Question

Vitamin D, whether ingested as a dietary supplement or produced naturally when sunlight falls upon the skin, is essential for strong, healthy bones. The bone disease rickets was largely eliminated during the 1950s, but now there is concern that a generation of children more likely to watch TV or play computer games than spend time outdoors is at increased risk. A recent study of 2392 children randomly selected found 16% of them deficient in vitamin D. (a) Construct a 90% confidence interval for the proportion of children who are deficient in vitamin D (use 4 decimals): ( , ) (b) Explain what 90% confidence means: 90% of repeated random samples of size 2392 will produce that will contain the of children that are deficient in vitamin D. (c) A different group of researchers are interested in conducting a follow-up study. If they need a margin of error no greater than 0.02 and a confidence level of 90%, how many children need to be sampled? (d) Does this study show that computer games are a likely cause of rickets? Yes. Since food manufacturers have increased the amount of vitamin D in their products over the years, it is likely the case that children playing too many computer games has caused the increase in rickets. Yes, because the bone disease was largely eliminated, and now we are 90% confident that only 16% of children have the deficiency. No, because the confidence interval says nothing about causation. No, because the study suggests that it is more likely the case that a change in children's diets over the years is the cause for the increase in rickets. Yes, because 16% is in the confidence interval for the proportion of children who are deficient in vitamin D.

Answer 1

a) At 90% confidence interval the critical value is z^* = 1.645

The 90% confidence interval for population proportion is

+/- z^* * sqrt((1 - )/n)

= 0.16 +/- 1.645 * sqrt(0.16 * (1 - 0.16)/2392)

= 0.16 +/- 0.0123

= 0.1477, 0.1723

b) we are 90% confident that that the true population proportion lies within the confidence limits 0.1477 and 0.1723.

c) Margin of error = 0.02

or, z^* * sqrt(0.16 * (1 - 0.16)/n) = 0.02

or, 1.645 * sqrt(0.5 * 0.5/n) = 0.02

or, n = (1.645 * sqrt(0.5 * 0.5)/0.02)^2

or, n = 1692that it is more likely the case that a change in children's

d) Yes, since food manufactures have increased the amount of vitamin D in their products over the years, it is likely the case that children playing too many computer games has caused the increase in rickets.