3.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, lessthan 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?

A) The probability that of 10 randomly selected students, less than four will be between 900 and 1,100 is 0.3823 .

B)  The probability that more than four students will be in this range 0.3669, the probability that exactly four students will be in this range is 0.2508 .

Step-by-step explanation:

About 40% of these students scores were between 900 and 1,100 which means that 0.40  is the probability.

Take "X" to 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 the following parameters;

a - Number of trials (number of randomly selected students (n) = 10

b - 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 it is seen in attached equation 1

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  attached equation 2 with the mathematical solution.

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?

See attached equation 3 and solution.

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?

See attached equation 4 and solution.

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