. An exam has 5 questions and each of them has 4 possible answers. A student who
ID: 3047886 • Letter: #
Question
. An exam has 5 questions and each of them has 4 possible answers. A student who takes this exam gets 3 points for each correct answer and loses 1 point for each wrong answer. Consider a student who answers all questions completely at random. Let X denote the number of correct answers and Y the number of points of this student at the end of the test. (A negative score is allowed) (a) Compute the variance of Y. What are its units? b) Compute the probability that Y takes a value within 1 standard deviation from its mean (c) Compute the probability that Y takes a value within 2 standard deviations from its mean: d) Compute the probability that Y takes a value within 3 standard deviations from its mean:Explanation / Answer
Suppose an exam has 5 MCQ with one correct answer. A student will get 3 points for every correct answer and so called penalty for each wrong answer. Define Y: Number of correct answers given by a student out of n=5. Assume that student give the answers randomly.
Then values assumed by X : 0,1,2,3,4,5.
Here we can say that probability of attempting the question correctly is probability of SUCCESS and is given by p=0.25.
Therefore X will follow Binomial distribution with parameters n=5 and p=0.25.
Now for each Value of X we will find corresponding value of Y i.e. points earned by the student.
Values of X
Values of Y
P[X=x] = P[Y=y]
YP[Y=y]
Y2P[Y=y]
0
-5
0.237305
-1.186523438
5.932617
1
-1
0.395508
-0.395507813
0.395508
2
3
0.263672
0.791015625
2.373047
3
7
0.087891
0.615234375
4.306641
4
11
0.014648
0.161132813
1.772461
5
15
0.000977
0.014648438
0.219727
Total
1
-5.55112E-17
15
P[ Mean – SD < Y < Mean + SD]
= P[-15 < Y < 15] =1
P[ Mean – 2 SD < Y < Mean +2 SD]
= P[-30 < Y < 30] =1
P[ Mean – 3 SD < Y < Mean + 3 SD]
= P[-45 < Y < 45] =1
Values of X
Values of Y
P[X=x] = P[Y=y]
YP[Y=y]
Y2P[Y=y]
0
-5
0.237305
-1.186523438
5.932617
1
-1
0.395508
-0.395507813
0.395508
2
3
0.263672
0.791015625
2.373047
3
7
0.087891
0.615234375
4.306641
4
11
0.014648
0.161132813
1.772461
5
15
0.000977
0.014648438
0.219727
Total
1
-5.55112E-17
15
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