# probability exam question 2

Steve, Katerina and Jess are three students who have agreed to take part in a psychology experiment. Each
student is to answer several sets of multiple-choice questions. Each set has the same number of questions,
n, where n is a number greater than 20. For each question there are four possible options (A, B, C or D), of
which only one is correct.

From these sets of 25 questions being completed by many students, it has been found that the time, in minutes, that any student takes to answer each set of 25 questions is another random variable, W, which is normally distributed with mean a and standard deviation b.
It turns out that, for Jess, Pr(Y = 18) = Pr(W = 20) and also Pr(Y = 22) = Pr(W = 25).
Calculate the values of a and b, correct to three decimal places.

I dont get this question
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Commented:
I don't either.
What is Y?  And what are 18, 20, 22, and 25?

W is probably time in minutes.
Is there anything here about the number of right and wrong answers?
Why do you bother to say n>20 if n is always equal to 25?
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Commented:
The random variable Y is probably addressed in an earlier question.
Hopefully you have found a and b for that normal distribution already.

Since Y and W are both normal curves, you can find the curve parameters  for W by scaling with the two data points given here.

Y might be the number of correct answers.
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Author Commented:
i dont quite get it.

a = 24.246, b = 2.500 rounded off to three decimal places.
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Author Commented:
What is Y?  And what are 18, 20, 22, and 25?

Each student is to answer several sets of multiple-choice questions. Each set has the same number of questions, n, where n is a number greater than 20. For each question there are four possible options (A, B, C or D), of which only one is correct.
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Commented:
Pr(Y = 18) = Pr(W = 20)

Explain please. What does Y=18 mean? What does W=20 mean?
Y is number of questions answered correctly maybe? W is minutes spent on exam right?

What are the mean and standard deviation for Y? Is it normally distributed too?
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Commented:
>> The random variable Y is probably addressed in an earlier question.

There isn't enough information here to solve this problem.
The missing information is almost certainly in an earlier question that presented something about the normal variable Y.

To solve this question, you have to know the parameters a and b for the normal curve describing Y.  And I think you may also need to know actual values for Pr(Y = 18) and Pr(Y = 22).

The usual parameters used to describe a normal distribution are mean and standard  deviation.    If I had to guess:  a ==> mean   and   b ==> SD

Please look back at the earlier questions.
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Author Commented:
Here is the exam, look at 3D
2012mmcas2-w.pdf
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Commented:
Question 3C gives you the information you need about Y.

The mean of Y is 25*(5/6) = 20.83      That is easy.
You also have to find the SD.                 I don't know that formula off hand.

From the data in 3D you know that Y(18-22) maps to W(20-25),
so a (center of the dist)  increases approx 3
and b (width of the dist)   increases by a factor of exactly 5/4 ==> 1.25

You still need to find SD of Y.
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Commented:
Y is not normally distributed so the mean and stddev of Y are useless.

Calculate the actual values of Pr(Y >= 18) and Pr(Y >= 22)

Also
Pr(Y >= 18) = Pr(W >= 20) and also Pr(Y >= 22) = Pr(W >= 25). (>= not =)

All you get from that is two points. But since there are only two unknowns, this is enough to solve.
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Author Commented:
OK, so how do you work it out?

i dont quite get it.

a = 24.246, b = 2.500 rounded off to three decimal places.
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Commented:
You know that Pr(Y >= 18) = Pr(W >= 20) and you know how to get Pr(Y >= 18), it's about .955268
Pr(Y>=22) = .381566

So you know that W is a normally distributed variable and
Pr(W >= 20) = .955268
Pr(W >= 25) = .381566

Flip them to get the CDF
CDF of W at 20 = 1-.955268 = 0.044732
CDF of W at 25 = 1-.381566 = 0.618434

Looking them up it a Z table gives
Z at 20 is -1.7
Z at 25 is 0.3

Remember that the formula for Z is Z=(W-mean)/stddev
So -1.7 = (20-a)/b and .3 = (25-a)/b
Solve that for a and b and you get
a = 24.25, b = 2.500

Note that you had a=24.246 which is the same thing only a bit more precise. My Z values were probably rounded too much.
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Author Commented:
where i am getting lost is where you get values from like

(20-a/)b=-1.69 from the answer ..  how can this happen?

Pr(Z = (20 – a)/b)) = 0.955268… so the inverse is -1.69???????????

please explain this to me as I am completely confused!
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Commented:
No -1.69 came from the Z table in the back of your book.
CDF is 0.044732 which is less than half, so it's not in the Z table.
But look up 1-0.044732 in the Z table and you get 1.69 so for 0.044732 it is -1.69
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