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Statistics, STD2 S5 2024 HSC 35

A random variable is normally distributed with mean 0 and standard deviation 1. The table gives the probability that this random variable is less than .

The probability values given in the table for different values of are represented by the shaded area in the following diagram.
 

The scores in a university examination with a large number of candidates are normally distributed with mean 58 and standard deviation 15.

  1. By calculating a -score, find the percentage of scores that are between 58 and 70.   (2 marks)
  2. Explain why the percentage of scores between 46 and 70 is twice your answer to part (a).   (1 mark)
  3. By using the values in the table above, find an approximate minimum score that a candidate would need to be placed in the top 10% of the candidates.   (2 marks)
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a.   

b.  

c.   

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a.   


 

♦♦ Mean mark (a) 36%.

b.  


 

♦♦♦ Mean mark (b) 19%.

c.   

 
   
   

 

♦♦ Mean mark (c) 30%.

Statistics, STD2 S5 2022 HSC 37

The life span of batteries from a particular factory is normally distributed with a mean of 840 hours and a standard deviation of 80 hours.

It is known from statistical tables that for this distribution approximately 60% of the batteries have a life span of less than 860 hours.

What is the approximate percentage of batteries with a life span between 820 and 920 hours?  (3 marks)

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♦♦ Mean mark 28%.

Statistics, STD2 S5 2022 HSC 13 MC

A random variable is normally distributed with mean 0 and standard deviation 1 . The table gives the probability that this random variable lies below for some positive values of .
 

The probability values given in the table are represented by the shaded area in the following diagram.

What is the probability that a normally distributed random variable with mean 0 and standard deviation 1 lies between 0 and 1.94 ?

  1. 0.0262
  2. 0.4738
  3. 0.5262
  4. 0.9738
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♦♦♦ Mean mark 22%.

Statistics, STD2 S5 2019 HSC 15 MC

The scores on an examination are normally distributed with a mean of 70 and a standard deviation of 6. Michael received a score on the examination between the lower quartile and the upper quartile of the scores.

Which shaded region most accurately represents where Michael's score lies?
 

A. B.
C. D.
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σ

♦♦♦ Mean mark 17%.

Statistics, STD2 S5 2017 HSC 29d

All the students in a class of 30 did a test.

The marks, out of 10, are shown in the dot plot.
 


 

  1. Find the median test mark.  (1 mark)

  2. The mean test mark is 5.4. The standard deviation of the test marks is 4.22.

     

    Using the dot plot, calculate the percentage of the marks which lie within one standard deviation of the mean.  (2 marks)

  3. A student states that for any data set, 68% of the scores should lie within one standard deviation of the mean. With reference to the dot plot, explain why the student’s statement is NOT relevant in this context.  (1 mark)

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♦ Mean mark 50%.
i.   
   
   

 

ii.   

♦♦ Mean mark 34%.

 

iii.   

♦♦♦ Mean mark 13%.

Statistics, STD2 S5 SM-Bank 2 MC

The pulse rates of a large group of 18-year-old students are approximately normally distributed with a mean of 75 beats/minute and a standard deviation of 11 beats/minute. 

The percentage of 18-year-old students with pulse rates less than 53 beats/minute or greater than 86 beats/minute is closest to

  1.  
  2.  
  3.  
  4.  
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COMMENT: Two applications of the 68-95-99.7% rule are required. A good strategy is to first draw a normal curve and shade the required areas.
   
   
 

core 2008 VCAA 6-7

 

 

Statistics, STD2 S5 2004 HSC 24c

The normal distribution shown has a mean of 170 and a standard deviation of 10.
 


 

  1. Roberto has a raw score in the shaded region. What could his -score be?  (1 mark)

  2. What percentage of the data lies in the shaded region?  (2 marks)

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i. 
 
 

 

 

 

 

ii.   

Statistics, STD2 S5 2014 HSC 24 MC

The weights of  10 000 newborn babies in NSW are normally distributed. These weights have a mean of 3.1 kg and a standard deviation of 0.35 kg.

How many of these newborn babies have a weight between 2.75 kg and 4.15 kg?

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♦ Mean mark 46%

 

 

 

Statistics, STD2 S5 2009 HSC 25d

In Broken Hill, the maximum temperature for each day has been recorded. The mean of these maximum temperatures during spring is 25.8°C, and their standard deviation is 4.2° C. 

  1. What temperature has a -score of  –1?    (1 mark)

  2. What percentage of spring days in Broken Hill would have maximum temperatures between 21.6° C and 38.4°C?

     

    You may assume that these maximum temperatures are normally distributed and that

  3.  

    • 68% of maximum temperatures have -scores between –1 and 1
    • 95% of maximum temperatures have -scores between –2 and 2
    • 99.7% of maximum temperatures have -scores between –3 and 3.   (3 marks)

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i.   

 

 

 

ii.   

♦ Mean mark 41%
MARKER’S COMMENT: Many students failed to use the answer to (d)(i), costing them valuable exam time.

 

 

 

 

Statistics, STD2 S5 2010 HSC 24c

The marks in a class test are normally distributed. The mean is 100 and the standard deviation is 10.

  1. Jason's mark is 115. What is his  -score?  (1 mark)

  2. Mary has a -score of 0. What mark did she achieve in the test?   (1 mark)

  3. What percentage of marks lie between 80 and 110?

     

    You may assume the following:

     

    • 68% of marks have a -score between –1 and 1

     

    • 95% of marks have a -score between  –2 and 2

     

    • 99.7% of marks have a -score between –3 and 3.   (2 marks) 

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i.     

MARKER’S COMMENT: Too may students had calculator errors in this question, giving away easy marks. BE CAREFUL!

 
 

 

ii.     

 

♦ Mean mark 42%
iii.   

 

 

 

 

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)

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♦ Mean mark 44%

σ

 

σ

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  -scores between  – 1 and 1
    • 95% of scores have  -scores between  – 2 and 2
    • 99.7% of scores have  -scores between  – 3 and 3.
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♦♦ Mean mark 24%