Question

In 1995, the Educational Testing Service in Princeton, New Jersey (which administers SAT exam) re-centered the scores so that the overall mean would be approximately 1,000 in the combined math and verbal scores for a “large standardized group”. In 1996, approximately 1.1 million college-bound high school students took the exam and registered a mean score of 1,013, with a standard deviation of 222. About 40 percent of these students’ scores were between 900 and 1,100.
a) Based on this estimate, what is the probability that of 10 randomly selected students, less than four will be between 900 and 1,100?
b) What is the probability that more than four students will be in this range? What is the probability that exactly four students will be in this range?
c) What is the probability that between three and five students will range between 900 and 1,100?

Answer 1

About 40 percent of these students’ scores were between 900 and 1,100. This is same as the probability that a randomly selected student's score is between 900 and 1,100 is 0.40

Let X be the number out of 10 randomly selected students, with score between 900 and 1,100. We can say that X has a Binomial distribution with parameters, number of trials (number of randomly selected students) n=10 and success probability ( the probability that a randomly selected student's score is between 900 and 1,100) p=0.40

We can write the probability that X=x students will be between 900 and 1,100 as

a) Based on this estimate, what is the probability that of 10 randomly selected students, less than four will be between 900 and 1,100?

the probability that of 10 randomly selected students, less than four will be between 900 and 1,100 is

ans: the probability that of 10 randomly selected students, less than four will be between 900 and 1,100 is 0.3823

b) What is the probability that more than four students will be in this range?

ans: the probability that more than four students will be in this range 0.3669

What is the probability that exactly four students will be in this range?

ans: the probability that exactly four students will be in this range is 0.2508

c) What is the probability that between three and five students will range between 900 and 1,100?

the probability that between three and five students (note: 3 and 5 inclusive) will range between 900 and 1,100

ans: the probability that between three and five students will range between 900 and 1,100 is 0.6665

Alternatively: the probability that between three and five students (note: not inclusive of 3 and 5 ) will range between 900 and 1,100 is

the probability that between three and five students (note: not inclusive of 3 and 5 ) will range between 900 and 1,100 is 0.2508