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Data Analysis, SM-Bank 056 MC

The dot plot below shows the times, in seconds, of 40 runners in the qualifying heats of their 800 m club championship.
 

The shape of this distribution is best described as

  1. positively skewed with one outlier.
  2. approximately symmetric with one outlier.
  3. approximately symmetric with no outliers.
  4. negatively skewed with one outlier.
Show Answers Only

\(A\)

Show Worked Solution

\(\text{Distribution is positively skewed (tail stretches to the right)} \)

\(\text{and 146 is a possible outlier}\)

\(\Rightarrow A\)

Data Analysis, SM-Bank 055

The stem plot below shows the distribution of mathematics test scores for a class of 23 students.
 


 

For this class:

  1. What was the range of test scores?  (1 mark)
  2. What was the mean test score, correct to 1 decimal place?  (2 marks)
  3. What was the median test mark?  (1 mark)
  4. What was the mode of the test scores?  (1 mark)
  5. A student sits the test late and scores a mark of 58. Describe the change, if any, in the range, the mean, the median and the mode.  (2 marks)
Show Answers Only

a.    \(49\)

b.    \(64.3\ \text{(1 d.p.)}\)

c.    \(68\)

d.    \(\text{Range → unchanged}\)

\(\text{Mean → reduced}\)

\(\text{Median → reduced}\)

\(\text{Mode → unchanged}\)

Show Worked Solution
a.    \(\text{Range}\) \(=89-40\)
    \(=49\)

 

b.   \(\text{Mean}\) \(=\dfrac{40+41+2\times 44+52+57+3\times 59+65+66+2\times 68+2\times 69+2\times 70+75+76+77+78+85+89}{23}\)
    \(=\dfrac{1480}{23}\)
    \(=64.347\dots\)
    \(\approx 64.3\ \text{(1 d.p.)}\)

 

c.    \(\text{Median}\) \(=\dfrac{23+1}{2}\ \text{score}\)
    \(=\text{12th score}\)
    \(=68\)

 
d.    \(\text{Range}\ \longrightarrow\ \text{stays the same}\)
 

\(\text{Mean}\) \(=\dfrac{1480+58}{24}\)
  \(=64.1\ \text{(1 d.p.)}\)
  \(\therefore\ \text{Mean is reduced}\)

 

\(\text{Median}\) \(=\dfrac{\text{12th score+13th score}}{2}\)
  \(=\dfrac{66+68}{2}\)
  \(=67\)
  \(\therefore\ \text{Median is reduced}\)

 

\(\text{Mode}\ \longrightarrow\ \text{stays the same}\)

 

Data Analysis, SM-Bank 054

The following ordered stem plot shows the areas, in square kilometres, of 27 suburbs of a large city.

\begin{array} {r|lll}
\textbf{Stem} & \textbf{Leaf} \\
\hline 1 & 5\ 6\ 7\ 8  \\
2 & 1\ 2\ 4\ 5 \ 6\ 8\ 9\ 9 \\
3 & 0\ 1\ 1\ 2\ 2\ 8\ 9 \\
4 & 0\ 4\ 7 \\
5 & 0\ 1 \\
6 & 1\ 9 \\
7 &  \\
8 & 4 \\
\end{array}		 \(\text{key:  }1|6=1.6\ \text{km}^2\)
  1. For these suburbs
    i.     What is the median, in square kilometres?  (1 mark)

    ii.    What is the range, in square kilometres?  (1 mark)

  2. What is the possible outlier?  (1 mark)
  3. Briefly describe the skewness of the data.  (1 mark)
Show Answers Only

a.    i.    \(3.1\ \text{km}^2\)

ii.   \(6.9\)

b.    \(8.4\ \text{km}^2\)

c.    \(\text{Positively skewed.}\)

Show Worked Solution
a.    i.     \(\text{Median}\) \(=\dfrac{27+1}{2}\)
      \(=\ \text{14 th score}\)
    \(\therefore\ \text{Median}\) \(=3.1\ \text{km}^2\)
       
  ii. \(\text{Range}\) \(=8.4-1.5\)
      \(=6.9\)

 
b.    \(8.4\ \text{km}^2\ \text{is a possible outlier}\)

c.    \(\text{The data is positively skewed as the tail is to the right.}\)

Data Analysis, SM-Bank 053

The following ordered stem plot shows the percentage of homes connected to broadband internet for 24 countries in 2007.
 

