Healthy 10-week-old domesticated kittens have average weight 24.5 oz., with a st

Question

Healthy 10-week-old domesticated kittens have average weight 24.5 oz., with a standard deviation of 5.5 oz. The distribution is approximately normal. A kitten is designated as dangerously underweight when, at 10 weeks, it weighs less than 10.15 OZ. What proportion of healthy kittens will have designated as dangerously underweight? What is the median weight of the kittens? What are the first and third quartiles of the kitten weight? (25% of the kittens weigh less than the first quartile and 75% of the kitten's weight less than the third quartile.)

Explanation / Answer

Mean ( u ) =24.5
Standard Deviation ( sd )=5.5
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a.
P(X < 10.25) = (10.25-24.5)/5.5
= -14.25/5.5= -2.5909
= P ( Z <-2.5909) From Standard Normal Table
= 0.0048                  
proportion of kittens 0.48%
b.
P ( Z < x ) = 0.5
Value of z to the cumulative probability of 0.5 from normal table is 0
P( x-u/s.d < x - 24.5/5.5 ) = 0.5
That is, ( x - 24.5/5.5 ) = 0
--> x = 0 * 5.5 + 24.5 = 24.5                  
c.
Q1 = 0.25
P ( Z < x ) = 0.25
Value of z to the cumulative probability of 0.25 from normal table is -0.674
P( x-u/s.d < x - 24.5/5.5 ) = 0.25
That is, ( x - 24.5/5.5 ) = -0.67
--> x = -0.67 * 5.5 + 24.5 = 20.7903                  

P ( Z < x ) = 0.75
Value of z to the cumulative probability of 0.75 from normal table is 0.674
P( x-u/s.d < x - 24.5/5.5 ) = 0.75
That is, ( x - 24.5/5.5 ) = 0.67
--> x = 0.67 * 5.5 + 24.5 = 28.2097                  

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