Q. 3.138 - Elementary Statistics

Question

The cheetah (Acinonyx jubatus) 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. according to the online document "Cheetah Conservation in Southern Africa" by J. Urbaniak, the cheetah is capable of speeds up to $72mph$The following table gives the top speeds, in miles per hour, arranged in increasing order, for a sample of $35$cheetahs.

The sample mean and sample standard deviation of these speeds are $59.53 mph$and $4.27 mph$, respectively. A histogram of the speeds is bell shaped.
a. Is it reasonable to apply the empirical rule to estimate the percentages of observations that lie within one, two, and three standard deviations to either side of the mean?
b. Use the empirical rule to estimate the percentages of observations that lie within one, two, and three standard deviations to either side of the mean.
c. 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.
d. Compare your answers in parts (b) and (c).

Step-by-Step Solution

Answer

Part(a) Yes, it is reasonable to apply empirical formula.

Part(b) $68\%, 95\%, 99.7\%$are the percentages of observations that lie within one, two, and three standard deviations to either side of the mean.

Part(c) The exact percentages of observations that lie within one, two, and three standard deviations to either side of the mean are$71.7\%, 90\%, 100\%$

Part(d) The empirical rule provides excellent estimates of the actual percentages.

Part(a) Step 1 : Given information

We are given data of speed of $35$ cheetahs.

Mean $\overline{x}=59.53 mph$

Standard deviation $s=4.27 mph$

Part(a) Step 2 : Simplify

Because the distribution of cheetah speed is essentially bell-shaped, it is possible to use the empirical approach to estimate the percentages of observations that lie within one, two, and $3$ standard deviations to any side of the mean.

Part(b) Step 1 : Given information

We are given data of speed of $35$ cheetahs.

Mean $\overline{x}=59.53 mph$

Standard deviation $4.27 mph$

Part(b) Step 2 : Simplify

We estimate that of the data are one standard deviation to each side of the mean, $68\%$whereas  two standard deviations to either side of the mean are $95\%$and $3$ standard deviations to any of the two side of the mean,   $99.7\%$ using the empirical rule.

Part(c) Step 1 : Given information

We are given data of speed of $35$cheetahs.

Mean $\overline{x}=59.53 mph$

Standard deviation $s=4.27 mph$

Part(c) Step 2 : Simplify

Referring to Table , we found that $25$ of the $35$ observations, or $71.7\%$. are within $1$ standard deviation to any side of the mean; $32$ of the $35$ observations, or $90\%$, are within $2$ standard deviations to any side of the mean; and $35$ of the $35$observations, or $100\%$, are within $3$ standard deviations to any side of the mean.

Part(d) Step 1 : Given information

We are given data of speed of $35$ cheetahs.

Mean $\overline{x}=59.53 mph$

Standard deviation $s=4.27 mph$

Part(d) Step 2 : Simplify

Comparing the percentages obtained in parts (b) and (c) 

$68\% , 95\%, 99.7\%$ vs $71.7\%, 95\%, 100\%$

 We can see that the empirical rule yields great approximations of the actual percentages in this example.
This result is unsurprising, given that cheetahs' speed distribution is nearly bell-shaped.