   CORE, FUR1 2013 VCAA 1-2 MC 
 

  1. How many of these countries had more than 22% of homes connected to broadband internet in 2007?  (1 mark)
  2. What was the median percentage of homes connected to broadband?  (1 mark)
  3. For these countries, what is the modal percentage of homes connected to broadband?  (1 mark)
Show Answers Only

a.    \(19\)

b.    \(29.5\)

c.    \(31\)

Show Worked Solution

a.    \(\text{There are 19 values greater than 22%}\)

 

b.     \(\text{Median}\) \(=\dfrac{\text{12th + 13th}}{2}\)
    \(=\dfrac{29+30}{2}\)
    \(=29.5\)

 
c.    \(\text{Mode} = 31\)

Data Analysis, SM-Bank 052 MC

The stem plot below displays the average number of decayed teeth in 12-year-old children from `31` countries.

\begin{array} {r|lll}
\textbf{Stem} & \textbf{Leaf} \\
\hline 0 & 2 \\
0 & 5\ 6\ 7\ 7 \ 8\ 9\  \\
1 & 0\ 0\ 0\ 0\ 1\ 4\ 4\ 4\\
1 & 5\ 6\ 7 \\
2 & 3\ 3\ 4 \\
2 & 7\ 7\ 8\ 8 \\
3 & 0\ 4 \\
3 & 5\ 6 \\
4 & 1 \\
4 & 7 \\
\end{array}		 \(\text{key:  }0|2=0.2\)

Based on this stem plot, the distribution of the average number of decayed teeth for these countries is best described as

  1. positively skewed with a median of 15 decayed teeth and a range of 45
  2. approximately symmetric with a median of 1.5 decayed teeth and a range of 4.5
  3. negatively skewed with a median of 1.5 decayed teeth and a range of 4.5
  4. positively skewed with a median of 1.5 decayed teeth and a range of 4.5
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Median = 16th value} = 1.5\)

\(\text{Range} = 4.7-0.2=4.5\)

\(\text{The clear tail to the upper end of values shows that the}\)

\(\text{data is positively skewed.}\)

\(\Rightarrow D\)

Data Analysis, SM-Bank 051

Ms Granger measured and recorded the heights of all the students in her class, to the nearest centimetre.

She made a dot plot to show the heights of these 32 children.
 

   
Student Heights (nearest cm)

 

  1. What fraction of the students' heights are greater than 145 centimetres and less than 150?  (2 marks)
  2. What is the range of the heights?  (1 mark)
  3. What is the median of the heights?  (1 mark)
  4. What would the median be if a new student arrived with a height of 135 cm?  (1 mark)
Show Answers Only

a.    \(\dfrac{1}{4}\)

b.    \(18\)

c.    \(147.5\ \text{cm}\)

d.    \(147\ \text{cm}\)

Show Worked Solution
a.    \(\text{Fraction}\) \(=\dfrac{\text{Students with height 145 – 150}}{\text{Total students}}\)
    \(=\dfrac{8}{32}\)
    \(=\dfrac{1}{4}\)

 
\(\text{(Note students with heights of 145 or 150 are not included)}\)
 

b.    \(\text{Range}=154-136=18\)
 

c.    \(\text{Median}\) \(=\dfrac{\text{16th score + 17th score}}{\text{2}}\)
    \(=\dfrac{147+148}{2}\)
    \(=147.5\ \text{cm}\)

 

d.    \(\text{Median }\) \(=\text{ 17th score}\)
    \(=147\ \text{cm}\)

Data Analysis, SM-Bank 050

Brandon made a dot plot to show the hours he worked over the last 16 weeks.
 

 
What is the mean number of hours that Brandon worked over that last 16 weeks?  (2 marks)

Show Answers Only

\(15.75\ \text{hours}\)

Show Worked Solution
\(\text{Mean}\ \) \(=\dfrac{\text{Sum of the scores}}{\text{Number of scores}}\)
  \(=\dfrac{2\times 13+3\times 14+3\times 15+3\times 16+2\times 17+3\times 19}{16}\)
  \(=\dfrac{252}{16}\)
  \(=15.75\ \text{hours}\)

Data Analysis, SM-Bank 049

Evie made a dot plot to show the distances she has swum in her training for a long distance ocean swim.
 

 
What is the mean distance that Evie has swum? Give your answer correct to 1 decimal place.  (2 marks)

Show Answers Only

\(21.6\ \text{km}\)

Show Worked Solution
\(\text{Mean}\ \) \(=\dfrac{\text{Sum of the scores}}{\text{Number of scores}}\)
  \(=\dfrac{18+19+2\times 20+2\times 21+22+3\times 24+25}{11}\)
  \(=\dfrac{238}{11}=21.636\dots\)
  \(\approx 21.6\ \text{km (1 d.p.)}\)

Data Analysis, SM-Bank 048 MC

Dante made a dot plot to show the distances he has run in his training for a half-marathon.
 

