The number of students taking the SAT has risen to an all-time high of more than 1.5 million (College Board, August 26, 2008). Students are allowed to repeat the test in hopes of improving the score that is sent to college and university admission offices. The number of times the SAT was taken and the number of students are as follows.

Number of Times Number of Students
1 721,769
2 601,325
3 166,736
4 22,299
5 6,730

A. Let x be a random variable indicating the number of times a student takes the SAT. Show the probability for this random variable?
B. What is the probability that a student takes the SAT more than one time?
C. What is the probability that a student takes the SAT three or more times?
D. What is the expected value of the number of times the SAT is taken? What is your interpretation of the expected value?
E. What is the variance and standard deviation for the number of times the SAT is taken?

a. The probability of the random variable is as follows

P(1) = 0.475

P(2) = 0.396

P(3) = 0.110

P(4) = 0.015

P(5) = 0.004 -

b. The probability that a student takes the SAT more than one time is calculated as 0.525

c. The probability that a student takes the SAT three or more times is calculated as 0.129

d. The Expected value E(x) is calculated as 1.7

This means that the number of students that took the SAT test are approximately 1.7 times

e.. The variance = 0.579

Step-by-step explanation:

a.

Calculating total number of students.

Total = 721,769 + 601,325 + 166,736 + 22,299 + 6,730

Total = 1,518,859

The probability of the random variable is calculated as

Number of observations/Number of total outcomes

So, the probability of the random variable is as follows

P(1) = 721,769/1,518,859

P(1) = 0.47520474250736901845398

P(1) = 0.475 --- Approximated

P(2) = 601,325/1,518,859

P(2) = 0.39590574240268517354145

P(2) = 0.396

P(3) = 166,736/1,518,859

P(3) = 0.10977714192034942018976

P(3) = 0.110 --- Approximated

P(4) = 22,299/1,518,859

P(4) = 0.01468141545726100974481

P(4) = 0.015 Approximated

P(5) = 6,730/1,518,859

P(5) = 0.00443095771233537806998

P(5) = 0.004 --- Approximated

b. The probability that a student takes the SAT more than one time is calculated as

P(2) ++P(5)

= 0.396 + 0.110 + 0.015 + 0.004

= 0.525

Alternatively, it can be calculated as 1 - P(1)

= 1 - 0.475

= 0.525

c. The probability that a student takes the SAT three or more times is calculated as

P(3) + P(4) + P(5)

= 0.110 + 0.015 + 0.004

= 0.129

d. What is the expected value of the number of times the SAT is taken?

What is your interpretation of the expected value?

The Expected value E(x) is calculated as

∑xf(x)

X --- f(x) xf(x) x - u (x-u)² (x-u)²f(x)

P(1) = 0.475 0.475 --- -0.677 0.457 0.218

P(2) = 0.396 --- 0.792 0.323 0.104 0.412

P(3) = 0.110 0.330 --- 1.323 1.750 0.192

P(4) = 0.015 0.06 2.323 5.400 0.792

P(5) = 0.004 --- 0.02 3.323 11.041 0.049

∑xf(x) = 1.677 = 1.7 Approximated

This means that the number of students that took the SAT test are approximately 1.7 times

E. What is the variance

Var (x) = ∑(x-u)²f(x)

Var(x) = 0.579

70,000 + 6,000 + 800 + 30

step-by-step explanation: