# Normal Family of Distributions ```Normal Family of Distributions
Now suppose we also know the population standard deviations of the scores on the two tests:
 ABC:  = 100,  = 10
 XYZ:  = 150,  = 15
Problems: 1. Use tables in textbook to solve the similar problem for the XYZ test.
Answer: solve 0.85=(X-150)/15 for X, getting X=162.75. So the Catalog will say you need a score
of 163.
2. Pilots for a certain type of aircraft must be under 5'7&quot;. If a normal population of men has
 = 5'9&quot; and  = 3.5&quot;, what percent of this population is excluded from piloting such aircraft.
Answer: Let X be the height of a man. P[X is at least 5x12+7=67 inches tall] is excluded.
P[ Z 
X  69 67  69

 0.57]  1  .2843  .7157.
3.5
3.5
Problems: Draw sketches for all. Use normal tables to work.
3. Find c such that P{–c &lt; Z  c} = .95 for standard normal Z.
Answer: P[Z&lt; -c]=.05/2=.025 gives –c= -1.96. So c=1.96.
4.
If X ~ NORM(200, 25), find P{190 &lt; X &lt; 230}.
190  200 X  200 230  200
P[


]  P[.4  Z  1.2]  P{Z  1.2]  P[ Z  .4]  .8849  .3446  .5403.
25
25
25
5. Find d such that P{–d &lt; Z  d} = .99 for Z ~ NORM(0, 1).
Answer: P[Z&lt;-d]=.005 gives –d=-2.60. So d=2.60.
6. If  = 100 and P{95 &lt; Y  105} = .6826, then find .