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Box Plots and 5-Number Summary, SM-Bank 014

The test results in English and Mathematics for a class were recorded and displayed in the box plots.
 

  1. What is the interquartile range for English?   (1 mark)

  2. Compare and contrast the two data sets by referring to the skewness of the distributions and the measures of their centres and spread.   (3 marks)

Show Answers Only

i.    `text{IQR}_text{(English)}\ = 80-50=30`

ii.  `text(Skewness)`

  • `text(English has greater negative skew)`
  • `text(Maths is more normally distributed)`

`text(Location and Spread)`

  • `text(English has a range of 85, Maths has 40.)`
  • `text{English has larger IQR than Maths (30 vs 15)}`
  • `text{Maths’ median (75) is higher than English (70)}`
  • `text{Same upper quartile marks (80)}`
  • `text(English has highest and lowest individual mark)`
Show Worked Solution

i.    `text{IQR}_text{(English)}\ = 80-50=30`

 

♦ Mean mark (ii) 35%
COMMENT: The best answers use the correct language of location and spread such as mean, median, interquartile range, standard deviation and skewness.

ii.  `text(Skewness)`

  • `text(English has greater negative skew)`
  • `text(Maths is more normally distributed)`

`text(Location and Spread)`

  • `text(English has a range of 85, Maths has 40.)`
  • `text{English has larger IQR than Maths (30 vs 15)}`
  • `text{Maths’ median (75) is higher than English (70)}`
  • `text{Same upper quartile marks (80)}`
  • `text(English has highest and lowest individual mark)`

Box Plots and 5-Number Summary, SM-Bank 013

Terry and Kim each sat twenty class tests. Terry’s results on the tests are displayed in the box plot shown in part (i).

  1. Kim’s  5-number summary for the tests is  67,  69,  71,  73,  75.

     

    Draw a box plot to display Kim’s results below that of Terry’s results.   (1 mark)
     
       

  2. What percentage of Terry’s results were below 69?     (1 mark)

  3. Describe the skew of Kim's results.   (1 mark)

Show Answers Only
i.    

ii.  `text(50%)`
 

iii.  `text{By inspection of box plot: Evenly skewed}`

Show Worked Solution
i. 

 
ii.  `text(50%)`
 

iii.  `text{By inspection of box plot: Evenly skewed}`

Statistics, STD2 S1 2016 HSC 22 MC

The box-and-whisker plots show the results of a History test and a Geography test.
 

In History, 112 students completed the test. The number of students who scored above 30 marks was the same for the History test and the Geography test.

How many students completed the Geography test?

  1. 8
  2. 50
  3. 56
  4. 112
Show Answers Only

`=> C`

Show Worked Solution

`text{In History} \ => \  text{Q}_3 = 30\ \text{marks}`

`:.\ text{Scoring over 30}\ = 25text(%) xx 112 = 28\ \text{students}`
 

`text{In Geography} \ => \ text{Median}\ = 30\ \text{marks}`

`:.\ text{Students completing Geography}\ =2 xx 28 = 56\ \text{students}`

`=> C`

Statistics, STD2 S1 2005 HSC 22 MC

Two groups of people were surveyed about their weekly wages. The results are shown in the box-and-whisker plots.
 

Which of the following statements is true for the people surveyed?

  1. The same percentage of people in each group earned more than $325 per week.
  2. Approximately 75% of people under 21 years earned less than $350 per week.
  3. Approximately 75% of people 21 years and older earned more than $350 per week.
  4. Approximately 50% of people in each group earned between $325 and $350 per week.
Show Answers Only

`B`

Show Worked Solution

`text{Option A: 50% of Under 21 group earned over $325 and 75%}`

`text{of Over 21 group did. NOT TRUE.}`
 

`text{Option B: 75% of Under 21 group earned below $350 is TRUE.}`
 

`text{Options C and D: can both be proven to be untrue using their}`

`text{median and quartile values.}`

`=>  B`

Statistics, STD2 S1 2008 HSC 10 MC

The marks for a Science test and a Mathematics test are presented in box-and-whisker plots.
 

 Which measure must be the same for both tests?

  1. Mean
  2. Range
  3. Median
  4. Interquartile range
Show Answers Only

`D`

Show Worked Solution

`text(IQR)=text(Upper Quartile)-text(Lower Quartile)`

`text{In both box plots, IQR = 3 intervals (against bottom scale)}`

`=>  D`

Statistics, STD2 S1 2010 HSC 27b

The graphs show the distribution of the ages of children in Numbertown in 2000 and 2010.
  

  1. In 2000 there were 1750 children aged 0–18 years.

     

    How many children were aged 12–18 years in 2000?   (1 mark)

  2. The number of children aged 12–18 years is the same in both 2000 and 2010.

     

    How many children aged 0–18 years are there in 2010?    (1 mark)

  3. Identify TWO changes in the distribution of ages between 2000 and 2010. In your answer, refer to measures of location or spread or the shape of the distributions.   (2 marks)

  4. What would be ONE possible implication for government planning, as a consequence of this change in the distribution of ages?   (1 mark)

Show Answers Only

i.    `875`

ii.    `3500`

iii.  `text{Changes in distribution (include 2 of the following):}`

  • `text(the lower quartile age is lower in 2010)`
  • `text(the median is lower in 2010)`
  • `text(the upper quartile age is lower in 2010)`
  • `text(the interquartile range is greater in 2010)`
  • `text(2010 is positively skewed while 2000 is negatively)`

iv.  `text(Implication for government planning:)`

`text(Since the children are getting younger in 2010,)`

  • `text(Approve and build more childcare facilities)`
  • `text(Build more school and public playgrounds)`
Show Worked Solution

i.    `text{Since the median = 12 years}`

♦ Mean mark (i) 45%

`=>\ text{50% of children are aged 12–18 years}`

`:.\ text{Children aged 12–18}\ = 50\text{%}\ xx 1750 = 875`

 

♦♦ Mean mark (ii) 25%

ii.   `text{Upper quartile (2010) = 12 years}`

`text{Children in upper quartile = 875 (from part (i))}`

`:.\ text{Children aged 0–18}\ =4 xx 875= 3500`
 

iii.  `text{Changes in distribution (include 2 of the following):}`

♦ Mean mark (iii) 35%
MARKER’S COMMENT: A number of students incorrectly identified “positive” skew as “negative” skew here.
  • `text(the lower quartile age is lower in 2010)`
  • `text(the median is lower in 2010)`
  • `text(the upper quartile age is lower in 2010)`
  • `text(the interquartile range is greater in 2010)`
  • `text(2010 is positively skewed while 2000 is negatively)`

iv.  `text(Implication for government planning:)`

♦ Mean mark (iv) 46%
MARKER’S COMMENT: Answers should reflect the 1 mark allocation.

`text(Since the children are getting younger in 2010,)`

  • `text(Approve and build more childcare facilities)`
  • `text(Build more school and public playgrounds)`