"Denkopdrachten" Normaalverdeling

We write N(m,s) to denote a normal dist. with mean m and
standard deviation s.  So the IQ test is N(100,16) and 
Dutch men's height is N(181,6) (in cm).

Then make sure you can do the following.

1. Given N(100,16) convert the following to Z-scores:

   92,   132, 164,  90

z= -0.5,   2,   4,  -0.625

  vergeet de min teken NIET!

2. Given N(181cm,6cm) convert the following heights to 
Z-scores:

  187 cm, 172 cm, 181 cm, 160 cm

z=  1,     -1.5     0      -3.5   

3. Given N(100,16), what percentile do the following
scores fall on?

     108,  96,   120,    88
    z= 0.5, -0.25   1.25  -0.75

%ile  69th, 40th,  89th,  23rd


4. Given that Dutch women aged 18-24 jr. have a normal height
distribution with mean 172cm and sd 6cm, what percentage of 
them will have heights in the following range?

  a. between 166 cm and 178 cm  (abbreviated: 166 <= h <= 178)

     =within 1 sd of mean, so 68%

  b. up to 175 cm (<= 175)

     corresponds to z=0.5, at 69th %-ile, so 69% are shorter

  c. more than 180 cm (> 180)

     corresponds to z=1.33, at 90th %-ile, so 10% are taller

  d. 160 <= h <= 180

     i. we calculate first how many are taller than 160 cm., which
    is -2 sd's below the mean, i.e. z_160 = -2.  We've memorized 
    that 95% is between -2 and +2 standard deviations, so 2.5% is 
    more than 2 above the mean (and 2.5% below -2 sd.)  This 
    means that 97.5 is taller than -2 sd's (160cm).
        but 10% is taller that 180 cm. (question 4c)
        so 87.5 is between 160 cm. and 180 cm.    

5. Only the "top p%" have a chance of winning a scholarship.

  a. What is the top 10% of a N(100,16) distribution?

  i. we check the tables to see find the z value above which 10%
    of the data occurs.  This is z=1.3

         z   = (x - m)/sd

         1.3 = (x-100)/16

         1.3 * 16 + 100 = x 

         x = 120.8 

  b. What is the top 5% of a N(500,100) distribution?

   i. as above, find the z value above which 5% fall, z=1.65

        z   = (x - m)/sd

        1.65 = (x - 500)/100
       
        1.65 * 100 + 500 = x

        x = 665

6. The bottom 5% of a satisfaction survey should be checked 
carefully.  The survey had a near normal distribution of 
N(3.1,0.7)   Which satisfaction scores will lead to further
checking?

   1. as in 5b, we read z=-1.65 from the Table A

        z    = (x - m)/sd

       -1.65 = (x - 3.1)/0.7

       -1.65 * 0.7 + 3.1 = x

        x = -1.155 + 3.1 = 1.945

      Scores below about 2 should be checked carefully.    
  

7. Mass-market clothing manufacturers ignore the 5% of the 
population with extreme sizes (the 2.5% very small and 2.5% 
very large).  What consequences does this have in the 
following cases?

  a. Men's shoe size is N(43,2.5)  What men have to buy
  shoes in specialty shops?

  We've memorized that the relevant z values are 2 and -2
  (and that 95% fall within 2 sd's of the mean).

  Using the z formula as above, we see that
 
         z =  (x - m)/sd 

         2 =  (x - 43)/2.5   and   -2 = (x' - 43)/2.5
        
         2 * 2.5 + 43 = x    and   -2 * 2.5 + 43 = x'
   
         x = 48, x'=38

  So men with feet smaller than size 38 or larger than 48
  need to shop in specialty shops.

  b. Women's hat size is N(6.75, 0.25).  Which women 
  are forced to buy expensive hats?

         z =  (x - m)/sd 

         2 =  (x - 6.75)/0.25  and  -2 = (x'- 6.75)/sd

         2* 0.25 + 6.75 = x    and  -2 * 0.25 + 6.75 = x'

         x = 6.25 and  x' = 7.25

   So women with hat size less than 6.25 or larger than 
   7.25 need special hats.