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Functions, MET1 2024 VCAA 5

The function    models the population of a town after years.

  1. Use the model to predict the population of the town after four years.   (1 mark)

  2. A new function, , models a population where  but decreases at half the rate of at any point in time.
  3. State a sequence of two transformations that maps to this new model .   (2 marks)

  4. In the town, 100 people were randomly selected and surveyed, with 60 people indicating that they were unhappy with the roads.
    1. Determine an approximate 95% confidence interval for the proportion of people in the town who are unhappy with the roads.
    2. Use    for this confidence interval.   (2 marks)
    3. A new sample of people results in the same sample proportion.
    4. Find the smallest value of to achieve a standard deviation of    for the sample proportion.   (1 mark)

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a.   

b.   

ci.   

cii. 

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a. 

b.    
 
 
 

♦♦♦ Mean mark (a) 17%.


 


 

ci.   

 
 

  

cii.  
 
 
 
 
   
♦♦ Mean mark (c.ii.) 33%.

Probability, MET2 2022 VCAA 3

Mika is flipping a coin. The unbiased coin has a probability of of landing on heads and of landing on tails.

Let be the binomial random variable representing the number of times that the coin lands on heads.

Mika flips the coin five times.

    1. Find .   (1 mark)
    2. Find (1 mark)
    3. Find , correct to three decimal places.   (2 marks)
    4. Find the expected value and the standard deviation for .   (2 marks)

The height reached by each of Mika's coin flips is given by a continuous random variable, , with the probability density function
  


  

where is the vertical height reached by the coin flip, in metres, between the coin and the floor, and and are real constants.

    1. State the value of the definite integral .   (1 mark)
    2. Given that    and  , find the values of and .   (3 marks)

    3. The ceiling of Mika's room is 3 m above the floor. The minimum distance between the coin and the ceiling is a continuous random variable, , with probability density function .
    4. The function is a transformation of the function given by , where is the minimum distance between the coin and the ceiling, and and are real constants.
    5. Find the values of and .   (1 mark)

  1. Mika's sister Bella also has a coin. On each flip, Bella's coin has a probability of of landing on heads and of landing on tails, where is a constant value between 0 and 1 .
  2. Bella flips her coin 25 times in order to estimate .
  3. Let be the random variable representing the proportion of times that Bella's coin lands on heads in her sample.
    1. Is the random variable discrete or continuous? Justify your answer.   (1 mark)
    2. If , find an approximate 95% confidence interval for , correct to three decimal places.   (1 mark)
    3. Bella knows that she can decrease the width of a 95% confidence interval by using a larger sample of coin flips.
    4. If , how many coin flips would be required to halve the width of the confidence interval found in part c.ii.?   (1 mark)

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a. i.        ii.        iii.    (3 d.p.)

a. iv   

b. i.      

b. ii.   

b. iii.

c. i.      ii.      iii.   

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a.i 


 

a.ii  By CAS:        

  
a.iii 

  
By CAS:
 

(3 decimal places)
  

a.iv

  

b.i  


♦♦ Mean mark (b.i) 40%.
MARKER’S COMMENT: Many students did not evaluate the integral or evaluated incorrectly.

b.ii  By CAS:


 


♦♦ Mean mark (b.ii) 47%.
MARKER’S COMMENT: Many students did not give exact answers.

b.iii 

and


♦♦♦♦ Mean mark (b.iii) 10%.
MARKER’S COMMENT: Many students did not attempt this question.

c.i    is discrete.

The number of coin flips must be zero or a positive integer so    is countable and therefore discrete.
 

c.ii 


 

c.iii To halve the width of the confidence interval, the standard deviation needs to be halved.

She would need to flip the coin 100 times


♦♦♦ Mean mark (c.iii) 30%.
MARKER’S COMMENT: Common incorrect answers were 0, 10, 11, 50 and 101.

Probability, MET2 2022 VCAA 10 MC

An organisation randomly surveyed 1000 Australian adults and found that 55% of those surveyed were happy with their level of physical activity.

An approximate 95% confidence interval for the percentage of Australian adults who were happy with their level of physical activity is closest to

  1. (4.1, 6.9)
  2. (50.9, 59.1)
  3. (52.4, 57.6)
  4. (51.9, 58.1)
  5. (45.2, 64.8)
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Show Worked Solution

, , with approximate 95% confidence interval () for

Confidence Interval  
   
   

 

Probability, MET2 2023 VCAA 4

A manufacturer produces tennis balls.

The diameter of the tennis balls is a normally distributed random variable , which has a mean of 6.7 cm and a standard deviation of 0.1 cm.

