Let  denote a normal random variable with mean 0 and standard deviation 1.

The random variable has the probability density function

   where 

The diagram shows the graph of  .
 

1. Complete the table of values for the given function, correct to four decimal places.  (1 mark)
    
   
    
2. Use the trapezoidal rule and 5 function values in the table in part i. to estimate
   

    

           (2 marks)
   

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3. The weights of Rhodesian ridgebacks are normally distributed with a mean of 48 kilograms and a standard deviation of 6 kilograms.
   

    

   Using the result from part ii., calculate the probability of a randomly selected Rhodesian ridgeback weighing less than 36 kilograms.  (2 marks)
   

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The diastolic measurement for blood pressure in 35-year-old people is normally distributed, with a mean of 75 and a standard deviation of 12.

1. A person is considered to have low blood pressure if their diastolic measurement is 63 or less.What percentage of 35-year-olds have low blood pressure?  (1 mark)
   

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2. Calculate the -score for a diastolic measurement of 57.  (1 mark)
   

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3. The probability that a 35-year-old person has a diastolic measurement for blood pressure between 57 and 63 can be found by evaluating
    
   
    
   where and are constants and where
    
   
    
   is the normal probability density function with mean 0 and standard deviation 1.
   

    

   By first finding the values and , calculate an approximate value for this probability by using the trapezoidal rule with 3 function values.  (3 marks)
   

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4. Hence, find the approximate probability that a 35-year-old person chosen at random has a diastolic measurement of 57 or less.  (1 mark)
   

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