 
What is the median of the distances Dante has run?

  1. \(2\)
  2. \(7\)
  3. \(21\)
  4. \(24\)
Show Answers Only

\(C\)

Show Worked Solution
\(\text{Median}\ \) \(=\ \text{6th score}\)
  \(=21\)

\(\Rightarrow C\)

Data Analysis, SM-Bank 047 MC

Angelica made a dot plot to show the distances she has run in her training for a marathon.
 

 
What is the range of the distances Angelica has run?

  1. \(3\)
  2. \(7\)
  3. \(21\)
  4. \(24\)
Show Answers Only

\(B\)

Show Worked Solution
\(\text{Range}\ \) \(=\ \text{high – low}\)
  \(=25-1\)
  \(=7\)

\(\Rightarrow B\)

Data Analysis, SM-Bank 046

The back-to-back ordered stem-and-leaf plot below shows the female and male smoking rates, expressed as a percentage, in 18 countries.
 

  1. For the 18 countries listed, what is the range of the male smoking rates?  (1 mark)
  2. For the 18 countries listed, what is the mode of the female smoking rates?  (1 mark)
  3. For the 18 countries listed, what is the difference between the medians of the female and male smoking rates?  (2 marks)
Show Answers Only

a.     \(30\%\)

b.     \(25\%\)

c.     \(5.5\%\)

Show Worked Solution

a.    \(\text{Range}=47-17=30\%\)

b.    \(\text{Female mode}=25\%\)

c.     \(\text{Female Median }\) \(=\ \text{average of 9th and 10th scores}\)
    \(=\dfrac{21+22}{2}=21.5\%\)

 

\(\text{Male Median }\) \(=\ \text{average of 9th and 10th scores}\)
  \(=27\%\)

  
\(\therefore\ \text{The difference in medians}\)

\(=27-21.5=5.5\%\)

Data Analysis, SM-Bank 044 MC

The total birth weight of a sample of 12 babies is 39.0 kg.

The mean birth weight of these babies, in kilograms, is

  1.  2.50
  2.  2.75
  3.  3.00
  4.  3.25
Show Answers Only

\(D\)

Show Worked Solution
\(\text{Mean}\) \(=\dfrac{\text{Total birth weight}}{\text{# babies}}\)
  \(=\dfrac{39.0}{12}\)
  \(=3.25\ \text{kg}\)

 
\(\Rightarrow D\)

Data Analysis, SM-Bank 043 MC

The total weight of nine oranges is 1.53 kg.

Using this information, the mean weight of an orange would be calculated to be closest to

  1. 115 g
  2. 153 g
  3. 162 g
  4. 170 g
Show Answers Only

\(D\)

Show Worked Solution
\(\text{Mean Weight}\) \(=\dfrac{\text{Total weight}}{\text{# Oranges}}\)
  \(=\dfrac{1.53}{9}\)
  \(= 0.17\ \text{kg}\)
  \(= 170\ \text{g}\)

 
\(\Rightarrow D\)

Data Analysis, SM-Bank 042

The following data represents the ages of people on a riverboat cruise in Europe.
 

28,  35,  39,  39,  28,  34,  40,  43,  51,  34,  35,  39,  40,  46,  60

 

  1. Organise the data into an ordered stem-and-leaf plot.  (2 marks)

  2. What is the median of the ages?  (1 mark)

  3. What is the modal age of the travellers?  (1 mark)

  4. Is this age data skewed or symmetrical?  (1 mark)

Show Answers Only

a.   

\begin{array} {r|lll} \textbf{Stem} & \textbf{Leaf}  \\ \hline  2 & 8\   8 \\  3 & 4\ 4\ 5\ 5 \ 9\ 9\ 9\  \\  4 & 0\ 0\ 3\ 6 \\ 5 & 1 \\ 6 & 0 \\ \end{array}

b.    `39`

c.    `39`

d.    `text(Skewed)`

Show Worked Solution

a.    `text(Ordered data:) \ 28,  28,  34,  34,  35,  35,  39,  39,  39,  40,  40,  43,  46,  51,  60`

\begin{array} {r|lll} \textbf{Stem} & \textbf{Leaf}  \\ \hline  2 & 8\   8 \\  3 & 4\ 4\ 5\ 5 \ 9\ 9\ 9\  \\  4 & 0\ 0\ 3\ 6 \\ 5 & 1 \\ 6 & 0 \\ \end{array}

b.    `text(Median)\ = \text(8th score)\ =\ 39`

c.    `text(Mode)\ = \ 39`

d.    `text(The data is skewed.)`

 

Data Analysis, SM-Bank 038

Jason recorded the following marks out of 100 in his last 8 class tests.
 

