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Statistics, 2ADV S3 EQ-Bank 2

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

The random variable `X` has the probability density function

`f(x) = 1/sqrt(2pi) e^((−x^2)/2)`   where  `x ∈ (−∞, ∞)`

The diagram shows the graph of  `y = f(x)`.
 

 
 

  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

     

         `int_(−2)^2 f(x)\ dx`   (2 marks)

  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)

Show Answers Only
i.   

ii.  `0.9369`

iii.  `3.155text(%)`

Show Worked Solution
i.   

 

ii.   

 

`int_(−2)^2 f(x)` `~~ 1/2[0.0540 + 2(0.2420 + 0.3989 + 0.2420) + 0.0540]`
  `~~ 1/2 xx 1.8738`
  `~~ 0.9369`

 

iii.   `mu = 48, sigma = 6`

`ztext(-score (36))` `= (x – mu)/sigma= (32 – 48)/6=-2`

`text(Shaded area = 93.69%)`

`text(By symmetry,)`

`P(X < 36\ text(kgs))` `= P(z < −2)`
  `= (100 – 93.69)/2`
  `= 3.155text(%)`

Statistics, 2ADV S3 EQ-Bank 17

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)

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

  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
     
    `qquad qquad int_a^b f(x)\ dx`
     
    where `a` and `b` are constants and where
     
    `qquad qquad f(x) = 1/(sqrt(2pi)) e^((−x^2)/2)`
     
    is the normal probability density function with mean 0 and standard deviation 1.

     

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

  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)

Show Answers Only
  1. `16text(% have low blood pressure)`
  2. `−1.5`
  3. `9.2text(%)`
  4. `6.8text(%)`
Show Worked Solution
i.    `ztext(-score)\ (63)` `= (x – mu)/σ`
    `= (63 – 75)/12`
    `= −1`

 

`:. 16text(% have low blood pressure)`

 

ii.    `ztext(-score)` `= (57 – 75)/12`
    `= −1.5`

 

iii.  `y = 1/sqrt(2pi) e^((−x^2)/2)`

 
`text(Area)` `= h/2(y_0 + 2y_1 + y_2)`
  `~~ 0.25/2 (0.1295 + 2 xx 0.1826 + 0.2420)`
  `~~ 0.0920`
  `~~ 9.2text(%)`

 

iv.   

 

`P(text{blood pressure}\ <= 57)` `= 16 – 9.2`
  `~~ 6.8text(%)`