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    S2 Heinemann exercise 3B

    (8) In a multiple choice examination candidate jones picks his answers to each question at random from a list of 3 answers provided, of which only one is correct. A candidate passing 18 or more questions coreectly passes the examination (a) For paper containing 45 questions use normal approx to find prob that jones passes (b) It is required that prob that Jones passes shouls be less than 0.005. Use normal approx to show paper should contain at most 31 questions

    exercise 2D
    9- Mode of a continous random variable X is 1 and this is a stationary point. Given that X takes values over the range [0,3] find possible expressions for (a) C.D.F of x AND (b) p.d.f of X
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    (8)
    (a)
    X
    = Number of correct answers
    ~ Bin(45, 1/3)

    Approximately,

    X
    ~ N(45*(1/3), 45*(1/3)*(2/3))
    = N(15, 10)

    Using a continuity correction,

    P(Jones passes)
    = P(X >= 18)
    \approx P(N(15, 10) >= 17.5)
    = 1 - Phi(2.5/sqrt(10))
    = 0.215

    (b)
    P(Jones passes if there are 32 questions)
    \approx P(N(32*(1/3), 32*(1/3)*(2/3)) >= 17.5)
    = P(N(32/3, 64/9) >= 17.5)
    = 1 - Phi((17.5 - 32/3) / sqrt(64/9))
    = 0.00520
    > 0.005

    P(Jones passes if there are 31 questions)
    \approx P(N(31*(1/3), 31*(1/3)*(2/3)) >= 17.5)
    = P(N(31/3, 62/9) >= 17.5)
    = 1 - Phi((17.5 - 31/3) / sqrt(62/9))
    = 0.00316
    < 0.005

    Therefore the test must contain at most 31 questions.
 
 
 
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