The recommended dietary allowance (RDA) of iron for adult females is 18 miligrams (mg) per day. The given iron intakes (mg) were obtained for 45 random adult females. At the $1 \%$ signicicance lovel, do the data suggest that adult females are, on average, getting less than the RDA of $18 \mathrm{mg}$ of iron? Assume that the population standard deviation is $4.6 \mathrm{mg}$. Preliminary data analyses indicate that applying the z-test is reasonable. (Note: $x=14.65 \mathrm{mg})$ Click here to view the ron intake data. Click here to view Poge 1 of the table of areas under the standard normat curve. Click here to virw Page 2 of the tatle of ateas under the standard nocmal curve. State the hypotheses for the one-mean z-test. \[ \begin{array}{l} \mathrm{H}_{6} \mu=18 \mathrm{ma} \\ \mathrm{H}_{\mathrm{a}} \mu<18 \mathrm{mg} \end{array} \] (Type integers or decimals. Do not round) Compute the value of the test slatistic \[ z=-4: 89 \] (Round to fwo decimal places as needed) Determine the P-value \[ P=0.000 \] (Round to three decimal places as needed) the nuft hypothesis. At the $1 \%$ significance level, the data sutficient evidonce to conclude that adult females are the RDA of iron, on average
Solution
Step 1 ：State the hypotheses for the one-mean z-test: \(H_0: \mu = 18mg\) and \(H_a: \mu < 18mg\).
Step 2 ：Compute the value of the test statistic: \(z = -4.89\).
Step 3 ：Determine the P-value: \(P = 0.000\).
Step 4 ：Interpret the results: The calculated z-score is approximately -4.89, which means that the sample mean is approximately 4.89 standard deviations below the population mean. This is a significant deviation, suggesting that the average iron intake of adult females is indeed less than the recommended dietary allowance (RDA) of 18mg.
Step 5 ：The P-value is 0.000, which is less than the significance level of 0.01. This means that the probability of observing a sample mean as extreme as 14.65mg, assuming the null hypothesis is true, is less than 1%. This is strong evidence against the null hypothesis.
Step 6 ：Make a conclusion: Therefore, we reject the null hypothesis and conclude that the average iron intake of adult females is less than the recommended dietary allowance (RDA) of 18mg. \(\boxed{\text{Final Answer: At the 1% significance level, the data provides sufficient evidence to conclude that adult females are getting less than the RDA of iron, on average.}}\)
