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Algebra, STD2 A4 2012 HSC 30c

In 2010, the city of Thagoras modelled the predicted population of the city using the equation

`P = A(1.04)^n`.

That year, the city introduced a policy to slow its population growth. The new predicted population was modelled using the equation

`P = A(b)^n`.

In both equations, `P` is the predicted population and `n` is the number of years after 2010.  

The graph shows the two predicted populations.
 

  1. Use the graph to find the predicted population of Thagoras in 2030 if the population policy had NOT been introduced.   (1 mark)

  2. In each of the two equations given, the value of `A` is 3 000 000.

     

    What does `A` represent?   (1 mark)

  3. The guess-and-check method is to be used to find the value of `b`, in  `P = A(b)^n`.

     

    (1) Explain, with or without calculations, why 1.05 is not a suitable first estimate for `b`.  (1 mark)

     

    (2) With  `n = 20`  and  `P = 4\ 460\ 000`, use the guess-and-check method and the equation  `P = A(b)^n`  to estimate the value of `b` to two decimal places. Show at least TWO estimate values for `b`, including calculations and conclusions.  (2 marks)

  4. The city of Thagoras was aiming to have a population under 7 000 000 in 2050. Does the model indicate that the city will achieve this aim?

     

    Justify your answer with suitable calculations.  (2 marks)

Show Answers Only
  1.  `6\ 600\ 000`
  2.  `text(The population in 2010.)`
  3.  `text{(1)  See Worked Solution}`

     

    `(2)\ \ b = 1.03, 1.02`

  4. `text(See Worked Solution)`
Show Worked Solution

i.    `text(2030 occurs at)\ \ n = 20\ \ text(on the)\ x text(-axis.)`

`text(Expected population (no policy) ) = 6\ 600\ 000`

COMMENT: Common ADV/STD2 content in new syllabus.

 

ii.   `A\ text(represents the population when)\ \ n=0` 

`text(which is the population in 2010.)`
 

♦ Mean mark part (ii) 48%

iii. (1)  `P = A(1.05)^n\ text(would be steeper and lie above)`

    `P = A(1.04)^n\ text(since)\ 1.05 > 1.04`
 

iii. (2)  `text(Let)\ \ b = 1.03`

`P` `= 3\ 000\ 000 xx 1.03^20`
  `= 5\ 418\ 000`

 
`text(Let)\ \ b = 1.02`

`P` `= 3\ 000\ 000 xx 1.02^20`
  `= 4\ 457\ 800`

 
`:. b = 1.02`
 

iv.  `text(In 2050,)\ n = 40`

`P` `= 3\ 000\ 000 xx 1.02^40`
  `= 6\ 624\ 119\ \ (text(nearest whole))`

 
`text(S)text(ince the population is below 7 million,)`

`text(the model will achieve the aim.)`

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 S5 2012 HSC 29b

A machine produces nails. When the machine is set correctly, the lengths of the nails are normally distributed with a mean of  6.000 cm  and a standard deviation of  0.040 cm.

To confirm the setting of the machine, three nails are randomly selected. In one sample the lengths are  5.950,  5.983 and  6.140.

The setting of the machine needs to be checked when the lengths of two or more nails in a sample lie more than 1 standard deviation from the mean.

Does the setting on the machine need to be checked? Justify your answer with suitable calculations.   (2 marks)

Show Answers Only

 `text(Settings need to be checked.)`

Show Worked Solution
♦ Mean mark 44%

`mu = 6.000,\ \ sigma = 0.040`

`text(Limits for  ±1 σ:)`

`6.000 + 0.040 = 6.040\ text{(upper)}`

`6.000-0.040 = 5.960\ text{(lower)}`

`text(Chosen nails:)\ \ 5.950` `=>\ text(outside limits)`
`5.983` `=>\ text(inside)`
`6.140` `=>\ text(outside)`

 

`text(S)text(ince 2 nails are outside 1 σ limits)`

`:.\ text(Settings need to be checked.)`

Statistics, STD2 S1 2012 HSC 28d

The test results in English and Mathematics for a class were recorded and displayed in the box-and-whisker 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 location 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%
MARKER’S COMMENT: Markers are looking for students to 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)`

Statistics, STD2 S1 2011 HSC 11 MC

The sets of data, `X` and `Y`, are displayed in the histograms.

2UG 2011 11

Which of these statements is true?

  1.   `X` has a larger mode and `Y ` has a larger range.
  2.   `X` has a larger mode and the ranges are the same.
  3.   The modes are the same and `Y` has a larger range.
  4.   The modes are the same and the ranges are the same.
Show Answers Only

`B`

Show Worked Solution
♦ Mean mark 47%

`text(Mode of)\ X=9`

`text(Range of)\ X=9-3=6`

`text(Mode of)\ Y=8`

`text(Range of)\ Y=11-5=6`

`:. X\ text(has a larger mode and ranges are the same)`

`=>B`

Statistics, STD2 S1 2011 HSC 7 MC

A set of data is displayed in this box-and-whisker 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

`B`

Show Worked Solution

`text{Since the median (155) is closer to the lower quartile (150) and range}`

`text{low (140) than the upper quartile (190) and range high (200), it is}`

`text{positively skewed.}`

`=>B`

♦ Mean mark 47%.

