate}
\item Over a period of time, the number of people $x$ leaving a hotel each morning was recorded. These data are summarised in the stem and leaf diagram below.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
Number leaving & & & & 3 & & 2 means 32 & Totals \\
\hline
2 & 7 & 9 & 9 & & & & (3) \\
\hline
3 & 2 & 2 & 3 & 5 & 6 & & (5) \\
\hline
4 & 0 & 1 & 4 & 8 & 9 & & (5) \\
\hline
5 & 2 & 3 & 3 & 6 & 6 & 6 & (7) \\
\hline
6 & 0 & 1 & 4 & 5 & & & (4) \\
\hline
7 & 2 & 3 & & & & & (2) \\
\hline
8 & 1 & & & & & & (1) \\
\hline
\end{tabular}
\end{center}
For these data,\\
(a) write down the mode,\\
(b) find the values of the three quartiles.
Given that $\Sigma x = 1335$ and $\Sigma x ^ { 2 } = 71801$, find\\
(c) the mean and the standard deviation of these data.
One measure of skewness is found using
$$\frac { \text { mean - mode } } { \text { standard deviation } } \text {. }$$
(d) Evaluate this measure to show that these data are negatively skewed.\\
(e) Give two other reasons why these data are negatively skewed.\\
\hfill \mbox{\textit{Edexcel S1 2006 Q1 [14]}}