How to do these questions?

Question

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?

 

I'm fine with an example of the equation, hopefully it would help me.

Answer 1

Hi Tatiana,

 

From the intro to the problem we know

mean = μ = 180 bpm

standard deviation = σ = 6 bpm

 

To solve any problem involving a normal distribution we need to calculated Z

         X - μ

Z = -----------

            σ

 

We also need a table of z-values

 

Now to the questions:

a) percentage of puppies with heartbeats less than 187 bpm

       187-180

Z = ---------- = 7/6 = 1.167

          6

 

We need to look up this value, 1.167 in a table of z values.

We get .8784

This means 87.84% of the puppies will have heartbeats per minute less than 187.

 

b) to determine the percentage of puppies with heartbeats between 170 and 187, we need to calculate 2 z-values and take the difference between the two.   We've already determined that 87.84% of the puppies will have 187 bpm or less.  We need to perform the same calculation for 170 bpm.

       170-180
Z = ---------- = -10/6 = -1.667
           6

We need to look up this value, -1.667 in a table of z values.
We get .0478
This means 4.78% of the puppies will have heartbeats per minute less than 170.

 

To find the percentage of puppies with heartbeats per minute between 170 and 187, we need to subtract the two percentages calculated.

87.84% - 4.78% = 83.06% of the puppies will have 170 to 187 bpm.

 

c) If 500 puppies are examined and 87.84% of them have 187 bpm or less, then

500 x .8784 = 439.2

Since we cannot have 0.2 of a puppy, then 439 puppies will have 187 or less bpm.

 

d) To find the heart rate for which only 10% of the puppies have a higher rate, we actually need to find the rate for which 90% of the puppies will have that heart rate or less.

 

Using the z-table, 90%, .9000, corresponds to a z-value of 1.28167

 

   X - 180

 ---------- = 1.28166666...

       6

 

Solving for X

X-180 = 7.69

X = 187.69

So 10% of the puppies will have an average heart rate of more than 187.69 bpm.