Answer to Question #198903 in Statistics and Probability for Fakeha

Question #198903

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.     


1
Expert's answer
2021-05-31T06:37:16-0400

X ~ Bin(n,p)

n=400p=14=0.2q=1p=10.2=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}=8

The probability that a sheer guess yields exactly 80 correct answers:

P(X=80)=P(79.5X80.5)=P(X80.5)P(X<79.5)=P(Z80.5808)P(Z<79.5808)=P(Z0.0625)P(Z<0.0625)=0.52510.4752=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 \\ = 0.0499


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