EXAMPLE 4.39

Calcium intake. To get enough calcium for optimal bone health, tablets containing calcium are often recommended to supplement the calcium in the diet. One study designed to evaluate the effectiveness of a supplement followed a group of young people for seven years. Each subject was assigned to take either a tablet containing 1000 milligrams of calcium per day (mg/d) or a placebo tablet that was identical except that it had no calcium.¹⁷ A major problem with studies like this one is compliance: subjects do not always take the treatments assigned to them.

In this study, the compliance rate declined to about 47% toward the end of the seven-year period. The standard deviation of compliance was 22%. Calcium from the diet averaged 850 mg/d with a standard deviation of 330 mg/d. The correlation between compliance and dietary intake was 0.68. Let’s find the mean and standard deviation for the total calcium intake. We let S stand for the intake from the supplement and D stand for the intake from the diet.

We start with the intake from the supplement. Because the compliance is 47% and the amount in each tablet is 1000 mg, the mean for S is

μ_S = 1000(0.47) = 470

Because the standard deviation of the compliance is 22%, the variance of S is

The standard deviation is

Be sure to verify which rules for means and variances are used in these calculations.

We can now find the mean and standard deviation for the total intake. The mean is

μ_S+D = μ_S + μ_D = 470 + 850 = 1320

the variance is

and the standard deviation is

The mean of the total calcium intake is 1320 mg/d and the standard deviation is 506 mg/d.