74,  65,  70,  72,  95,  68,  70,  64
 

  1. Which one of his marks is an outlier?  (1 mark)

  2. If the outlier is removed, by how many marks does the mean change?  (2 marks)

  3. Explain why it would be more appropriate to use the median rather than the mean when including the outlier in Jason's marks.  (2 marks)

Show Answers Only

a.   `95`

b.   `72.25 \-\69 = 3.25\ text(marks)`

c.   “

Show Worked Solution

a.   `text(The test mark of 95 is significantly different from the other marks)`

`:.\  95\ text(is an outlier)`
 

b.  `text(Initial Mean)`

`text(Mean)` `=(74 + 65 + 70 + 72 + 95 + 68 + 70 + 64)/8`  
  `= 578/8`  
  `= 72.25`  

 
`text(Mean without outlier)`

`text(New Mean)` `=(74 + 65 + 70 + 72  + 68 + 70 + 64)/7`
  `= 483/7`
  `= 69`

`:.\ text(The mean decreases by)\ 3.25\ text(marks)`

c.   `text(Ordered marks):\  64, \ 65, \ 68, \  70, \ 70, \ 72, \ 74, \ 95 `

`:.\ text(When 95 is included, the median is 70 where as the mean is 72.25.)`

`72.25\ text(lies between his 6th and 7th scores and is, therefore, not a)`

`text(good measure of centre for Jason’s marks.)`

Data Analysis, SM-Bank 036

The ages of boys competing in an inter-school futsal competition are shown in the frequency distribution table below.
 

\begin{array} {|c|c|}
\hline \textbf{Age (years)} & \textbf{Frequency} \\
\hline 13 & 4 \\
\hline 14 & 6  \\
\hline 15 & 11\\
\hline 16 & 6\\
\hline 17 & 3\\
\hline \end{array} 

  1. How many boys took part in the competition?  (1 mark)

  2. Calculate the mean age of the competitors, correct to the nearest whole number.  (2 marks)

  3. State the median age of the competitors?  (2 marks)

Show Answers Only

\begin{array} {ll} \textbf{a.} &  30 \\ \textbf{b.} & 15 \text{ years} \\ \textbf{c.} & 15 \text{ years} \end{array}

Show Worked Solution
a.   `text(  Number of boys)` `= 4 + 6 + 11 + 6 + 3`
    `= 30`

 

b.   `text(  Mean age of boys)` `= (13 xx 4 + 14 xx 6 + 15 xx 11 + 16 xx 6 + 17 xx 3)/30`
    `= (52 + 84 + 165 + 96 + 51)/30`
    `= 448/30`
    `= 14.9333…..`
    `~~ 15\ text(years (nearest whole number))`

 

c.   `text(  Median age of boys )` `=  text(average of 15th and 16th scores)`
    `= 15\ text(years, as both the 15th and 16th scores occur in 15 years)`

Data Analysis, SM-Bank 035

A data set has a range of 50 and a mean of 20.

Give an example of a dataset using 4 numbers that satisfies this condition.  (2 marks)

Show Answers Only

`0,  10,  20,  50`

`text(Note: other answers are possible.)`

Show Worked Solution

`0,  10,  20,  50`

`text(Range) = 50 \ -\ 0 = 50`

`text(Sum of the numbers) = 20 xx 4 = 80`

`text(Mean) = (0 + 10 + 20 +50)/4\ = 80/4 \= 20`

`text(Note: other answers are possible.)`

Data Analysis, SM-Bank 029

The local nursery is selling advanced orange trees. The heights of the trees are displayed in the dot plot below.
 

What is the mean height of these trees?  (2 marks)

Show Answers Only

`173`

Show Worked Solution
`text(Mean)\ ` `=\ text(Average of the heights)`
  `= (170 xx 2 + 171 xx 2 + 172 xx 2 + 174 xx 3 + 175 + 176 + 177)/12`
  `= 2076/12`
  `= 173`

Data Analysis, SM-Bank 028

Find the range for this set of scores.  (1 mark)

`74, \ 89, \ 65, \ 64, \ 87, \ 88, \ 68, \ 74, \ 72`

Show Answers Only

`25`

Show Worked Solution

`text(Range) = text(Highest score) \-\text(Lowest score)   = 89 \-\64 = 25`

Data Analysis, SM-Bank 025

A shop sells children's shoes in sizes from 5 to 9. The sizes of the last 100 shoes sold is shown in the table below.

