Answer:
0.00001 ; 92
Step-by-step explanation:
Given that:
Number of possible answers / options per question = 4 ; correct answers per question = 1
P(choosing the correct answer) ; p = 1/4 = 0.25
Probability of getting atleast 30 questions right :
Mean, m = np = 60 * 0.25 = 15
Standard deviation, s = sqrt(np(1-p)) = sqrt(60 * 0.25 * 0.75) = 3.354
Using binomial approximation :
P(x ≥ 29.5) = (29.5 - 15) / 3.354 = 4.32
P(Z ≥ 4.32) = 0.00001 (Z probability calculator)
To get no more than 35%
P(x ≤ 0.35) ; normal approximation P(x ≤ 0.35n + 0.5)
m= n * 1/4 = n/4
Variance = np(1-p) = n * 1/4 * 3/4 = 3n/16
X ~(n/4, 3n/16)
P(x ≤ 0.35) = [(0.35n + 0.5 - 0.25n) / sqrt(0.1875n)] = Zcritical 99%
Zcritical at 99% = 2.326
0.1n + 0.5 / sqrt(0.1875n) = 2.326
0.1n + 0.5 = 2.326 * sqrt(0.1875n)
Square both sides :
(0.1n + 0.5)² = (2.326*sqrt0.1875n)²
Quadratic relation obtained :
0.01n² - 0.914427 + 0.25n = 0
Solving using the quadratic. Formula :
-b ± [sqrt(b² - 4ac) / 2a]
a = 0.01 ; b = - 0.914427 ; c = 0.25
Output = 91.17 or 0.27
Hence, n = 92
