Question

A history professor decides to give a 12-question true-false quiz. She wants to choose the passing grade such that the probability of passing a student who guesses on every question is less than 0.10. What score should be set as the lowest passing grade? Group of answer choices

Answer 1

we can set the 9 as a benchmark to be the score for the passing grade so that probability of passing a student who guesses every question is less than 0.10

Explanation:

From the given information;

Sample size n = 12

the probability of passing a student who guesses on every question is less than 0.10

In a alternative - response question (true/false) question, the probability of answering a question correctly = 1/2 = 0.5

Let X be the random variable that is represent number of correct answers out of 12.

The X
`\sim` BInomial (12, 0.5)

The probability mass function :

`P(X = k) = (n!)/(k!(n-k)!) * p^k* (1-p)^(n-k)`

`P(X = 12) = (12!)/(12!(12-12)!) * 0.5^(12)* (1-0.5)^(12-12)`

P(X = 12) = 2.44 × 10⁻⁴

`P(X = 11) = (12!)/(11!(12-11)!) * 0.5^(11)* (1-0.5)^(12-11)`

P(X =11 ) = 0.00293

`P(X = 10) = (12!)/(10!(12-10)!) * 0.5^(10)* (1-0.5)^(12-10)`

P(X = 10) = 0.01611

`P(X = 9) = (12!)/(9!(12-9)!) * 0.5^(19)* (1-0.5)^(12-9)`

P(X = 9) = 0.0537

`P(X = 8) = (12!)/(8!(12-8)!) * 0.5^(8)* (1-0.5)^(12-8)`

P(X = 8) = 0.12085

`P(X = 7) = (12!)/(7!(12-7)!) * 0.5^(7)* (1-0.5)^(12-7)`

P(X = 7) = 0.19335

.........

We can see that,a t P(X = 9) , the probability is 0.0537 which less than 0.10 but starting from P(X = 8) downwards the probability is more than 0.01

As such, we can set the 9 as a benchmark to be the score for the passing grade so that probability of passing a student who guesses every question is less than 0.10