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Statistics, STD2 S1 SM-Bank 5 MC

 The stem plot below shows the height, in centimetres, of 20 players in a junior football team.
 

A player with a height of 179 cm is considered an outlier because 179 cm is greater than

  1. 162 cm
  2. 169 cm
  3. 173 cm
  4. 175.5 cm
Show Answers Only

`D`

Show Worked Solution

`Q_1 = (148 + 148)/2 = 148`

`Q_3 = (158 + 160)/2 = 159`

`IQR = 159 – 148 = 11`

`text{Upper fence}` `= Q_3 + 1.5 xx IQR`
  `= 159 + 1.5 xx 11`
  `= 175.5`

 
`=> D`

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 SM-Bank 2 MC

The dot plots show the height of students in Year 9 and Year 12 in a school. They are drawn on the same scale.
 


 

Which statement about the change in heights when comparing Y9 to Y12 is correct?

  1. The mean increased and the standard deviation decreased.
  2. The mean decreased and the standard deviation decreased.
  3. The mean increased and the standard deviation increased.
  4. The mean decreased and the standard deviation increased.
Show Answers Only

`A`

Show Worked Solution

`text{Mean has increased (Y9 to Y12)}`

`text(The Year 12 data is more tightly distributed)`

`text(around the mean.)`

`:.\ text{Standard deviation has decreased (Y9 to Y12)}`

`=> A`

Statistics, STD2 S1 2018 HSC 6 MC

A set of data is displayed in this dot plot.
 


 

Which of the following best describes this set of data?

  1. Symmetrical
  2. Positively skewed
  3. Negatively skewed
  4. Normally distributed
Show Answers Only

`text(C)`

Show Worked Solution

`text(Data is skewed.)`

♦ Mean mark 43% (a surprisingly poor result!)

`text(S)text(ince the “tail” is on the left had side, the)`

`text(data is negatively skewed.)`

`=>\ text(C)`

Statistics, STD2 S1 2016 HSC 29c

The ages of members of a dance class are shown in the back-to-back stem-and-leaf plot.
 

2ug-2016-hsc-q29_2
 

Pat claims that the women who attend the dance class are generally older than the men.

Is Pat correct? Justify your answer by referring to the median and skewness of the two sets of data.  (3 marks)

Show Answers Only

`text(Women:)`

`text(The median is 55 in a data)`

`text(set that is negatively skewed.)`

`text(Men:)`

`text(The median is 45 in a data)`

`text(set that is positively skewed.)`

`:.\ text(Pat is correct.)`

Show Worked Solution

`text(Women:)`

♦ Mean mark 44%.

`text(The median is 55 in a data)`

`text(set that is negatively skewed.)`

`text(Men:)`

`text(The median is 45 in a data)`

`text(set that is positively skewed.)`

`:.\ text(Pat is correct.)`

Statistics, STD2 S1 2015 HSC 19 MC

The table shows the life expectancy (expected remaining years of life) for females at selected ages in the given periods of time.

2015 19 mc

In 1975, a 45‑year‑old female used the information in the table to calculate the age to which she was expected to live. Twenty years later she recalculated the age to which she was expected to live.

What is the difference between the two ages she calculated?

  1.    2.7 years
  2.    3.1 years
  3.    3.7 years
  4.    5.8 years
Show Answers Only

`D`

Show Worked Solution

`text(In 1975, her life expectancy)`

♦ Mean mark 39%.

`=\ text(age + remaining years)`

`= 45 + 34`

`= 79`

`text(In 1995,  her life expectancy)`

`= 65 + 19.8`

`= 84.8`

`:.\ text(Difference)` `= 84.8 − 79`
  `= 5.8\ text(years)`

`⇒ D`

Statistics, STD2 S1 2005 HSC 24d

The sector graph shows the proportion of people, as a percentage, living in each region of Sumcity. There are 24 000 people living in the Eastern Suburbs.
 

2UG-2005-24d1
 

  1. Show that the total number of people living in Sumcity is  160 000.  (1 mark)

Jake used the information above to draw a column graph.
 

2UG-2005-24d2

  1. The column graph height is incorrect for one region.

     

    Identify this region and justify your answer.   (2 marks)

Show Answers Only
  1. `160\ 000`
  2. `text(Western Suburbs population)`
Show Worked Solution

i.   `text(Let the population of Sumcity =)\ P`

`text(15%)× P` `= 24\ 000`
`:.P`  `= (24\ 000)/0.15` 
  `= 160\ 000\ …\ text(as required)` 

 

ii.  `text(Western Suburbs population)`

`= text(10%) × 160\ 000`

`= 16\ 000`

`text(The column graph has this population as)`

`text(12 000 people which is incorrect.)`

Statistics, STD2 S1 2005 HSC 24a

  1. Draw a stem-and-leaf plot for the following set of scores.

  2.  