  1. Find  , correct to four decimal places.   (1 mark)
  2. Find the minimum diameter of a tennis ball that is larger than 90% of all tennis balls produced.
  3. Give your answer in centimetres, correct to two decimal places.   (1 mark)

Tennis balls are packed and sold in cylindrical containers. A tennis ball can fit through the opening at the top of the container if its diameter is smaller than 6.95 cm.

  1. Find the probability that a randomly selected tennis ball can fit through the opening at the top of the container.
  2. Give your answer correct to four decimal places.   (1 mark)
  3. In a random selection of 4 tennis balls, find the probability that at least 3 balls can fit through the opening at the top of the container.
  4. Give your answer correct to four decimal places.   (2 marks)

A tennis ball is classed as grade A if its diameter is between 6.54 cm and 6.86 cm, otherwise it is classed as grade B.

  1. Given that a tennis ball can fit through the opening at the top of the container, find the probability that it is classed as grade A.
  2. Give your answer correct to four decimal places.   (2 marks)
  3. The manufacturer would like to improve processes to ensure that more than 99% of all tennis balls produced are classed as grade A.
  4. Assuming that the mean diameter of the tennis balls remains the same, find the required standard deviation of the diameter, in centimetres, correct to two decimal places.   (2 marks)
  5. An inspector takes a random sample of 32 tennis balls from the manufacturer and determines a confidence interval for the population proportion of grade A balls produced.
  6. The confidence interval is (0.7382, 0.9493), correct to four decimal places.
  7. Find the level of confidence that the population proportion of grade A balls is within the interval, as a percentage correct to the nearest integer.   (2 marks)

A tennis coach uses both grade A and grade B balls. The serving speed, in metres per second, of a grade A ball is a continuous random variable, , with the probability density function
 


 

  1. Find the probability that the serving speed of a grade A ball exceeds 50 metres per second.
  2. Give your answer correct to four decimal places.   (1 mark)
  3. Find the exact mean serving speed for grade A balls, in metres per second.   (1 mark)

The serving speed of a grade B ball is given by a continuous random variable, , with the probability density function .

A transformation maps the graph of to the graph of , where .

  1. If the mean serving speed for a grade B ball is metres per second, find the values of and .   (2 marks)
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a.   

b.   

c.   

d.   

e.   

f.   

g.   

h.   

i.   

j.   

Show Worked Solution

a.    


 

b.     

 

c.  


 

d.   

 

e.   
   
 
   
   

  

f.    


♦♦ Mean mark (f) 35%.
MARKER’S COMMENT: Incorrect responses included using . Many answers were accepted in the range as max value of SD was not asked for provided sufficient working was shown.

g.  

    
   
 
 

 

 


 


♦♦ Mean mark (g) 25%.
MARKER’S COMMENT: was often calculated incorrectly with frequently seen.

h.  

i.  

j. 
 


 

 

 

 
 
 
 

♦♦♦ Mean mark (j) 10%.

Statistics, MET1 2023 VCAA 6

Let be the random variable that represents the sample proportion of households in a given suburb that have solar panels installed.

From a sample of randomly selected households in a given suburb, an approximate 95% confidence interval for the proportion of households having solar panels installed was determined to be (0.04, 0.16).

  1. Find the value of that was used to obtain this approximate 95% confidence interval.   (1 mark)

Use to approximate the 95% confidence interval.

  1. Find the size of the sample from which this 95% confidence interval was obtained.   (2 marks)
  2. A larger sample of households is selected, with a sample size four times the original sample.
  3. The sample proportion of households having solar panels installed is found to be the same.
  4. By what factor will the increased sample size affect the width of the confidence interval?   (1 mark)
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a.   

b.   

c.   

Show Worked Solution

a.   


♦ Mean mark (a) 52%.
b.   
 
 
 
 
 

♦♦ Mean mark (b) 38%.

c.   

→

→


♦♦♦ Mean mark (c) 23%.

Statistics, MET2 2021 VCAA 3 MC

A box contains many coloured glass beads.

A random sample of 48 beads is selected and it is found that the proportion of blue-coloured beads in this sample is 0.125

Based on this sample, a 95% confidence interval for the proportion of blue-coloured glass beads is

  1. (0.0314, 0.2186)
  2. (0.0465, 0.2035)
  3. (0.0018, 0.2482)
  4. (0.0896, 0.1604)
  5. (0.0264, 0.2136)
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Show Worked Solution


  


 

 

 

 

Statistics, MET1-NHT 2018 VCAA 8

Let    be the random variable that represents the sample proportions of customers who bring their own shopping bags to a large shopping centre.

From a sample consisting of all customers on a particular day, an approximate 95% confidence interval for the proportion    of customers who bring their own shopping bags to this large shopping centre was determined to be  .

  1. Find the value of    that was used to obtain this approximate 95% confidence interval.   (1 mark)

  2. Use the fact that    to find the size of the sample from which this approximate 95% confidence interval was obtained.   (2 marks)

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a.   

