605: discussion9

ex3 pg338

A true-false examination has 48 questions. June has probability 3/4 of answering a question correctly. April just guesses on each question. A passing score is 30 or more correct answers. Compare the probability that June passes the exam with the probability that April passes it.

the example is true-false with 48 questions. the answer of each individual is independent. with the total question number greater than 30. the distribution of it should follow the normal distribution.

# standard deviation:
sd = sqrt(48*0.5*0.5)
sd

## [1] 3.464102

# mean
mu = 48 * 0.5
mu 

## [1] 24

# the question that June will expect to get correct:
ex_june = 48 * 3/4
ex_june

## [1] 36

# the question that April will expect to get correct:
ex_april = 48 * 0.5
ex_april

## [1] 24

we know that the passing score is greater or equal to 30. So we need to calculate the difference between expected value and 30 and see how many standard deviation is away from 30.

# z-score for June
z_june = (ex_june - 30) / sd
z_june

## [1] 1.732051

# z-score for April
z_april = (ex_april - 30) / sd
z_april

## [1] -1.732051

then we calculate the probability using z-score above

pnorm(z_june)

## [1] 0.9583677

pnorm(z_april)

## [1] 0.04163226

So, according to the probabilities we compute from z-score, we get that June has 95.8% pass but April only has 4.2%.