Question

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 1 797,000 2 650,000 3 137,000 4 30,000 5 33,900 a. Let x be a random variable indicating the number of times a student takes the SAT. Show the probability distribution for this random variable. Round your answers to four decimal places. x f(x) 1 2 3 4 5 b. What is the probability that a student takes the SAT more than one time? Round your answer to four decimal places. c. What is the probability that a student takes the SAT three or more times? Round your answer to four decimal places. d. What is the expected value of the number of times the SAT is taken? Round your interim calculations and final answer to four decimal places. What is your interpretation of the expected value? The input in the box below will not be graded, but may be reviewed and considered by your instructor. e. What is the variance and standard deviation for the number of times the SAT is taken? Round your interim calculations and final answer to four decimal places. Variance Standard deviation

Answer 1

(a)

To find the probability distribution of X we need to divide Number of Students in each X by the total number of students.

Following table shows the probability distribution:

Number of Times, X	Number of Students, f	P(x), f / 1647900
1	7,97,000	0.4836
2	6,50,000	0.3944
3	1,37,000	0.0831
4	30,000	0.0182
5	33,900	0.0206
Total	16,47,900	0.9999

(b)

The probability that a student takes the SAT more than one time is

P(X=1) = 0.4836

(c)

The probability that a student takes the SAT three or more times is

P(X >= 3) = P(X=3) + P(X=4) + P(X=5) = 0.0831 + 0.0182 + 0.0206 = 0.1219

(d-e)

Following table shows the calculations for mean, variance and SD:

Number of Times, X	Number of Students, f	P(x), f / 1647900	xP(x)	x^2P(x)
1	7,97,000	0.4836	0.4836	0.4836
2	6,50,000	0.3944	0.7888	1.5776
3	1,37,000	0.0831	0.2493	0.7479
4	30,000	0.0182	0.0728	0.2912
5	33,900	0.0206	0.103	0.515
Total	16,47,900	0.9999	1.6975	3.6153

So, mean is