Response:
a) S ~ N (0, 48)
b) P(S > 10) = 0.0745
Detailed explanation:
Given Information:-
- Total number of questions, n = 100
- Each question has 5 options
- The probability of correctly guessing each answer is independent.
- Points for a correct answer = +4
- Points for an incorrect answer = -1
Inquiries:-
a) Determine????(S).
b) Determine P(S>10). Represent your response as a mathematical formula, then utilize the code cell below to calculate its numerical value, providing both the calculation and its result.
Solution:-
- The probability (p) for answering a question correctly is:
p (correct answer) = 1/5 = 0.2
- The expected number of correct and incorrect answers can be calculated as follows:
(Expected correct answers) = n*p = 100*0.2 = 20
(Expected incorrect answers) = n*(1-p) = 100*0.8 = 80
- The anticipated score for correct answers will be:
Sc(u) = (Points for a correct answer)*(Expected correct answers)
Sc(u) = (+4)*(20)
Sc(u) = 80 points
The anticipated score for incorrect answers will be:
Si(u) = (Points for an incorrect answer)*(Expected incorrect answers)
Si(u) = (-1)*(80)
Si(u) = -80 points.
- The average score a student might achieve would be S(u):
S(u) = Sc(u) + Si(u)
S(u) = 80 - 80 = 0
- The variance for both correct and incorrect answers can be calculated as:
Var(correct answers) = n*p*q = 100*0.2*0.8 = 16
Var(incorrect answers) = n*p*q = 100*0.2*0.8 = 16
- The variance of points for correct answers can be expressed as:
Sc(Var) = Var(correct answer) * (Points for a correct answer)
Sc(Var) = 16*(+4) = +64 points
- The variance of points for incorrect answers can be expressed as:
Si(Var) = Var(incorrect answer) * (Points for an incorrect answer)
Si(Var) = 16*(-1) = -16 points
- Since the probabilities of correct guesses are independent, according to the independence principle:
S(Var) = Sc(Var) + Si(Var)
= 64 - 16
= +48 points
- The standard deviation for the score distribution (s.d) is:
S(s.d) = √S(Var) = √48 = 6.9282
- Therefore, the anticipated score (S) from guessing on the MCQ test would yield a mean of u = 0 points and s.d = + 48 points.
- The random variable (S) can be approximated using normal distribution as follows:
S ~ N (0, 48)
- To find the required probability P(S>10).
Calculate the Z-value for S = 10 points:
Z-value = ( S - u ) / s.d
= ( 10 - 0 ) / 6.9282
= 1.4434
Consult 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
