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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 3100 children randomly selected found 23​% of them deficient in vitamin D. 

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.
b. Explain carefully what the interval means.
c. Explain what​ "98% confidence" means.
d. Does the study show that computer games are a likely cause of​ rickets?


 

The confidence interval of the population proportion.  Confidence interval formula for proportions.

p_hat +/- Z alpha * Standard deviation

standard deviation = sqrt(0.23 * (1-0.23)/3100) , Margin of error = 2.33 * 0.007558375  = 0.017611

The 98% confidence interval = 0.23 +/- 0.018 = (0.212 , 0.248)

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 3100 will produce confidence intervals that contain the true proportion of children that are deficient in vitamin D.

This interval does not say anything about causation. 

 

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