Q. 3.135 - Elementary Statistics

Question

 Iron is essential to most life forms and to normal human physiology. It is an integral part of many proteins and enzymes that maintain good health. Recommendations for iron are provided in Dietary Reference Intakes, developed by the Institute of Medicine of the National Academy of Sciences, The recommended dietary allowance (RDA) of iron for adult females under the age of $51$ is $18$ milligrams (mg) per day. The iron intakes during a $24$-hour period for a random sample of $45$ adult females under the age of $51$ have a mean of $14.7mg$ and  standard deviation of $3.1 mg$.

a. Construct a graph .
b. Apply Chebyshev's rule with $k=2$ to make pertinent statements about the observations in the sample.
c. Repeat part (b) with $k=3$.

Step-by-Step Solution

Answer

Part(a) Required figure is given below.

Part(b) At least $34$adult females have recommended dietary allowance (RDA) of iron
Part(c) At least $40$ adult females have recommended dietary allowance (RDA) of iron.

Part(a) Step 1 : Given information

We are given that sample has $45$ adult females.

Mean $\overline{x}=14.7$ 

Standard deviation $s=3.1$

Part(a) Step 2 : Simplify

According to chebyshev's Rule

$$\begin{gathered} \overline{x}=14.7 \\ \overline{x}-s=11.6 \\ \overline{x}-2s=8.5 \\ \overline{x}-3s=5.4 \\ \overline{x}+s=17.8 \\ \overline{x}+2s=20.9 \\ \overline{x}+3s=24.0 \end{gathered}$$

Required graph is

Part(b) Step 1 : Given information

We are given that sample has $45$ adult females

Mean $\overline{x}=14.7$ 

Standard deviation $s=3.1$

Part(b) Step 2 : Simplify

According to Chebyshev's rule ,if

$k=2$, then at least $75\%$ of the adult females in the sample within two standard deviations to either side of the mean.

 Now,

$45\times 0.75\approx 34$

 Now, two standard deviations to either side of the mean is from 

$8.5$ to $20.9$ milligram.

Interpretation- At least $34$ out of $45$ adult females in the sample have recommended dietary allowances of iron between $8.5 mg$and $20.9 mg$

Part(c) Step 2 : Given information

We are given that sample has $45$adult females.

Mean $\overline{x}=14.7$

Standard deviation $s=3.1$

Part(c) Step 2 : Simplify

According to Chebyshev's rule ,if

$k=3$, then at least $89\%$of the adult females are within two standard deviations to either side of the mean.

 Now,

$45\times 0.89\approx 40$

 Now, two standard deviations to either side of the mean is from 

$5.4$ to $24.0$.

Interpretation- At least $40$ out of $45$ adult females in the sample have recommended dietary allowance of iron between $5.4 mg$ and $24.0 mg$