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Amcote
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Homework Statement
A multiple choice test consists of a series of questions, each with four possible answers.
How many questions are needed in order to be 99% confident that a student who guesses blindly at each question scores no more than 35% on the test?
Homework Equations
So I know that this is a binomial setting with p=0.25 and 'n' is what we are trying to solve for.
for binomial, μ=n*p, σ=sqrt(n*p(1-p))
P(B(n,0.25)≤0.35*n)=0.99
And because of the binomial setting, we must use a correction factor, in this case '+0.5'
Z= (x - μ)/σ
The Attempt at a Solution
First I should say I know the answer is suppose to be n≥92
So how I start this problem is I use the standardizing formula
Z= (x - μ)/σ
which in this case would be
Z= (0.35*n + 0.5 - n*0.25)/(sqrt(n*0.25*0.75)
This simplifies to
Z=(0.1*n + 0.5)/(sqrt(n)*sqrt(0.1875))
I think what I have done so far is correct. But where I get confused is finding a value for Z,
I thought what I have to do is something like :
P(B(n,0.25)≤0.35*n)=Φ((0.1*n + 0.5)/(sqrt(n)*sqrt(0.1875))) = 0.99
so, Φ((0.1*n + 0.5)/(sqrt(n)*sqrt(0.1875))) = 0.99
look up 0.01 on the table which gives me Φ(2.33)=Φ((0.1*n + 0.5)/(sqrt(n)*sqrt(0.1875)))
and so I should be able to set those equal and solve for n:
2.33 = ((0.1*n + 0.5)/(sqrt(n)*sqrt(0.1875))
When I solve for n I get a quadratic formula but neither answers I get is the correct answer.
Any help would be appreciated.
Thanks!
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