A teacher coaches their school's table tennis team.

The teacher has an adjustable ball machine that they use to help the players practise.

The speed, measured in metres per second, of the balls shot by the ball machine is a normally distributed random variable .

The teacher sets the ball machine with a mean speed of 10 metres per second and standard deviation of 0.8 metres per second.

1. Determine  , correct to three decimal places.   (1 mark)
   

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2. Find the value of , in metres per second, which 80% of ball speeds are below. Give your answer in metres per second, correct to one decimal place.   (1 mark) 
   

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The teacher adjusts the height setting for the ball machine. The machine now shoots balls high above the table tennis table.

Unfortunately, with the new height setting, 8% of balls do not land on the table.

Let    be the random variable representing the sample proportion of the balls that do not land on the table in random samples of 25 balls.

3. Find the mean and the standard deviation of  .   (2 marks)
   

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4. Use the binomial distribution to find  , correct to three decimal places.   (2 marks) 
   

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The teacher can also adjust the spin setting on the ball machine.

The spin, measured in revolutions per second, is a continuous random variable  with the probability density function
 

5. Find the maximum possible spin applied by the ball machine, in revolutions per second.   (1 mark)
   

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