Aussie Maths & Science Teachers: Save your time with SmarterEd
The heights of students in a class were recorded.
The results for this class are displayed in the cumulative frequency graph shown.
The shortest student in this class is 130 cm and the tallest student is 180 cm.
Construct a box-plot for this class in the space below. (3 marks)
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\(Q_1(7.5 \ \text{students })=135\)
\(Q_3(22.5 \ \text{students })=160\)
\(\text{Median (15 students )}=140\)
A weather station records daily maximum temperatures
The five-number summary for the distribution of maximum temperatures for the month of February is displayed in the table below.
\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \ \rule[-1ex]{0pt}{0pt} & \textbf{Temperature (°C)} \\
\hline
\rule{0pt}{2.5ex} \text{Minimum} \rule[-1ex]{0pt}{0pt} & 16 \\
\hline
\rule{0pt}{2.5ex} Q_1 \rule[-1ex]{0pt}{0pt} & 21 \\
\hline
\rule{0pt}{2.5ex} \text{Median} \rule[-1ex]{0pt}{0pt} & 25 \\
\hline
\rule{0pt}{2.5ex} Q_3 \rule[-1ex]{0pt}{0pt} & 31 \\
\hline
\rule{0pt}{2.5ex} \text{Maximum} \rule[-1ex]{0pt}{0pt} & 39 \\
\hline
\end{array}
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| a. |  |
b. `IQR = Q_3-Q_1=31-21=10`
| `text(Lower fence)` | `= Q_1-1.5 xx IQR` |
| `= 21-1.5 xx 10` | |
| `= 6` |
| `text(Upper fence)` | `= Q_3 + 1.5 xx IQR` |
| `= 31 + 1.5 xx 10` | |
| `= 46` |
`text{Since 6 < 16 (minimum) and 46 > 39 (maximum)}`
`=>\ \text{There are no outliers.}`
The five-number summary below was determined from the sleep time, in hours, of a sample of 59 types of mammals.
\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \textbf{Statistic} \rule[-1ex]{0pt}{0pt} & \textbf{Sleep time (hours)} \\
\hline
\rule{0pt}{2.5ex} \text{minimum} \rule[-1ex]{0pt}{0pt} & \text{2.5} \\
\hline
\rule{0pt}{2.5ex} \text{first quartile} \rule[-1ex]{0pt}{0pt} & \text{8.0} \\
\hline
\rule{0pt}{2.5ex} \text{median} \rule[-1ex]{0pt}{0pt} & \text{10.5} \\
\hline
\rule{0pt}{2.5ex} \text{third quartile} \rule[-1ex]{0pt}{0pt} & \text{13.5} \\
\hline
\rule{0pt}{2.5ex} \text{maximum} \rule[-1ex]{0pt}{0pt} & \text{20.0} \\
\hline
\end{array}
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a. `IQR = Q_3-Q_1 = 13.5-8.0 = 5.5`
| `text(Lower fence)` | `= Q_1-1.5 xx IQR` |
| `= 8-1.5 xx 5.5` | |
| `= -0.25` |
| `text(Upper fence)` | `= Q_3 + 1.5 xx IQR` |
| `= 13.5 + 1.5 xx 5.5` | |
| `= 21.75` |
`text(S) text(ince) \ -0.25 < 2.5 \ text{(minimum value) and} \ 21.75 > 20.0 \ text{(maximum value)}`
`=> \ text(no outliers)`
b.
Two netball teams, Team A and Team B, each played 15 games in a tournament. For each team, the number of goals scored in each game was recorded.
The frequency table shows the data for Team A.
The data for Team B was analysed to create the box-plot shown.
Compare the distributions of the number of goals scored by the two teams. Support your answer with the construction of a box-plot for the data for Team A. (5 marks)
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`text(Team A: High = 28, Low = 19,)\ Q_1 = 23, Q_3 = 27,\ text{Median = 26}`
`text(Team A’s distribution is negatively skewed while)`♦♦ Mean mark 28%.
`text(Team B’s distribution is slightly positively skewed.)`
`text(The standard deviation of Team A’s distribution is)`
`text(smaller than Team B, as both its IQR and range is)`
`text(smaller.)`
`text(Team B is a more successful team at scoring goals)`
`text(as each value in its 5-point summary is higher than)`
`text(Team A’s equivalent value.)`
The heights of the 60 members of a choir were recorded. These results were grouped and then displayed as a cumulative frequency histogram and polygon.
The shortest person in the choir is 140 cm and the tallest is 190 cm.
Draw an accurate box-and-whisker plot to represent the data. (3 marks)
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`text(Low) = 140`
`text(High) = 190`
`text(Median) = 150\ \ \ \ text{(# People = 30)}`
`Q_1 = 145\ \ \ \ text{(# People = 15)}`
`Q_3 = 170\ \ \ \ text{(# People = 45)}`
`text(Box and Whisker)`
Terry and Kim each sat twenty class tests. Terry’s results on the tests are displayed in the box-and-whisker plot shown in part (i).
Draw a box-and-whisker plot to display Kim’s results below that of Terry’s results. (1 mark)
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Justify your answer by referring to the summary statistics and the skewness of the distributions. (4 marks)
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| i. |  |
ii. `text(50%)`
iii. `text(Terry’s results are more positively skewed than)`
`text(Kim’s and also have a higher limit high.)`
`text(However, Kim’s results are more consistent,)`
`text(showing a tighter IQR. They also have a)`
`text(significantly higher median than Terry’s and)`
`text(are evenly skewed.)`
`:.\ text(Kim’s results were better.)`