An exam consists of 30 multiple choice questions 4 An exam c

An exam consists of 30 multiple choice questions...

4. An exam consists of 30 multiple choice questions with 4 possible answers, only one of which is correct. Each correct answer gets 4 points, whereas each wrong answer gets -1 points. A student answers all questions completely at random and independently of one another. Let X he the number of his correct answers in the exam. (a) Explain why the distribution of X can be approximated by a normal distribution. What are the parameters of this normal distribution? (b) Compute. exactly and using the normal approximation, the probability that the student answers correctly at most 4 questions. (c) Compute approximately. using the normal approximation, the probability that. the student scores more than 10 points.

Solution

Normal Approximation to Binomial Distribution
a)
Mean ( np ) =30 * 0.25 = 7.5
Standard Deviation ( npq )= 30*0.25*0.75 = 2.3717
Normal Distribution = Z= X- u / sd                   
b)
P(X <= 4) = (4-7.5)/2.3717
= -3.5/2.3717= -1.4757
= P ( Z <-1.4757) From Standard NOrmal Table
= 0.07                  

c)
P(X > 10) = (10-7.5)/2.3717
= 2.5/2.3717 = 1.0541
= P ( Z >1.054) From Standard Normal Table
= 0.1459