EXAMPLE 5.22

Falling asleep in class. In the survey of 4513 college students described in Example 5.4, 46% of the respondents reported falling asleep in class due to poor sleep. You randomly sample 10 students in your dormitory, and eight state that they fell asleep in class during the last week due to poor sleep. Relative to the survey results, is this an unusually high number of students?

To answer this question, assume that the students’ actions (falling asleep or not) are independent, with the probability of falling asleep equal to 0.46. This independence assumption may not be reasonable if the students study and socialize together or if there is a loud student in the dormitory who keeps everyone up. We’ll assume this is not an issue here, so the number X of students who fell asleep in class out of 10 students has the B(10, 0.46) distribution.

We want the probability of classifying at least eight students as having fallen asleep in class. Using software, we find

P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10)

= 0.0263 + 0.0050 + 0.0004 = 0.0317

We would expect to find eight or more students falling asleep in class about 3% of the time or in fewer than one of every 30 surveys of 10 students. This is a pretty rare outcome and falls outside the range of the usual chance variation due to random sampling.