\begin{array} {|l|c|c|c|c|c|c|}
\hline \textbf{Shoe Size} & 5 & 6 & 7 & 8 & 9  \\
\hline \textbf{Frequency}  & 14 & 21 & 30 & 23 & 12 \\
\hline \end{array}

  1. What is the median shoe size?  (1 mark)

  2. What is the mean shoe size?  Give your answer correct to the nearest whole number.  (2 marks)

  3. What is the range of shoe sizes?  (1 mark)

  4. What is the modal shoe size?  (1 mark)

  5. Briefly explain why the shop owner would be particularly interested in the modal shoe size?  (2 marks)

Show Answers Only
  1. `7`
  2. `6.98`
  3. `4`
  4. `7`
  5. `text(The shop keeper would be primarily interested in profit,)`
    `text(and the most popular shoe size will generate the most profit.)`
    `text(Also for stock purposes it would be an advantage to have the)`
    `text(most popular size in stock.)`
Show Worked Solution

a.    `text(Median) \ = \text(Average of 50th and 51st scores) \ = 7`

b.    `text(Mean)` `=(5 xx 14 + 6 xx 21 + 7 xx 30 + 8 xx 23 + 9 xx 12)/100`  
  `=(70 + 126 + 210 + 184 +108)/100`  
  `=698/100 = 6.98`  

 
c.    `text(Range) \ = \text(highest score) \-\ text(lowest score) \ = 9 \-\ 5 = 4`

d.    `text(Modal shoe size) \ = \text(The shoe size with the highest frequency) \ = 7`

\begin{array} {ll}
\textbf{e.}&\text{The shop keeper would be primarily interested in profit,}\ \\
&\text{and the most popular shoe size will generate the most profit.}\ \\
&\text{Also for stock purposes it would be an advantage to have the}\ \\ 
&\text{most popular size in stock.}\end{array}

Data Analysis, SM-Bank 023

Justify why adding a score of 15 to the set of scores below will not change the mode of 7.  (2 marks)

`4, \ 5, \ 7, \ 7, \ 7, \ 10, \ 10, \ 11, \ 12, \ 15`

Show Answers Only

`text(Adding a score of 15 will increase the number of  15’s to 2.)`

`text(Therefore, the mode remains 7 as there are three 7’s.)`

Show Worked Solution

`text(Adding a score of 15 will increase the number of  15’s to 2.)`

`text(Therefore, the mode remains 7 as there are three 7’s.)`

Data Analysis, SM-Bank 024

Brendan scored the following marks in 4 class tests.

`15, \ 16, \ 16, \ 17 `

Explain the effect on his mean mark if he received a mark of 11 in his final class test.

Justify your answer with calculations.  (2 marks)

Show Answers Only

`text(Initial Mean = 16)`

`text(New Mean = 15)`

`:.\ text(Mean decreases as a lower mark is added.)`

Show Worked Solution
`text(Initial Mean)` `=(15 + 16 + 16 + 17)/4`  
  `= 64/4`  
  `= 16`  

 

`text(New Mean)` `=(15 + 16 + 16 + 17 + 11)/5`
  `= 75/5`
  `= 15`

 

`:.\ text(Mean decreases as a lower mark is added.)`

Data Analysis, SM-Bank 021 MC

A survey asked 9 students in Year 8 how many siblings they had.
 

\begin{array} {|c|c|}\hline \textbf{Number of Siblings}  & 0,  2,  2,  2,  3,  4,  4,  5,  5\\ \hline \end{array}

 
Which of the following is true for this data?

  1. mean > median > mode
  2. mean > median < mode
  3. mean = median = mode
  4. mean = median > mode
Show Answers Only

`D`

Show Worked Solution

`text(Mode = 2)`

`text(Median = 5th number = 3)`

`text(Mean)` `= (0 + 2 + 2 + 2 + 3 + 4 + 4 + 5 + 5)/9`
  `= 27 ÷ 9`
  `= 3`

 

`:.\ text(mean = median > mode)`

`=>D`

Data Analysis, SM-Bank 020

An AFL team has 10 players with the following heights in centimetres.

`180, \ 175, \ 163, \ 192, \ 200, \ 193, \ 195, \ 195, \ 188, \ 195`

What is the range of heights?  (1 mark)

Show Answers Only

`37\ text(centimetres)`

Show Worked Solution
`text(Range)` `=  200\-\163`
  `=  37\ text(centimetres)`

Data Analysis, SM-Bank 019

A vet measured the length of 21 dogs that came through his clinic.