 
 

 

b.   

Statistics, MET2-NHT 2019 VCAA 3

Concerts at the Mathsland Concert Hall begin minutes after the scheduled starting time. is a random variable that is normally distributed with a mean of 10 minutes and a standard deviation of four minutes.

  1. What proportion of concerts begin before the scheduled starting time, correct to four decimal places?   (1 mark)

  2. Find the probability that a concert begins more than 15 minutes after the scheduled starting time, correct to four decimal places.   (1 mark)

If a concert begins more than 15 minutes after the scheduled starting time, the cleaner is given an extra payment of $200. If a concert begins up to 15 minutes after the scheduled starting time, the cleaner is given an extra payment of $100. If a concert begins at or before the scheduled starting time, there is no extra payment for the cleaner.

Let be the random variable that represents the extra payment for the cleaner, in dollars.

    1. Using your responses from part a. and part b., complete the following table, correct to three decimal places.   (1 mark)

       

    2. Calculate the expected value of the extra payment for the cleaner, to the nearest dollar.   (1 mark)

    3. Calculate the standard deviation of , correct to the nearest dollar.   (1 mark)

The owners of the Mathsland Concert Hall decide to review their operation. They study information from 1000 concerts at other similar venues, collected as a simple random sample. The sample value for the number of concerts that start more than 15 minutes after the scheduled starting time is 43.

    1. Find the 95% confidence interval for the proportion of the concerts that begin more than 15 minutes after the scheduled starting time. Give values correct to three decimal places.   (1 mark)

    2. Explain why this confidence interval suggests that the proportion of concerts that begin more than 15 minutes after the scheduled starting time at the Mathsland Concert Hall is different from the proportion at the venue in the sample.   (1 mark)

The owners of the Mathsland Concert Hall decide that concerts must not begin before the scheduled starting time. They also make changes to reduce the number of concerts that begin after the scheduled starting time. Following these changes, is the random variable that represents the number of minutes after the scheduled starting time that concerts begin. The probability density function for is
 


 

where is the the time, in minutes, after the scheduled starting time.

  1. Calculate the expected value of .   (2 marks) 
    1. Find the probability that a concert now begins more than 15 minutes after the scheduled starting time.   (1 mark)

    2. Find the probability that each of the next nine concerts begins more than 15 minutes after the scheduled starting time and the 10th concert begins more than 15 minutes after the scheduled starting time. Give your answer correct to four decimal places.   (2 marks)

    3. Find the probability that a concert begins up to 20 minutes after the scheduled starting time, given that it begins more than 15 minutes after the scheduled starting time. Give your answer correct to three decimal places.   (2 marks)

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  3. i.   

     ii.   
     iii. 
  4. i.   
    ii.    

     

  5.  
  6. i.   
    ii.   
    iii.  

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a.   

 

 

b.   
 

 

c.i.

ii. 
 
 

 

iii. 
 

 

 
 
 
 

 

 
d.i. 

 

 
ii.   

 

e.   
 

 

f.i.   
 

 
ii. 
 

 
iii. 

 
 

 
 
 

Probability, MET2-NHT 2019 VCAA 19 MC

A random sample of computer users was surveyed about whether the users had played a particular computer game. An approximate 95% confidence interval for the proportion of computer users who had played this game was calculated from this random sample to be (0.6668, 0.8147).

The number of computer users in the sample is closest to

  1.      5
  2.    33
  3.  135
  4.  150
  5.  180
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Statistics, MET2 2019 VCAA 4

The Lorenz birdwing is the largest butterfly in Town A.

The probability density function that describes its life span, , in weeks, is given by
 


 

  1. Find the mean life span of the Lorenz birdwing butterfly.  (2 marks)

  2. In a sample of 80 Lorenz birdwing butterflies, how many butterflies are expected to live longer than two weeks, correct to the nearest integer?  (2 marks)

  3. What is the probability that a Lorenz birdwing butterfly lives for at least four weeks, given that it lives for at least two weeks, correct to four decimal places?  (2 marks)

The wingspans of Lorenz birdwing butterflies in Town A are normally distributed with a mean of 14.1 cm and a standard deviation of 2.1 cm.

  1. Find the probability that a randomly selected Lorenz birdwing butterfly in Town A has a wingspan between 16 cm and 18 cm, correct to four decimal places.  (1 mark)

  2. A Lorenz birdwing butterfly is considered to be very small if its wingspan is in the smallest 5% of all the Lorenz birdwing butterflies in Town A.

     

    Find the greatest possible wingspan, in centimetres, for a very small Lorenz birdwing butterfly in Town A, correct to one decimal place.  (1 mark)

Each year, a detailed study is conducted on a random sample of 36 Lorenz birdwing butterflies in Town A.