     

    `21\ \ \ 45\ \ \ 29\ \ \ 27\ \ \ 19\ \ \ 35\ \ \ 23\ \ \ 58\ \ \ 34\ \ \ 27`  (2 marks)

  3. What is the median of the set of scores?   (1 mark)

  4. Comment on the skewness of the set of scores.   (1 mark)

Show Answers Only
  1.  
  2. `28`
  3. `text(The data has a tail that stretches to the right)`

  4.  

    `:.\ text(Data is positively Skewed.)`

Show Worked Solution
i.    HSC 2005 24a

 

ii.  `text(10 scores)`

`:.\ text(Median)` `= text{(5th + 6th)}/2`
  `= (27 + 29)/2`
  `= 28`

 

iii.  `text(The data has a tail that stretches to the right)`

`:.\ text(Data is positively skewed.)`

Statistics, STD2 S1 2006 HSC 8 MC

Which of these graphs best represents positively skewed data with the smaller standard deviation?
 

2UG-2006-8abMC

2UG-2006-8cdMC

Show Answers Only

`C`

Show Worked Solution

`text(By elimination)`

`text(Positive skew when the tail on the`

`text(right side is longer.)`

`:.\ text(NOT)\ B\ text(or)\ D`

`text(A smaller standard deviation occurs)`

`text(when data is clustered more closely.)`

`:.\ text(NOT)\ A\ text(where data is more widely spread.)`

`=>  C`

Statistics, STD2 S1 2006 HSC 4 MC

A set of scores is displayed in a stem-and-leaf plot.
 

 2UG-2006-4MC

 
What is the median of this set of scores?

  1.   28
  2.   30
  3.   33
  4.   47
Show Answers Only

`C`

Show Worked Solution

`text(10 scores)`

`text(Median)` `= text{5th + 6th}/2`
  `= (28 + 38)/2`
  `= 33`

`=>  C`

Statistics, STD2 S1 2004 HSC 8 MC

This sector graph shows the distribution of 116 prizes won by three schools: X, Y and Z.
 

 
How many prizes were won by School X?

  1.   26
  2.   32
  3.   81
  4.   99
Show Answers Only

`B`

Show Worked Solution

`text(Centre angle of School X sector)`

`= 100^@\ text{(by measurement)}`
 

`:.\ text(Prizes won by school X)`

`= 100/360 xx 116`

`= 32.22\ …`

`=> B`

Statistics, STD2 S1 2007 HSC 24d

Barry constructed a back-to-back stem-and-leaf plot to compare the ages of his students.
 

 

  1. Write a brief statement that compares the distribution of the ages of males and females from this set of data.   (1 mark)

  2. What is the mode of this set of data?   (1 mark)

  3. Liam decided to use a grouped frequency distribution table to calculate the mean age of the students at Barry’s Ballroom Dancing Studio. 

     

    For the age group 30 - 39 years, what is the value of the product of the class centre and the frequency?   (2 marks)

  4. Liam correctly calculated the mean from the grouped frequency distribution table to be 39.5.

     

    Caitlyn correctly used the original data in the back-to-back stem-and-leaf plot and calculated the mean to be 38.2. 

     

    What is the reason for the difference in the two answers?   (1 mark)

Show Answers Only
  1. `text(More males attend than females and a higher proportion)`
    `text(of those are younger males, with the distribution being)`
    `text(positively skewed. Female attendees are generally older)`
    `text(and have a negatively skewed distribution.)`
  2. `text(Mode) = 64\ \ \ text{(4 times)}`
  3. `172.5`
  4. `text(The difference in the answers is due to the class)`
  5. `text(centres used in group frequency tables distorting)`
  6. `text(the mean value from the exact data.)`
Show Worked Solution
i. `text(More males attend than females and a higher proportion)`
  `text(of those are younger males, with the distribution being)`
  `text(positively skewed. Female attendees are generally older)`
  `text(and have a negatively skewed distribution.)`

 

ii. `text(Mode) = 64\ \ \ text{(4 times)}`

 

iii. `text(Class centre)` `= (30 + 39)/2`
    `= 34.5`
  `text(Frequency) = 5`

 
`:.\ text(Class centre) xx text(frequency)`

`= 34.5 xx 5`

`= 172.5`
 

iv. `text(The difference in the answers is due to the class)`
  `text(centres used in group frequency tables distorting)`
  `text(the mean value from the exact data.)`

Statistics, STD2 S1 2008 HSC 23a

You are organising an outside sporting event at Mathsville and have to decide which month has the best weather for your event. The average temperature must be between 20°C and 30°C, and average rainfall must be less than 80 mm.

The radar chart for Mathsville shows the average temperature for each month, and the table gives the average rainfall for each month.
 

VCAA 2008 23a
 

  1. If you consider only the temperature data, there are a number of possible months for holding the event. Name ONE of these months.  (1 mark)

  2. If both rainfall and temperature data are considered, which month is the best month for the sporting event?   (1 mark)

Show Answers Only
  1. `text(One of Feb, Mar, Nov, Dec)`
  2. `text(November)`
Show Worked Solution

i.  `text(One of Feb, Mar, Nov, Dec)`
 

ii.  `text(November)`

Statistics, STD2 S1 2008 HSC 3 MC

The stem-and-leaf plot represents the daily sales of soft drink from a vending machine.

