Answer:
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