The vet recorded the length of each dog.
  

  1.  What is the median length?  (1 mark)

  2. The vet measured and recorded a new dog with a length of 41 centimetres. What is the new median length?  (1 mark)

Show Answers Only

a.   `text(52 cm)`

b.   `text(51.5 cm)`

Show Worked Solution

a.   `text(The median of 21 data points is the 11th data point.)`

`:.\ text(The median is 52 cm.)`

 

b.   `text(The median of 22 data points is the average of the 11th and 12th data point.)`

`text(Average)` `=(51 +52)/2`
  `=51.5\ text(cm)`

 
`:.\ text(The new median is 51.5 cm.)`

Data Analysis, SM-Bank 018

Five students do a standing long jump at their athletics carnival and the length of their jumps, in centimetres, are recorded in the table below.
 

 
If Lenny's distance is removed from the data, what happens to the mean distance that is jumped from this group? (1 mark)

Show Answers Only

`text(Decreases)`

Show Worked Solution

`text(The mean decreases because the longest distance is removed from the data set.)`

Data Analysis, SM-Bank 017

A school's drama class puts on a play over five nights.

The play is open to the public and the numbers of tickets sold are shown in the table below.
 

The cost of each ticket was $15.

What was the mean amount of money collected from ticket sales per night?  (2 marks)

Show Answers Only

`$2850`

Show Worked Solution
`text(Mean number of tickets sold)` `= (210 + 170 + 180 + 170 + 220)/5`
  `= 190`

 

`:.\ text(Mean tickets sales)` `= 190 xx 15`
  `= $2850`

Data Analysis, SM-Bank 016

This table summarises the time Tutty spent training her parrot over five days.
 

 
What was the average (mean) time for training the parrot each day?  (2 marks)

Show Answers Only

`text(56 minutes)`

Show Worked Solution
`text(Average)` `= (25 + 55 + 60 + 94 + 46)/5`
  `= 280/5`
  `= 56\ text(minutes)`

Data Analysis, SM-Bank 015

A policeman is recording the speed of 25 cars travelling on a highway using a speed gun.

The results are shown in the stem-and-leaf plot.

  1. What is the median speed?  (1 mark)

  2. What is the range of the speeds?  (1 mark)

Show Answers Only

a.   `119\ text(km/hr)`

b.  `49\ text(km/hr)`

Show Worked Solution

a.   `25\ text(data points)`

`text(Median is the 13th data point)`

`:.\ text(Median) = 119\ text(km/hr)`
 

b.   `text(Range)` `= text(Highest) \ – \ text(Lowest)`
    `= 139\ – \ 90`
    `=49\ text(km/hr)`

 

Data Analysis, SM-Bank 014 MC

The Stars and the Thunder are playing cricket in a 20 over competition.

The stem-and-leaf plots show the number of runs each side has scored in their last 15 games.
 

   

Which statement is true about the data.

  1. The Stars had the lowest run score.
  2. The Stars scored over 135 runs more times than the Thunder.
  3. The median score for The Stars is higher than the median score for The Thunder.
  4. The range of scores for The Stars is smaller than the range of scores for The Thunder.
Show Answers Only

`D`

Show Worked Solution

`text{Range The Stars} = 149\ – \ 140\ =\ 39`

`text{Range The Thunder} = 148\ – \ 108\ =\ 40`

`:.\ text(Correct statement is:)`

`text{The range of scores for The Stars is smaller than the}`

`text{range of scores for The Thunder (39 vs 40).}`

`=>D`

 

Data Analysis, SM-Bank 013

Curly measures the position of glaciers in the Antarctic.

His measurements showed that in 1 full year, a glacier moved 88 cm.

On average, how many centimetres did the glacier move per day?  (2 marks)

Show Answers Only

`0.24\ text(cm)`

Show Worked Solution

`text(Average daily movement of glacier)`

`=88/365`

`= 0.24\ text(cm)`

Data Analysis, SM-Bank 012 MC

In Wadonga, there are 29 538 people.

Each day, the average person uses 168 litres of water.

Which of these gives the best estimate for the total number of litres of water used in Wadonga each day?

  1. `30\ 000 xx 200`
  2. `30\ 000 xx 100`
  3. `30\ 000 ÷ 200`
  4. `30\ 000 ÷ 100`
Show Answers Only

`A`

Show Worked Solution
`text(Total litres)` `=\ text(litres per person × total people)`  
  `= 30\ 000 xx 200`  
     

`text{(168 is closer to 200 than 100 and will give a better estimate.)}`

`=>A`

Data Analysis, SM-Bank 011 MC

Five students throw the javelin at their athletics carnival and the length of their throws, to the nearest metre, are recorded in the table below.
 