If the range of sales is 43, what is the value of  2008 3 mc  ?

 
 

  1.    `4` 
  2.    `5`
  3.    `24`
  4.    `25`
Show Answers Only

`A`

Show Worked Solution

`text(Range = High) – text(Low) = 43`

`:.\ 67 – text(Low)` `= 43`
`text(Low)` `= 24`

`:.\ N = 4`

`=>  A`

Statistics, STD2 S1 2011 HSC 25d

Data was collected from 30 students on the number of text messages they had sent in the previous 24 hours. The set of data collected is displayed.
 

2UG 2011 25d

  1. What is the outlier for this set of data? (1 mark)

  2. What is the interquartile range of the data collected from the female students? (1 mark)

Show Answers Only
  1. `71`
  2. `9`
Show Worked Solution

i.   `text(Outlier is 71)`

♦♦ Mean mark 34%
COMMENT: Ensure you can quickly and accurately find quartile values using stem and leaf graphs!

ii.   `text{Lower quartile = 9   (4th female data point)}`

`text{Upper quartile = 20   (11th female data point)}`

`:.\ text{Interquartile range (female)}=20-11=9`

Statistics, STD2 S1 2013 HSC 26f

Jason travels to work by car on all five days of his working week, leaving home at 7 am each day. He compares his travel times using roads without tolls and roads with tolls over a period of 12 working weeks.

He records his travel times (in minutes) in a back-to-back stem-and-leaf plot.
 

2013 26f
 

  1. What is the modal travel time when he uses roads without tolls?  (1 mark)

  2. What is the median travel time when he uses roads without tolls?   (1 mark)

  3. Describe how the two data sets differ in terms of the spread and skewness of their distributions.   (2 marks)

Show Answers Only
  1. `52\ text(minutes)`
  2. `50.5\ text(minutes)`
  3. `text(Spread)`
  4. `text{Times without tolls have a tighter spread (range = 22)}`
  5. `text{than times with tolls (range = 55).}`

  6.  

    `text(Skewness)`

  7. `text(Times without tolls shows virtually no skewness while`
  8. `text(times with tolls are positively skewed.)`
Show Worked Solution

i.  `text(Modal time) = 52\ text(minutes)`

♦ Mean mark 36%
MARKER’S COMMENT: Finding a median proved challenging for many students. Take note!

 

ii.  `text(30 times with no tolls)`

`text(Median)` `=\ text(Average of 15th and 16th)`
  `=(50 + 51)/2`
  `= 50.5\ text(minutes)`

 

♦ Mean mark 39%

 

 iii.  `text(Spread)`

`text{Times without tolls have a much tighter}`

`text{spread (range = 22) than times with tolls}`

`text{(range = 55).}`

`text(Skewness)`

`text(Times without tolls shows virtually no skewness)`

`text(while times with tolls are positively skewed.)`

Statistics, STD2 S1 2010 HSC 16 MC

This back-to-back stem-and-leaf plot displays the test results for a class of 26 students.
 

2010 Q16 MC  
 

What is the median test result for the class?

  1.    `44`
  2.    `46`
  3.    `48`
  4.    `49`
Show Answers Only

`B`

Show Worked Solution
♦♦ Mean mark 35%

`text(26 results given in the data)`

  `=>text(Median is average of)\ 13^text(th)\ text(and)\ 14^text(th)`

`:.\ text(Median)` `=(45+47)/2`
  `=46`

`=>B`

Statistics, STD2 S1 2009 HSC 24a

The diagram below shows a stem-and-leaf plot for 22 scores. 
 

2UG-2009-24a
 

  1.  What is the mode for this data?   (1 mark)

  2.  What is the median for this data?     (1 mark)

Show Answers Only
  1. `78`
  2. `46`
Show Worked Solution

i.   `text(Mode) = 78`

 

ii.    `22\ text(scores)`

`=>\ text(Median is the average of 11th and 12th scores)`
 

`:.\ text(Median)` `= (45 + 47)/2`
  `= 46`

Statistics, STD2 S1 2010 HSC 1 MC

The results of a survey are displayed in the dot plot.

What is the range of this data?
 

2010 1 MC 
 

  1.    7
  2.    8
  3.    9
  4.    10
Show Answers Only

`C`

Show Worked Solution
`text(Range)` `=text(High)-text(Low)`
  `=9-0`
  `=9`

`=>  C`

Statistics, STD2 S1 2012 HSC 1 MC

A set of 15 scores is displayed in a stem-and-leaf plot.
 

2012 1 mc 
 

 What is the median of these scores?

  1.    7 
  2.    8
  3.   77
  4.   78
Show Answers Only

`D`

Show Worked Solution

`text(15 scores)\ \ =>\ \ text(Median is 8th)`

`:.\ \ text(Median is)\ \ 78`

`=>  D`