Question

A calculus instructor uses computer aided instruction and allows students to take the midterm exam as many times as needed until a passing grade is obtained. Following is a record of the number of students in a class of 20 who took the test each number of times. Number of time test was taken (x) 1 2 3 4 Number of students 7 6 4 3

What is the expected value of the number of tests taken?

What is the variance?

What is the standard deviation?

Answer 1

Given:

No of tests, x	Number of students, f
1	7
2	6
3	4
4	3
Sum	20

Probability distribution

The probability distribution for the number of tests taken is obtained by calculating the relative frequency for the number of tests taken.

The relative frequency is obtained by dividing the frequency by total frequency.

No of tests	Number of students, f	Relative freq, P(X)
1	7	7/20=0.35
2	6	6/20=0.3
3	4	4/20=0.2
4	3	3/20=0.15
Sum	20

Expected value

The expected value for the number of tests taken is obtained using the following formula,

No of tests, x	Relative freq, P(X)	x*P(X=x)
1	0.35	0.35
2	0.3	0.6
3	0.2	0.6
4	0.15	0.6
	Sum	2.15

Variance

The variance for the number of tests taken is obtained using the following formula,

No of tests, x	Relative freq, P(X)	x2*P(X=x)
1	0.35	0.35
2	0.3	1.2
3	0.2	1.8
4	0.15	2.4
	Sum	5.75

Standard deviation

The variance for the number of tests taken is obtained using the following formula,