Statistics, STD2 S5 2010 HSC 4 MC

Which of the following frequency histograms shows data that could be normally distributed?
 

Capture1

Capture2 

Show Answers Only

`A`

Show Worked Solution

`text(Normally distributed data have a frequency)`

`text(histogram graph that is shaped like a bell.)`

`=>  A`

Probability, STD2 S2 2012 HSC 26e

The dot plot shows the number of push-ups that 13 members of a fitness class can do in one minute.

2012 26e

  1.  What is the probability that a member selected at random from the class can do more than 38 push-ups in one minute?   (1 mark)

  2.  A new member who can do 32 push-ups in one minute joins the class.

     

    Does the addition of this new member to the class change the probability calculated in part (i)? Justify your answer.    (1 mark)

Show Answers Only
  1. `7/13`
  2. `text{Yes (See Worked Solutions)}`
Show Worked Solution
i.  `P` `= text(# Members > 38 push-ups)/text(Total members)`
  `= 7/13`

 
ii.   `text(Yes.)`

`Ptext{(+ New member)}` `= text(Members > 38 push-ups)/text(Total members)`
  `= 7/14≠ 7/13`
MARKER’S COMMENT: The most successful candidates used the fraction `7/14` in their part (ii) answer rather than relying solely on words.

Statistics, STD2 S1 2012 HSC 26d

Greg needs to conduct a statistical inquiry into how much time people aged 18–25 years have spent accessing social media websites in the last two weeks. He has decided to survey a sample of students from his university.

The process of statistical inquiry includes the following steps, which are NOT in order.
 

2012 26d 

  1. Using the letters A, B, C, D, E and F, list the steps in the most appropriate order for Greg to conduct his statistical inquiry.   (2 marks)

  2. Greg conducts his statistical inquiry.

     

    At which step in the process would he have drawn this graph?    (1 mark) 

2012 26d2

Show Answers Only
  1. `text(B, E, C, F, D, A)`
  2. `text(F)`
Show Worked Solution

i.   `text(B, E, C, F, D, A)`

ii.   `text(F)`

Measurement, STD2 M6 2012 HSC 20 MC

Town `B` is 80 km due north of Town `A` and 59 km from Town `C`.

Town `A` is 31 km from Town `C`.
 

2012 20 mc
 

 What is the bearing of Town `C` from Town `B`?  

  1.   `019^@`
  2.   `122^@` 
  3.   `161^@` 
  4.   `341^@` 
Show Answers Only

`C`

Show Worked Solution
♦ Mean mark 49%

`text(Using the cosine rule:)`

`cos\ /_B` `= (a^2 + c^2 -b^2)/(2ac)`
  `= (59^2 + 80^2 -31^2)/(2 xx 59 xx 80)`
  `= 0.9449…`
`/_B` `= 19^@\  text((nearest degree))`

 

`:.\ text(Bearing of Town C from B) = 180-19= 161^@`

`=>  C`

Probability, STD2 S2 2012 HSC 17 MC

A spinner with different coloured sectors is spun 40 times. The results are recorded in the table.

 What is the relative frequency of obtaining the colour orange? 

  1.    `3/20`  
  2.    `1/5`  
  3.    `6`  
  4.    `8` 
Show Answers Only

`A`

Show Worked Solution
♦♦ Mean mark 34%
COMMENT: Note that relative frequency is the frequency of an event divided by the total frequencies (i.e. the probability).
`text(Total frequency)` `= 40\ text(spins)`
`text(Orange freq.)` `= 40\-(2 + 4 + 6 + 10 +12)`
  `=6`
`:.\ text(Relative freq.)` `= 6/40 = 3/20`

 
`=>  A`

Statistics, STD2 S4 2012 HSC 11 MC

Which of the following relationships would most likely show a negative correlation?

  1. The population of a town and the number of hospitals in that town. 
  2. The hours spent training for a race and the time taken to complete the race. 
  3. The price per litre of petrol and the number of people riding bicycles to work. 
  4. The number of pets per household and the number of computers per household. 
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Increased hours training should reduce the time}\)

\(\text{to complete a race.}\)

\(\Rightarrow B\)

♦ Mean mark 43%.

Financial Maths, STD2 F4 2012 HSC 9 MC

Tracy invests some money for 2 years at 4% per annum, compounded quarterly. 

2012 9 mc

Which figure from the table should Tracy use to calculate the value of her investment at the end of 2 years?

