A standardized test consists of 100 multiple-choice questions. Each question has five possible answers, only one of which is correct. Four points are awarded for each correct answer. To discourage guessing, one point is taken away for every answer that is not correct (this includes answers that are missing).

The company that creates the test has to understand how well a student could do just by random guessing. Suppose a student answers each question by picking one of the five choices at random independently of the choices on all other questions. Let S be the student's score on the test.

a) Find (S).

b) Find P(S>10). Write your answer as a math expression, then use the code cell below to find its numerical value and provide it along with your math expression.

a) S ~ N ( 0 , 48 )

b) P ( S > 10 ) = 0.0745

Step-by-step explanation:

Given:-

- We have n = 100 MCQs

- 5 options for every MCQs

- probability to guess each MCQ correct is independent from one another.

- Right Answer points= +4

- Wrong answer points= -1

Find:-

a) Find (S).

b) Find P(S>10). Write your answer as a math expression, then use the code cell below to find its numerical value and provide it along with your math expression.

Solution:-

- The probability (p) of guessing a correct answer for each question is:

                             p ( correct answer ) = 1 / 5 = 0.2

- The mean number of correct and incorrect answers can be determined by:

                             ( Mean correct answers) = n*p = 100*0.2 = 20

                             ( Mean incorrect answers) = n*(1-p) = 100*0.8 = 80

- The mean score for correct answers would be:

                            Sc ( u ) = (Points for right answer)*(Mean correct answers)

                            Sc ( u ) = ( +4 )*(20)

                            Sc ( u ) = 80 points

The mean score for incorrect answers would be:

                            Si ( u ) = (Points for wrong answer)*(Mean incorrect answers)

                            Si ( u ) = ( -1)*(80)

                            Si ( u ) = -80 points.

- The mean score attained by a student would be S (u):

                           S (u) = Sc(u) + Si(u)

                           S (u) = 80 - 80 = 0

- The variance of the correct and incorrect answers can be determined by:

                           Var ( correct answers ) = n*p*q = 100*0.2*0.8 = 16

                           Var ( in-correct answers ) = n*p*q = 100*0.2*0.8 = 16

- The variance of points of correct answers can be:

                           Sc (Var) = Var ( correct answer ) * (Points for right answer)

                           Sc (Var) = 16*(+4) = +64 points

- The variance of points of incorrect answers can be:    

                          Si (Var) = Var ( incorrect answer ) * (Points for wrong answer)

                          Si (Var) = 16*(-1) = -16 points  

- Since the probabilities of guessing correct answers are independent. Then as per law of independence:

                         S ( Var ) =  Sc (Var) + Si (Var)

                                       = 64 - 16

                                       = +48 points

- The standard deviation for the distribution (s.d) of points (S) is:

                         S ( s.d ) = √S (Var)  = √48 = 6.9282            

- The number of points (S) attained by a student by guessing on the test containing MCQs would have a mean u = 0 points and s.d = + 48 points.

- The random variable (S) can be modeled by normal distribution as follows:

                         S ~ N ( 0 , 48 )      

- To find the required probability P(S>10).

Compute the Z-value of S = 10 points:

                        Z - value =  ( S - u ) / s.d

                                        =  ( 10 - 0 ) / 6.9282

                                        = 1.4434

Use the standardized Z-table for normal distribution:

                       P ( Z > 1.4434 ) = 0.0745

The probability is:

                       P ( S > 10 ) = P ( Z > 1.4434 ) = 0.0745

a. -0.3 > -0.33