A multiple choice quiz has 400 questions, each question with five possible answers of which one is correct answer. What is the probability that a sheer guess yields exactly 80 correct answers using normal approximation.
X ~ Bin(n,p)
n=400p=14=0.2q=1−p=1−0.2=0.8mean=n×p=400×0.2=80σ=npq=400×0.2×0.8=8n=400 \\ p=\frac{1}{4}=0.2 \\ q=1-p=1-0.2=0.8 \\ mean = n \times p = 400 \times 0.2 =80 \\ \sigma = \sqrt{npq}=\sqrt{400 \times 0.2 \times 0.8}=8n=400p=41=0.2q=1−p=1−0.2=0.8mean=n×p=400×0.2=80σ=npq=400×0.2×0.8=8
The probability that a sheer guess yields exactly 80 correct answers:
P(X=80)=P(79.5≤X≤80.5)=P(X≤80.5)−P(X<79.5)=P(Z≤80.5−808)−P(Z<79.5−808)=P(Z≤0.0625)−P(Z<−0.0625)=0.5251−0.4752=0.0499P(X=80) = P(79.5≤X≤80.5) \\ = P(X≤80.5) -P(X<79.5) \\ =P(Z≤ \frac{80.5-80}{8}) -P(Z< \frac{79.5-80}{8}) \\ =P(Z≤0.0625) -P(Z< -0.0625) \\ = 0.5251 -0.4752 \\ = 0.0499P(X=80)=P(79.5≤X≤80.5)=P(X≤80.5)−P(X<79.5)=P(Z≤880.5−80)−P(Z<879.5−80)=P(Z≤0.0625)−P(Z<−0.0625)=0.5251−0.4752=0.0499
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