EXAMPLE 19Binomial mean, variance, and standard deviation
SAT Scores and AP Exam Scores
The College Board reports (2014) that 90% of students taking the Natural Sciences Subject SAT exam have taken high school chemistry. Suppose we take a sample of 100 students.
- Find the mean or expected number of Natural Sciences exam takers who have taken a chemistry course.
- Calculate the variance and standard deviation of the number of Natural Sciences exam takers who have taken a chemistry course.
- In our sample of 100, would it be unusual to observe 80 Natural Sciences exam takers who have taken a chemistry course?
Solution
The binomial random variable here is X = the number of Natural Sciences SAT exam takers who have taken a chemistry course, with sample size , probability of success , and probability failure .
- The mean or expected number who have taken a chemistry course is .
- , expressed in “students squared.” Then .
We use the -score method (Section 6.1, page 320) to determine whether would be unusual. The -score for 80 is:
335
According to the -score method of identifying outliers, Natural Sciences SAT exam takers having taken a chemistry course would be unusual because it is an outlier, with .
NOW YOU CAN DO
Exercises 49–52.