Refer to the March, 2013 National Science Foundation study of girls participation in science, technology, engineering or mathematics (STEM) programs, Exercise 1.1 (p.5). Recall that the study found that of girls surveyed who participated in a STEM program, 27% felt that the program increased their interest in science. Assume that this figure applies to all girls who have participated in a STEM program. Now, consider a sample of 20 girls randomly selected from all girls who have participated in a STEM program. Let Y represent the number of girls in the sample who feel that the program increased their interest in science.
a. Demonstrate that Y is a binomial random variable.
b. Give the mean and variance of Y, and interpret the results practically.
c. Find the probability that fewer than half of the 20 sampled girls feel that the STEM program increased their interest in science.