An Internet article reports on results of a study completed in 2004 by psychologists at universities in Washington and Oregon: "65 Percent of Children Have Had an Imaginary Companion." "The researchers originally recruited 152 preschoolers, ages 3 and 4, and their parents several years ago. Each child and parents were interviewed separately about imaginary companions. [. . .]
Three years later, 100 of those children (50 girls and 50 boys) and their parents volunteered for the newly published study. The children and their parents again were interviewed separately about imaginary companions. [. . .] Children were considered to have imaginary companions if they said they had one and provided a description of it."19 Some examples cited in a newspaper report of the study included Rose (an invisible 9-year-old squirrel), Skateboard Guy (an invisible 11-year-old who lived in a boy's pocket and popped up at boring moments to do tricks on his skateboard), and Elephant (a 7-inch-tall pachyderm that wore a tank top and shorts and sometimes was mean).
a. Is there reason to believe that the 100 children (with parents) who volunteered to continue in the study 3 years after the initial interview were no different from the 52 children who abstained? If not, would you expect those 100 children to be more likely or less likely to have imaginary companions?
b. The researchers focused on children up to the age of 7. Use the reported proportion (0.65) and the sample size (100) to construct a 95% confidence interval for the proportion of all such children who have an imaginary companion, rounding your margin of error to the nearest hundredth.
c. There are roughly 20,000,000 children aged three to seven in the United States. Use your confidence interval in part (b) to give a range for the number of imaginary friends they have altogether. (Note that some children may have more than one imaginary friend, and others may share the same imaginary friend, but we will disregard those here.)