65. Many medical trials randomly assign patients to either an active treatment or a placebo. These trials are always double-blind. Sometimes the patients can tell whether they are getting the active treatment. This defeats the purpose of blinding. Reports of medical research usually ignore this problem. Investigators looked at a random sample of 97 articles reporting on placebo-controlled randomized trials in the top five general medical journals. Only 7 of the 97 discussed the success of blinding. Give a 95% confidence interval for the proportion of all such articles that discuss the success of blinding. [Dean Fergusson et al., Turning a blind eye: The success of blinding reported in a random sample of randomised, placebo-controlled trials, British Medical Journal, 328 (2004): 432–436.]
65.
The sample proportion of successes is . That is, there were 7.2% successes in the sample. The approximate 95% confidence interval is calculated as follows:
We are 95% confident that the true proportion of articles that discuss the success of blinding is between 0.020 and 0.124 (that is, 2.0% to 12.4%).
The distribution of the sample proportion is approximately normal with mean (i.e., 10%) and standard deviation
or 3%. Notice that 7% is 1 standard deviation below the mean. By the 68 part of the 68-95-99.7 rule, 68% of all samples will have between 7% and 13% that discuss blinding. Half of the remaining 32% of all samples lie on either side. So 16% of samples will have fewer than 7% articles that discuss blinding. That is, the probability is about 0.16.