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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 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 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 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 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 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 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 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 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`

Statistics, STD2 S1 2007 HSC 24a

Consider the following set of scores:

`3, \ 5, \ 5, \ 6, \ 8, \ 8, \ 9, \ 10, \ 10, \ 50.` 

  1. Calculate the mean of the set of scores.   (1 mark)

  2. What is the effect on the mean and on the median of removing the outlier?   (2 marks)

Show Answers Only
  1. `11.4`
  2. `text{If the outlier (50) is removed, the mean}`

     

    `text(would become lower.)`

  3.  

    `text(Median will NOT change.)`

Show Worked Solution

i.  `text(Total of scores)`

`= 3 + 5 + 5 + 6 + 8 + 8 + 9 + 10 + 10 +50`

`= 114`
 

`:.\ text(Mean) = 114/10 = 11.4`

 

ii.  `text(Mean)`

`text{If the outlier (50) is removed, the mean}`

`text(would become lower.)`
 

`text(Median)`

`text(The current median (10 data points))`

`= text(5th + 6th)/2 = (8 + 8)/2 = 8`

`text(The new median (9 data points))`

`=\ text(5th value)`

`= 8`
 

`:.\ text(Median will NOT change.)`

Statistics, STD2 S1 2008 HSC 23f

Christina has completed three Mathematics tests. Her mean mark is 72%.

What mark (out of 100) does she have to get in her next test to increase her mean mark to 73%?   (2 marks)

Show Answers Only

`76`

Show Worked Solution

`text(Total marks in 3 tests)`

`= 3 xx 72`

`= 216`

`text(We need 4-test mean) = 73`

`text(i.e.)\ \ \ ` `text{Total Marks (4 tests)}-:4` `= 73`
  `text(Total Marks)\ text{(4 tests)}` `= 292`

 

`:.\ text(4th test score)` `= 292 – 216`
  `= 76`

Statistics, STD2 S1 2008 HSC 13 MC

The height of each student in a class was measured and it was found that the mean height was 160 cm.

Two students were absent. When their heights were included in the data for the class, the mean height did not change.

Which of the following heights are possible for the two absent students?

  1.    155 cm and 162 cm
  2.    152 cm and 167 cm
  3.    149 cm and 171 cm
  4.    143 cm and 178 cm
Show Answers Only

`C`

Show Worked Solution

`text(S) text(ince the mean doesn’t change)`

`=>\ text(2 absent students must have a)`

`text(mean height of 160 cm.)`

`text(Considering each option given,)`

`(149 + 171) -: 2 = 160`

`=>  C`

Statistics, STD2 S1 2011 HSC 17 MC

The heights of the players in a basketball team were recorded as 1.8 m, 1.83 m, 1.84 m, 1.86 m and 1.92 m. When a sixth player joined the team, the average height of the players increased by 1 centimetre.

What was the height of the sixth player?

  1.   1.85 m
  2.   1.86 m
  3.   1.91 m
  4.   1.93 m
Show Answers Only

`C`

Show Worked Solution
`text(Old Mean)` `=(1.8+1.83+1.84+1.86+1.92)-:5`
  `=9.25/5`
  `=1.85\ \ text(m)`

 

`text{S}text{ince the new mean = 1.86m  (given)}`

`text(New Mean)` `=text(Height of all 6 players) -: 6`
`:.1.86` `=(9.25+h)/6\ \ \ \ (h\ text{= height of new player})`
`h` `=(6xx1.86)-9.25`
  `=1.91\ \ text(m)`

`=> C`

Statistics, STD2 S1 2009 HSC 21 MC

The mean of a set of ten scores is 14. Another two scores are included and the new mean is 16.

What is the mean of the two additional scores?

  1.    4
  2.    16
  3.    18
  4.    26
Show Answers Only

`D`

Show Worked Solution
♦♦♦ Mean mark 28%.

`text(If ) bar x\ text(of 10 scores = 14)`

  `=>text(Sum of 10 scores)= 10 xx 14 = 140`

`text(With 2 additional scores,)\ \ bar x = 16 `

  `=>text(Sum of 12 scores)= 12 xx 16 = 192`

`:.\ text(Value of 2 extra scores)` `= 192\-140`
  `= 52`

 

`:.\ text(Mean of 2 extra scores)= 52/2 = 26`

`=>  D`

Statistics, STD2 S1 2013 HSC 15 MC

The frequency histogram shows the number of goals scored by a football team in each game in a season.
 

2013 15 mc

 
What is the mean number of goals scored per game by this team?

  1.    4
  2.    4.5
  3.    5
  4.    5.5
Show Answers Only

`C`

Show Worked Solution

`text(Total number of goals scored)`

`=(3xx3)+(4xx7)+(5xx5)+(6xx1)+(7xx0)+(8xx4)`

`=9+28+25+6+0+32`

`=100`

`text(Number of games)=3+7+5+1+4=20`

`:.\ text(Mean goals per game)=100/20=5`

`=>\ C`