The cheetah is the fastest land mammal and is highly specialized to run down pre

Question

The cheetah is the fastest land mammal and is highly specialized to run down prey. The cheetah often exceeds speeds of 60 miles per hour (mph) and is capable of speeds above 72 mph. The accompanying table contains a sample of the top speeds of 35 cheetahs. The sample mean and sample standard deviation of these speeds are

59.5359.53mph and 4.164.16 mph, respectively. A histogram of the speeds is bell-shaped. Complete parts (a) through (d) below.

Use the data to obtain the exact percentages of observations that lie within one, two, and three standard deviations to either side of the mean.

Explanation / Answer

Empirical rule                  
Given that the mean (u) of a normal probability distribution is 59.53 and the
standard deviation (sd) is 4.16                  
About 68% of the area under the normal curve is within one standard deviation of the mean. i.e. (u ± 1s.d)
So to the given normal distribution about 68% of the observations lie in between
= [59.53 ± 4.16]
= [ 59.53 - 4.16 , 59.53 + 4.16]
= [ 55.37 , 63.69 ]                  
About 95% of the area under the normal curve is within two standard deviation of the mean. i.e. (u ± 2s.d)
So to the given normal distribution about 95% of the observations lie in between
= [59.53 ± 2 * 4.16]
= [ 59.53 - 2 * 4.16 , 59.53 + 2* 4.16]
= [ 51.21 , 67.85 ]                  
About 99.7% of the area under the normal curve is within two standard deviation of the mean. i.e. (u ± 3s.d)
So to the given normal distribution about 95% of the observations lie in between
= [59.53 ± 3 * 4.16]
= [ 59.53 - 3 * 4.16 , 59.53 + 3* 4.16]
= [ 47.05 , 72.01 ]                  

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