  1.   1.020 
  2.   1.082
  3.   1.083
  4.   1.369
Show Answers Only

`C`

Show Worked Solution
♦♦♦ Mean mark 24%. The lowest MC mean mark in the 2012 exam.
COMMENT: A process to follow: 1-convert the annual rate to the rate per compounding period. 2-calculate the number of compounding periods.

`text(4% annual)`

`=>\ text(4%)/4=text(1% compounded each quarter)`

`=>n=8\ \ \ \ \ \ text{(8 quarters in 2 years)}`

 

`:.\ text{Factor = 1.083   (from table)}`

`=>  C`

Statistics, STD2 S1 2012 HSC 7 MC

The Pi Company has two bakeries. The radar chart displays the monthly sales for the bakeries.
 

2012 7 mc 

  
What was the difference in sales in June between the two bakeries?

  1.   $7.50
  2.   $17.50
  3.   $7500
  4.   $17 500
Show Answers Only

`D`

Show Worked Solution
`text(Bakery 1 Sales in June)` `= $17\ 500`
`text(Bakery 2 Sales in June)` `=$35\ 000`
`:.\ text(Difference in Sales)` `= 35\ 000 \-17\ 500`
  `= 17\ 500`

`=>  D`

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`

Statistics, STD2 S5 2013 HSC 20 MC

There are  60 000  students sitting a state-wide examination. If the results form a normal distribution, how many students would be expected to score a result between 1 and 2 standard deviations above the mean?

You may assume for normally distributed data that:

    • 68% of scores have  `z`-scores between  – 1 and 1
    • 95% of scores have  `z`-scores between  – 2 and 2
    • 99.7% of scores have  `z`-scores between  – 3 and 3.
  1. `8100`
  2. `16\ 200`
  3. `20\ 400`
  4. `28\ 500`
Show Answers Only

`A`

Show Worked Solution
♦♦ Mean mark 24% 

`text(S)text(ince 68% between)\ z=1\ text(and)\ -1`

`  =>  text(34% between)\  z=0\ text(and)\   1`

`text(Likewise, 95% between)\ z=2\ text(and)  -2`

`  =>  text(47.5% between)\ z=0\ text(and)\ 2`
 

`:.\ text(% between)\ z=1\ text(and)\ 2`

`=47.5-34`

`=13.5 text(%)`

 
`:.\ text(Students between)\ \ z=1\  text(and)\ \ z=2`

`=13.5 text(%)xx60\ 000`

`=8100`

`=>\ A`

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`

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`

Probability, STD2 S2 2013 HSC 10 MC

Students studying vocational education courses were surveyed about their living arrangements.

One of these students is selected at random.

 What is the probability that this student is male and living with his parent(s)?

  1. 31%
  2. 40%
  3. 56%
  4. 77%
Show Answers Only

`A`

Show Worked Solution

`text(Number of males living with their parents is = 155)`.

`:.\ P=155/505=0.30693…%`

`=>\ A`

Statistics, STD2 S1 2013 HSC 8 MC

A high school has 100 students in each year group, Year 7 to Year 12. A survey is to be conducted to determine the average number of text messages sent per month by students at the school.

Which of the following would provide the most representative sample for this survey?

  1. All Year 7 students
  2. All physics students in Year 11 and 12
  3. 20 students chosen at random from each year group
  4. 120 students chosen at random from the school roll
Show Answers Only

`C`

Show Worked Solution

`text(The best sample would have an equal amount)`

`text(of people in each year randomly selected.)`

`=>\ C`

Probability, STD2 S2 2013 HSC 7 MC

In an experiment, a standard six-sided die was rolled 72 times. The results are shown in the table.
 

Which number on the die was obtained the expected number of times?

  1.    1
  2.    2
  3.    3
  4.    6
Show Answers Only

`B`

Show Worked Solution

`text(Probability of rolling a specific number)=1/6`

`:.\ text(After 72 rolls, a specific number is expected)`

 `1/6xx72=12\ text(times.)`

`=>\ B`

Statistics, STD2 S1 2013 HSC 6 MC

A survey was conducted where people were asked which of two brands of smartphones they preferred. The results were:

  • 48% preferred Brand X
  • 52% preferred Brand Y

A graph displaying the data is to be included in a magazine article. The editor of the magazine wishes to ensure that the graph is not misleading in any way.

Which graph should the editor choose to include in the article?
 

2013 6 mc1

2013 6 mc2

Show Answers Only

`D`

Show Worked Solution

`D\ text(is the best as it starts at zero on the)\ y text(-axis)`

`text(and has the same column widths.)`

Statistics, STD2 S4 2013 HSC 2 MC

Which graph best shows data with a correlation closest to 0.3?
 

2013 2 mc1

2013 2 mc2

 

Show Answers Only

`A`

Show Worked Solution

`A\ text(is correct since the data slopes bottom)`

`text{left to top right (i.e. it’s positive).}`

`D\ text(also slopes correctly but exhibits a higher)`

`text(correlation co-efficient.)`

`=>A`