The heart rates for puppies are normally distributed with a

The heart rates for puppies are normally distributed, with a mean heart rate of 180 beats per minute (60-160 beats per minute for most adult dogs) with a standard deviation of 6 bpm (beats per minute).

a. What percentage of puppies will have heart rates of less than 187 bpm?

b. What percentage of puppies will have heart rates between 170 bpm and 187 bpm?

c. If 500 puppies are examined, how many would you expect to have a heart rate of less than 187 bpm?

d. 10% of all puppies will have a heart rate greater than what value?

Solution

Mean ( u ) =180
Standard Deviation ( sd )=6
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
              
a)
P(X < 187) = (187-180)/6
= 7/6= 1.1667
= P ( Z <1.1667) From Standard Normal Table
= 0.8783                  
b)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 170) = (170-180)/6
= -10/6 = -1.6667
= P ( Z <-1.6667) From Standard Normal Table
= 0.04779
P(X < 187) = (187-180)/6
= 7/6 = 1.1667
= P ( Z <1.1667) From Standard Normal Table
= 0.87833
P(170 < X < 187) = 0.87833-0.04779 = 0.8305                  

c) If 500 puppies are examined
P(X < 187) = (187-180)/6/ Sqrt ( 500 )
= 7/0.2683= 26.0875
= P ( Z <26.0875) From Standard NOrmal Table
= 1                  

d)
P ( Z > x ) = 0.1
Value of z to the cumulative probability of 0.1 from normal table is 1.28
P( x-u/ (s.d) > x - 180/6) = 0.1
That is, ( x - 180/6) = 1.28
--> x = 1.28 * 6+180 = 187.692