In 1995, the Educational Testing Service in Princeton, New Jersey (which adminis

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. Based on this estimate, what is the probability that of 10 randomly selected students, less than four will be between 900 and 1,100? 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? What is the probability that between three and five students will range between 900 and 1,100?

Explanation / Answer

Here p = 0.4
n = 10

This is binomial distribution example.
Formula to find probability as per binomial distribution is

P(X = r) = nCr * p^r * (1-p)^(n-r)

less than four will be between 900 and 1,100
P(X < 4) = 10C0 * 0.4^0 * 0.6^10 + 10C1 * 0.4^1 * 0.6^9 + 10C2 * 0.4^2 * 0.6^8 + 10C3 * 0.4^3 * 0.6^7 = 0.3823

more than four students will be in this range
P(X > 4) = 1 - P(X < 4) - P(X = 4)
= 1 - 0.3823 - 10C4 * 0.4^4 * 0.6^6
= 0.3669

exactly four students will be in this range
P(X = 4) = 10C4 * 0.4^4 * 0.6^6 = 0.7492

between three and five students will range between 900 and 1,100
P(X = 3) + P(X = 4) + P(X = 5)
= 10C3 * 0.4^3 * 0.6^7 + 10C4 * 0.4^4 * 0.6^6 + 10C5 * 0.4^5 * 0.6^5
= 0.6665

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