Iron Deficiency? 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 following iron intakes, in milligrams, were obtained during a 24 -hour period for 45 randomly selected adult females under the age of 51.

15.0	18.1	14.4	14.6	10.9	18.1	18.2	18.3	14
16.0	12.6	16.6	20.7	19.8	11.6	12.8	15.6	11
15.3	9.4	19.5	18.3	14.5	16.6	11.5	16.4	12
14.6	11.9	12.5	18.6	13.1	12.1	10.7	17.3	12
17.0	6.3	16.8	12.5	16.3	14.7	12.7	16.3	11

At the significance level, do the data suggest that adult females der the age of 51 are, on average, getting less than the RDA of iron? Assume that the population standard deviation is 4 . (Note: x =14.68mg.)

Step by step solution

01

Step 1. Given information.

given,

15.0	18.1	14.4	14.6	10.9	18.1	18.2	18.3	14
16.0	12.6	16.6	20.7	19.8	11.6	12.8	15.6	11
15.3	9.4	19.5	18.3	14.5	16.6	11.5	16.4	12
14.6	11.9	12.5	18.6	13.1	12.1	10.7	17.3	12
17.0	6.3	16.8	12.5	16.3	14.7	12.7	16.3	11

02

Step 2. The given information is related to a deficiency of iron.

Let μ be the mean of iron intake by adult females under the age of 51.

Given that the population standard deviation is,

σ=4.2mg

Let's n=45 be the size of the sample.

Given that the Sample mean is,

$\overline{x}$=14.68mg

03

Step 3. The setup of null and alternative hypothesis is:

The null hypothesis is:

$H_0:\mu =18$

The alternative hypothesis is:

$H_a:\mu <18\mathrm{mg}$

Let $\alpha =0.01$be the 19 levels of significance.

04

Step 4. Compute the fest statistic:

$\begin{array}{r}z=\frac{\overline{x}-{\mu }_0}{\frac{\sigma }{\sqrt{n}}}\\ =\frac{14.68-18}{\frac{4.2}{\sqrt{45}}}\\ =\frac{-3.32}{0.6261}\\ =-5.303\end{array}$

P-value is:

$\begin{array}{r}P(Z\le -5.303)=1-P(Z\le 5.303)\\ =1-0.9999\\ =0.0000\end{array}$

Therefore, the P-value is 0.0000.

05

Step 5. Conclusion:

Since the P-value of 0.0000 is less than a 1% level of significance.

Therefore, it is enough evidence to reject the null hypothesis.

Therefore, it is concluded that adult females under the age of 51, on average, get less than the RDA at 18mg of iron.

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