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Statistics, 2ADV S3 2024 HSC 25

A function is defined as

where is a constant.

  1. Find the value of such that is a probability density function.   (2 marks)

  2. By first finding a formula for the cumulative distribution function, sketch its graph.   (2 marks)

  3. Find the value of the median of the probability density function . Give your answer correct to 3 decimal places.   (2 marks)

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a.   

b.   

c.   

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a.   

 
 
 
 

  
b.   


 

♦♦ Mean mark (b) 27%.

c.   

   
 
 
 
 
 
 

 

   
   
♦♦ Mean mark (c) 38%.

Statistics, 2ADV S3 2023 HSC 29

A continuous random variable has probability density function given by
 

 

  1. Find the mode of .  (2 marks)

  2. Find the cumulative distribution function for the given probability density function.  (2 marks)

  3. Without calculating the median, show that the mode is greater than the median.  (2 marks)

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a.   

b.   

c.   

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a.    

♦ Mean mark (a) 45%.


  

b.    
   

 


 

 
c.   

 
   

 

♦♦♦ Mean mark (c) 17%.

Statistics, 2ADV S3 2021 HSC 33

People are given a maximum of six hours to complete a puzzle. The time spent on the puzzle, in hours, can be modelled using the continuous random variable which has probability density function

The graph of the probability density function is shown below. The graph has a local maximum.
 

  1. Show that  .  (2 marks)

  2. Show that the mode of is two hours.  (2 marks)

  3. Show that  .  (2 marks)

  4. The Intelligence Quotient (IQ) scores of people are normally distributed with a mean of 100 and standard deviation of 15.
  5. It has been observed that the puzzle is generally completed more quickly by people with a high IQ.
  6. It is known that 80% of people with an IQ greater than 130 can complete the puzzle in less than two hours.
  7. A person chosen at random can complete the puzzle in less than two hours.
  8. What is the probability that this person has an IQ greater than 130? Give your answer correct to three decimal places.  (2 marks)

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a. 

b. 

♦♦♦ Mean mark part (b) 24%.

♦♦ Mean mark part (c) 30%.
c.   
   
   
   
   
   

 

d.  

♦♦ Mean mark part (d) 25%.

Statistics, 2ADV S3 2021 HSC 30

The number of hours for which light bulbs will work before failing can be modelled by the random variable with cumulative distribution function

Jane sells light bulbs and promises that they will work for longer than exactly 99% of all light bulbs.

Find how long, according to Jane’s promise, a light bulb bought from her should work. Give your answer in hours, rounded to two decimal places.  (2 marks)

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♦♦ Mean mark 22%.
 
 

 

Statistics, 2ADV S3 EQ-Bank 1

A probability density function can be used to model the lifespan of a termite, , in weeks, is given by
 


 

  1. Show that the value of    is  .  (2 marks)

  2. Find the cumulative distribution function.  (2 marks)

  3. Find the probability that a termite's lifespan is greater than 5 weeks.  (1 mark)

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i.   
 
 
 
 

 

ii.   
   
   
   
   
   
   

 

 

iii.   
   
   
   

Statistics, 2ADV S3 EQ-Bank 16

A probability density function is defined by
 


 

  1. Find the value of  .  (2 marks)

  2. Sketch the probability density function.  (1 mark)

  3. Find an expression for the cumulative distribution function.  (2 marks)

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  2.  
Show Worked Solution

i.   

 

ii.   

 

iii.   


 

COMMENT: Understand why you can check your equation here by confirming