Angelman syndrome is a rare genetic disease in which children are delayed developmentally and exhibit unusual symptoms such as inappropriate and prolonged laughter, difficulty in speaking or inability to speak, and seizures. Imagine that a researcher obtained vocabulary data for six children with Angelman syndrome and wants to develop an estimate of the mean vocabulary score of the population of children with Angelman syndrome. (Although those with Angelman syndrome often cannot speak, they are usually able to understand at least some simple language and they may learn to communicate with sign language.) The General Social Survey (GSS) asks children the meaning of ten words using a multiple-choice format; the GSS data have a mean of 6.1, with a standard deviation of 2.1. The fictional data for the six children with Angelman syndrome are: 0, 1, 1, 2, 3, and 4. Write each of these six numbers on a separate, small piece of paper.

a. Put the six pieces of paper in a bowl or hat, and then pull six out, one at a time, replacing each one and mixing them up before pulling the next. List the numbers and take the mean. Repeat this procedure two more times so that you have three lists and three means.

b. We did this 20 times and got the following 20 means:

1.833 1.167 2.000 2.333 1.333

1.333 2.000 1.667 1.667 1.667

1.500 1.000 1.500 1.667 1.833

1.500 1.667 2.333 2.167 2.000

Determine the 90% confidence interval for these means.

c. Why is bootstrapping a helpful technique in this particular situation?