Q. 3.139 - Elementary Statistics

Question

Malnutrition and Poverty. R. Reifen et al. studied various nutritional measures of Ethiopian school children and published their findings in the paper "Ethiopian-Born and Native Israeli School Children Have Different Growth Patterns" . The study, conducted in Azezo, North West Ethiopia, found that malnutrition is prevalent in primary and secondary school children because of economic poverty. The weights, in kilograms (kg), of $60$ randomly selected male Ethiopian-born school children aged $12-15$ years old are presented in increasing order in the following table.
The sample mean and sample standard deviation of these weights are $45.30$kg and $4.16$kg, respectively.

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 Ethiopian-born school children 

Mean $\overline{x}=45.30kg$

Standard deviation $s=4.16kg$

Part(a) Step 2 : Simplify

As we know we can apply empirical rule if data is roughly bell shaped. Therefore,

Yes, it is 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 because the distribution of the sample of randomly selected male Ethiopian-born school children  is roughly bell shaped.

Part(b) Step 1 : Given information

We are given data of Ethiopian-born school children 

Mean $\overline{x}=45.30 kg$

Standard deviation  $s=4.16 kg$

Part(b) Step 2 : Simplify

According to empirical formula by property 1, 2, 3,

we can estimate that $68\%$ of the observations lie within one standard deviation to either side of the mean, and $95\%$ lie within two standard deviations to either side of the mean, whereas  $99.7\%$ lie within three standard deviations to either side of the mean.

Part(c) Step 1 : Given information

We are given data of Ethiopian-born school children 

Mean $\overline{x}=45.30 kg$

Standard deviation  $s=4.16 kg$

Part(c) Step 2 : Simplify

According to Table given in the question and with the help of empirical rule,

We found that $43$ of the $60$ observations, or $71.7\%$. lie within one standard deviation to either side of the mean; $54$ of the $60$ observations, or $90\%$, lie within two standard deviations to either side of the mean; and $60$ of the $60$ observations, or $100$, lie within three standard deviations to either side of the mean.

Part(d) Step 1 : Given information

We are given data of Ethiopian-born school children 

Mean $\overline{x}=45.30 kg$

Standard deviation  $s=4.16 kg$

Part(d) Step 2 : Simplify

On comparing the percentages obtained in parts (b) and (c) 

$68\%, 95\%, 99.7\%$ vs $71.7\%, 90\%, 100\%$

 we can say that, in this case, the empirical rule provides excellent estimates of the actual percentages. we are already expecting this result therefore, this result is not surprising because the distribution of  speed of cheetahs is close to bell shaped.