If Monica's distance is removed from the data, what happens to the mean distance that is thrown from this group?

  1. It increases.
  2. It decreases.
  3. It stays the same.
  4. It is impossible to tell from the information given.
Show Answers Only

`A`

Show Worked Solution

The mean increases because the shortest distance is removed from the data set.

`=>A`

Data Analysis, SM-Bank 010 MC

The heights, in centimetres, of David's hockey side are displayed in the dot plot below.
 

Which of the following statements is true about this data?

  1. The median and the mode are both 174 and the mean is 174.5.
  2. The mean and the median are both 174 and the mode is 177.
  3. The mean is 175, the mode is 174 and the median is 174.5.
  4. The mean, median and mode are all equal to 174.
Show Answers Only

`D`

Show Worked Solution

`text(Data from the dot plot:)` `\ 171, \ 172, \ 172, \ 173, \ 174, \ 174, \ 174, \ 174, \ 176, \ 177, \ 177`

`text(Median)\ ` `=\ text(Middle or 6th score)`
  `= 174`

 

`text(Mode)\ ` `=\ text(The most frequent score)`
  `= 174`

 

`text(Mean)\ ` `=\ text(Average of the scores)`
  `= (171 + 172 + 172 + 173 + 174 + 174 + 174 + 174 + 176 + 177 + 177)/11`
  `= 1914/11`
  `= 174`

 
∴ The mean, median and mode are all equal to 174.

`=>D`

Data Analysis, SM-Bank 009

A group of 10 students scored the following marks in an English exam.

 `87, \ 56, \ 86, \ 84, \ 89, \ 89, \ 87, \ 88, \ 90, \ 94`

  1. Calculate the mean mark for the exam.  (1 mark)

  2. After receiving his mark back for the exam, Marcus told his friend:

`text(“My mark of 84 was much better than the average, so I did really well.”)`

  Comment briefly on Marcus' statement.  (2 marks)

Show Answers Only
  1. `85`
  2. `text(Marcus’ mark of 84 was below the average of 85 and he did not do well)`
    `text(compared to the other students as he received the second lowest mark.)`
Show Worked Solution
a.  `text(Mean)` `= (87 + 56 + 86 + 84 + 89 + 89 + 87 + 88 + 90 + 94)/10`
  `= (850)/10`
  `= 85`

 
b.  `text(Marcus’ mark of 84 was below the average of 85 and he did not do well)`
     `text(compared to the other students as he received the second lowest mark.)`

 

Data Analysis, SM-Bank 008 MC

A group of friends met with their 7 children. The ages of the children were:

 `9, \ 4, \ 5, \ 6, \ 11, \ 3, \ 4`

The median of the children's ages is:

  1. `6`
  2. `5`
  3. `4`
  4. `8`
Show Answers Only

`B`

Show Worked Solution

`text(Ordered data:)` `\ 3, \ 4, \ 4, \ 5, \ 6, \ 9, \ 11`

`text(Median)\ ` `=\ text(Middle score)`
  `=5`

 
`=>B`

Data Analysis, SM-Bank 007

Bailey's soccer coach recorded the number of goals scored during the last 6 games of the season.

  `3, \ 7, \ 6, \ 3, \ 1, \ 4`

Find:

  1. the median number of goals scored.  (1 mark)

  2. the mean number of goals scored.  (1 mark)

Show Answers Only

i.  `3.5`

ii.  `4`

Show Worked Solution

i.  `text(Ordered data:)` `\ 1, \ 3, \ 3, \ 4, \ 6, \ 7`

`text(Median)\ ` `=\ text(Average of 2 middle scores)`
  `= (3+4)/2`
  `= 7/2`
  `=3.5`

 

ii.  `text(Mean) ` `= (3+7+6+3+1+4)/6`  
  `= 24/6`  
  `= 4`  

 

Data Analysis, SM-Bank 006

In the two weeks leading up to a half marathon, Rogan ran the following distances, in kilometres.

`15, \ 21, \ 17, \ 9, \ 17, \ 25, \ 11`

What was his mean distance, in kilometres?  Give your answer correct to 2 decimal places?  (2 marks)

Show Answers Only

`16.43` km

Show Worked Solution
`text(Mean)` `= (15 + 21 + 17 + 9 + 17 + 25 + 11)/7`
  `= (115)/7`
  `= 16.4285…`
  `~~ 16.43` km

Data Analysis, SM-Bank 005 MC

The points scored by an AFL team in their first 13 games of the season is recorded.

