SOLUTION: If you solve the question below, I'll be appreciated. An exam consists of 42 multiple-choice questions. Each question has a choice of five answers, only one of which is correct.

Question 1205509: If you solve the question below, I'll be appreciated.
An exam consists of 42 multiple-choice questions. Each question has a choice of five answers, only one of which is correct. For each correct answer, a candidate gets 1 mark, and no penalty is applied for getting an incorrect answer. A particular candidate answers each question purely by guess-work.
Using Normal approximation to Binomial distribution with continuity correction, what is the estimated probability this student obtains a score greater than or equal to 10? Please use R to obtain probabilities and keep at least 6 decimal places in intermediate steps.
A. 0.6643
B. 0.2089
C. 0.4059
D. 0.3357
E. 0.5650
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
answer is selection D.
i did it two ways.
first way was direct, using the binomial distribution formula, in excel.
that formula is p(x) = p^x * q^(n-x) * c(n,x)
all values of p(10) to p(42) are summed up to get the probability of x being greater than or equal to 10.
the probability of x being greater than or equal to 10 was equal to 0.324376129
i then used normal approximation to the binomial.
p(x >= 10) = selection D = 0.3357
the normal approximation won't be exact, but it'll be close.

here are the results using the normal distribution calculator at https://davidmlane.com/hyperstat/z_table.html
here are the results.

the mean was set at 10
the standard deviation was set at sqrt(42 * .2 * .8) = 2.592296

here is a reference on normal approximation of the binomial.
https://stats.libretexts.org/Courses/Las_Positas_College/Math_40%3A_Statistics_and_Probability/06%3A_Continuous_Random_Variables_and_the_Normal_Distribution/6.04%3A_Normal_Approximation_to_the_Binomial_Distribution

with proportions, the mean is equal to n * p.
in this case it was 42 * .2 = 8.4
the standard error is equal to sqrt(n * p * q) = sqrt(42 * .2 * .8).

let me know if you have any questions.
theo

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