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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)
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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 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 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`

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`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)

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`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)

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`text(Decreases)`

Show Worked Solution

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

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 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`

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 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
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`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 2011 HSC 14 MC

A data set of nine scores has a median of 7.

The scores  6, 6, 12 and 17  are added to this data set.

What is the median of the data set now?

  1. 6
  2. 7
  3. 8
  4. 9
Show Answers Only

`B`

Show Worked Solution

`text(S)text(ince an even amount of scores are added below and)`

`text(above the existing median, it will not change.)`

`=>B`

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 14 MC

The July sales prices for properties in a suburb were:

$552 000,  $595 000,  $607 000,  $607 000,  $682 000, and  $685 000.

On 1 August, another property in the same suburb was sold for over one million dollars.

If the property had been sold in July, what effect would it have had on the mean and median sale prices for July?

  1.    Both the mean and median would have changed.
  2.    Neither the mean nor the median would have changed.
  3.    The mean would have changed and the median would have stayed the same.
  4.    The mean would have stayed the same and the median would have changed.
Show Answers Only

`C`

Show Worked Solution

`text(Mean increases because new house is sold above)`

`text(the existing average.)`

`text(Initial median)= (607\ 000+607\ 000)/2=607\ 000` 

`text(New median)=607\ 000\ \ \  text{(4th value in a list of 7)}`

`=>\ C`