`78, \ 84, \ 63, \ 75, \ 98, \ 105, \ 92, \ 75, \ 84, \ 96, \ 84, \102, \100`

In the 14th game, they scored 61.

Which of these values would increase?

  1. `text(mode)`
  2. `text(range)`
  3. `text(mean)`
  4. `text(median)`
Show Answers Only

`B`

Show Worked Solution

`text(Consider each option:)`

`text(Mode – unchanged at 84)`

`text(Range – increases from 42 to 44)`

`text(Mean – decreases from 87.38 to 85.5)`

`text{Median – unchanged at 84}`

`=>B`

Data Analysis, SM-Bank 004 MC

Jacqui's basketball team has 5 players.

The height of each player is listed below (in cm):
 

`186, 190, 164, 190, 175`
  

What is the median height of these players?

  1. `181\ text(cm)`
  2. `164\ text(cm)`
  3. `186\ text(cm)`
  4. `190\ text(cm)`
Show Answers Only

`C`

Show Worked Solution

`text(Heights in order are:)`

`164, 175, 186, 190, 190`

`:.\ text(Median) = 186\ text(cm)`

`=> C`

Data Analysis, SM-Bank 003

This table shows the number of people who visited a war memorial on weekdays over 4 weeks.
 

 
 

  1. What was the range of people visiting the war memorial on Monday?  (1 mark)

  2. What was the mean number of people who attended the war memorial on Fridays?  (1 mark)

  3. What was the median number of people who visited the war memorial during week 3?  (1 mark)

  4. What is the modal number of visitors to the war memorial during the four week period?  (1 mark)

Show Answers Only

a.   `44`

b.   `26`

c.   `39`

d.   `22`

Show Worked Solution
a.   `text(Range on Mondays)` `= 81 \ -\ 37`
  `= 44`

 

b.   `text(Mean on Fridays)` `=(22 + 32+28+22)/4`
  `=104/4`
  `=26`

  
c.   `text(Week 3 data in order:   28,  37,  39,  53,  72)`

`text(Median Week 3)` `=\ text(middle score)`
  `=\  39`

 
d.   `text(Modal number of visitors) = 22`

Data Analysis, SM-Bank 002

The mean (average) of four numbers is 26.

One more number is added and the mean number becomes 27.

What is the number that was added?  (2 marks)

Show Answers Only

`31`

Show Worked Solution

`text(Total of the first 4 numbers,)`

`26 xx 4 = 104`

`text(Total including the 5th number added,)`

`27 xx 5 = 135`

`:.\ text(The number added)` `=135-104`
  `=31`

Statistics, STD2 S1 2021 HSC 3 MC

The stem-and-leaf plot shows the number of goals scored by a team in each of ten netball games.
  

What is the mode of this dataset?

  1.  5
  2.  18
  3.  25
  4.  29
Show Answers Only

`C`

Show Worked Solution

`\text{Mode}  -> \text{data point with highest frequency}`

`\text{Mode}  = 25`

`=> C`

Statistics, STD2 S1 2020 HSC 7 MC

Which histogram best represents a dataset that is positively skewed?

 
 
Show Answers Only

`A`

Show Worked Solution

♦♦ Mean mark 32%.

`text(Positive skew occurs when the tail on the)`

`text{histogram is longer on the right-hand}`

`text{(positive) side.}`

`=> \ A`

Statistics, STD2 S1 2018 HSC 11 MC

A set of data is summarised in this frequency distribution table.
 

 
Which of the following is true about the data?

  1. Mode = 7, median = 5.5
  2. Mode = 7, median = 6
  3. Mode = 9, median = 5.5
  4. Mode = 9, median = 6
Show Answers Only

`text(B)`

Show Worked Solution

`text{Mode = 7  (highest frequency of 9)}`

`text(Median = average of 15th and 16th data points.)`

`:.\ text(Median = 6)`

`=>\ text(B)`

Statistics, STD2 S1 2006 HSC 12 MC

The mean of a set of 5 scores is 62.

What is the new mean of the set of scores after a score of 14 is added?

  1.   38
  2.   54
  3.   62
  4.   76
Show Answers Only

`B`

Show Worked Solution

`text(Mean of 5 scores) = 62`

`:.\ text(Total of 5 scores) = 62 xx 5 = 310`

`text(Add a score of 14)`

`text(Total of 6 scores) = 310 + 14 = 324`

`:.\ text(New mean)` `= 324/6`
  `= 54`

`=>  B`