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 14.7 mg and a standard deviation of 3.7 mg. complete parts (a) through (c) below.
a. construct a graph that shows the mean and one, two, and three standard deviations to either side of the mean.
\\(\bar{x}-3s\\) \\(\bar{x}-2s\\) \\(\bar{x}-s\\) \\(\bar{x}\\) \\(\bar{x}+s\\) \\(\bar{x}+2s\\) \\(\bar{x}+3s\\)
3.6 7.3 11.0 14.7 18.4 22.1 25.8
(type integers or decimals rounded to one decimal place as needed.)
b. apply chebyshevs rule with k = 2 to make pertinent statements about the observations in the sample.
at least
of the 45 adult females in the sample have daily iron intakes between \\(\square\\)mg and \\(\square\\)mg.
(round to place as needed. use ascending order.)
40
43
45
34
30

Explanation:

Step1: Recall Chebyshev's rule formula

The formula for Chebyshev's rule is $1-\frac{1}{k^{2}}$, which gives the proportion of data within $k$ standard - deviations of the mean. Here, $k = 2$.

Step2: Calculate the proportion of data

Substitute $k = 2$ into the formula: $1-\frac{1}{2^{2}}=1 - \frac{1}{4}=\frac{3}{4}=0.75$ or $75\%$.

Step3: Calculate the lower and upper bounds

The mean $\bar{x}=14.7$ mg and the standard deviation $s = 3.7$ mg. The lower bound is $\bar{x}-ks=14.7-2\times3.7=14.7 - 7.4 = 7.3$ mg. The upper bound is $\bar{x}+ks=14.7 + 2\times3.7=14.7+7.4 = 22.1$ mg.

Answer:

At least $75\%$ of the 45 adult females in the sample have daily iron intakes between $7.3$ mg and $22.1$ mg.