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Use a computer program to compute the lower 10 th percentile of a \(t\) distribution with 54 df.

Iron-deficiency anemia is an important nutritional health problem in the United States. A dietary assessment was performed on 51 boys 9 to 11 years of age whose families were below the poverty level. The mean daily iron intake among these boys was found to be 12.50 mg with standard deviation 4.75 mg. Suppose the mean daily iron intake among a large population of \(9-\) to 11 -year-old boys from all income strata is \(14.44 \mathrm{mg}\). We want to test whether the mean iron intake among the low-income group is different from that of the general population. State the hypotheses that we can use to consider this question.

The standard deviation of daily iron intake in the larger population of \(9-\) to 11 -year-old boys was \(5.56 \mathrm{mg}\). We want to test whether the standard deviation from the low-income group is comparable to that of the general population. State the hypotheses that we can use to answer this question.

Left ventricular mass (LVM) is an important risk factor for subsequent cardiovascular disease. A study is proposed to assess the relationship between childhood blood pressure levels and LVM in children as determined from echocardiograms. The goal is to stratify children into a normal bp group \((<80\) th percentile for their age, gender, and height) and an elevated bp group \((\geq 90\) th percentile for their age, gender, and height) and compare change in LVM between the 2 groups. Before this can be done, one needs to demonstrate that LVM actually changes in children over a 4-year period. To help plan the main study, a pilot study is conducted where echocardiograms are obtained from 10 random children from the Bogalusa Heart Study at baseline and after 4 years of follow-up. The data are given in Table 7.6. Since this was a pilot study, the main question of interest is how many subjects would be needed to detect an increase of \(10 \mathrm{g}\) in mean LVM over 4 years using a twosided test with \(\alpha=0.05\) and power \(=80 \%\) ? Hint. Assume that the estimated variance of change in \(L\) VM in the pilot study is the true variance of change in LVM.

An investigator wants to test a new eye drop that is supposed to prevent ocular itching during allergy season. To study the drug, she uses a contralateral design whereby for each participant one eye is randomized (using a random-number table) to get active drug (A) while the other eye gets placebo (P). The participants use the eye drops three times a day for a 1-week period and then report their degree of itching in each eye on a 4-point scale \((1=\text { none }, 2=\text { mild, } 3=\text { moderate, } 4=\) severe) without knowing which eye drop is used in which eye. Ten participants are randomized into the study. What is the principal advantage of the contra-lateral design?

What is the 25 th percentile of a \(\chi^{2}\) distribution with 20 degrees of freedom? What symbol is used to denote this value?