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 2600 children randomly selected found 21​% of them deficient in vitamin D. Complete parts a through c below.

​a) Construct a​ 98% confidence interval for the true proportion of children who are deficient in vitamin D. Select the correct choice below and fill in any answer boxes within your choice.

A. (___, ___)

​(Round to three decimal places as​ needed.)

B. The conditions of a confidence interval are not satisfied.

​b) Explain carefully what the interval means.

A. We are​ 98% confident that the proportion of people deficient in vitamin D falls inside the confidence interval bounds.

B. We are​ 98% confident that the proportion of children deficient in vitamin D falls inside the confidence interval bounds.

C. We are​ 98% confident that the proportion of vitamin D deficiency in a randomly sampled child falls within the confidence interval bounds.

D. We are​ 98% confident that the proportion of children deficient in vitamin D is 0.21.

​c) Explain what​ "98% confidence" means.

About 98% of random samples of size 2600 will produce ____ (pick: sample means, a true proportion, confidence intervals) that contain the ____ (pick: confidence interval, true proportion, number) of children that are deficient in vitamin D.

Answer 1

A)
sample proportion, = 0.21
sample size, n = 2600
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.21 * (1 - 0.21)/2600) = 0.008

Given CI level is 98%, hence α = 1 - 0.98 = 0.02
α/2 = 0.02/2 = 0.01, Zc = Z(α/2) = 2.33

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.21 - 2.33 * 0.008 , 0.21 + 2.33 * 0.008)
CI = (0.191 , 0.229)

B)
B. We are​ 98% confident that the proportion of children deficient in vitamin D falls inside the confidence interval bounds.

c)
About 98% of random samples of size 2600 will produce a confidence intervals that contain the true proportion of children that are deficient in vitamin D.