A Lorenz birdwing butterfly is considered to be very large if its wingspan is greater than 17.5 cm. The probability that the wingspan of any Lorenz birdwing butterfly in Town A is greater than 17.5 cm is 0.0527, correct to four decimal places.

    1. Find the probability that three or more of the butterflies, in a random sample of 36 Lorenz birdwing butterflies from Town A, are very large, correct to four decimal places.  (1 mark)

    2. The probability that or more butterflies, in a random sample of 36 Lorenz birdwing butterflies from Town A, are very large is less than 1%.

       

      Find the smallest value of , where is an integer.  (2 marks)

    3. For random samples of 36 Lorenz birdwing butterflies in Town A, is the random variable that represents the proportion of butterflies that are very large.
    4. Find the expected value and the standard deviation of , correct to four decimal places.  (2 marks)

    5. What is the probability that a sample proportion of butterflies that are very large lies within one standard deviation of 0.0527, correct to four decimal places? Do not use a normal approximation.  (2 marks)

  2. The Lorenz birdwing butterfly also lives in Town B.

     

    In a particular sample of Lorenz birdwing butterflies from Town B, an approximate 95% confidence interval for the proportion of butterflies that are very large was calculated to be (0.0234, 0.0866), correct to four decimal places.

     

    Determine the sample size used in the calculation of this confidence interval.  (2 marks)

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a.   
   
   

 

b.   
   
   

 

 
 

 

c.   
   
   

 

d.   

 

e.   

 

 

f.i.   

 

f.ii.   
 

 

f.iii.   
 

 

f.iv.       

 

g.   

Probability, MET2 2016 VCAA 3*

A school has a class set of 22 new laptops kept in a recharging trolley. Provided each laptop is correctly plugged into the trolley after use, its battery recharges.

On a particular day, a class of 22 students uses the laptops. All laptop batteries are fully charged at the start of the lesson. Each student uses and returns exactly one laptop. The probability that a student does not correctly plug their laptop into the trolley at the end of the lesson is 10%. The correctness of any student’s plugging-in is independent of any other student’s correctness.

  1. Determine the probability that at least one of the laptops is not correctly plugged into the trolley at the end of the lesson. Give your answer correct to four decimal places.   (2 marks)

  2. A teacher observes that at least one of the returned laptops is not correctly plugged into the trolley.
  3. Given this, find the probability that fewer than five laptops are not correctly plugged in. Give your answer correct to four decimal places.   (2 marks)

The time for which a laptop will work without recharging (the battery life) is normally distributed, with a mean of three hours and 10 minutes and standard deviation of six minutes. Suppose that the laptops remain out of the recharging trolley for three hours.

  1. For any one laptop, find the probability that it will stop working by the end of these three hours. Give your answer correct to four decimal places.   (2 marks)

A supplier of laptops decides to take a sample of 100 new laptops from a number of different schools. For samples of size 100 from the population of laptops with a mean battery life of three hours and 10 minutes and standard deviation of six minutes,  is the random variable of the distribution of sample proportions of laptops with a battery life of less than three hours.

  1. Find the probability that . Give your answer correct to three decimal places. Do not use a normal approximation.   (3 marks)

It is known that when laptops have been used regularly in a school for six months, their battery life is still normally distributed but the mean battery life drops to three hours. It is also known that only 12% of such laptops work for more than three hours and 10 minutes.

  1. Find the standard deviation for the normal distribution that applies to the battery life of laptops that have been used regularly in a school for six months, correct to four decimal places.   (2 marks)

The laptop supplier collects a sample of 100 laptops that have been used for six months from a number of different schools and tests their battery life. The laptop supplier wishes to estimate the proportion of such laptops with a battery life of less than three hours.

  1. Suppose the supplier tests the battery life of the laptops one at a time.
  2. Find the probability that the first laptop found to have a battery life of less than three hours is the third one.   (1 mark)

The laptop supplier finds that, in a particular sample of 100 laptops, six of them have a battery life of less than three hours.

  1. Determine the 95% confidence interval for the supplier’s estimate of the proportion of interest. Give values correct to two decimal places.   (1 mark)

  2. The supplier also provides laptops to businesses. The probability density function for battery life, (in minutes), of a laptop after six months of use in a business is
     

     


     

  3. Find the mean battery life, in minutes, of a laptop with six months of business use, correct to two decimal places.   (1 mark)

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Show Worked Solution

a.  

 

 
 

 

 b.   

MARKER’S COMMENT: Early rounding was a common mistake, producing 0.9312.

 

c.   

MARKER’S COMMENT: Some working must be shown for full marks in questions worth more than 1 mark.

 

d.   

♦ Mean mark part (d) 33%.

 
 
 

 

e.   

 

f.   
   

 

g.   

 

h.