CHEN 3010
Applied Data Analysis
Fall 2008
Name:________________________________
Midterm Examination 2
¾
120 minutes
¾
100 points (points for each problem shown in parentheses beside the problem number)
¾
open book and notes
¾
calculator allowed (and needed)
¾
do all work on blank sheets or graph paper provided and start each problem on a new
sheet
¾
order your work sheets and staple them behind this exam document when you turn in
your exam
¾
time will be of the essence on this exam – attempt the problems you know best first, and
move from problem to problem quickly – if you get stuck on a problem, move on and
return to it later – you will not likely have time to "relearn" the material during the exam
On my honor, as a University of Colorado at Boulder student, I have neither given nor
received unauthorized assistance on this work.
______________________________________
(sign on this line)
1) (15 pts) You work for a small startup pharmaceutical company and are preparing
batches of an important protein drug to be used in FDA trials.
You need 4 successful
batches to ship to the FDA.
The company claims that the probability of getting a
successful batch is 0.25 (p=0.25).
You believe that this is low and the
actual
probability
of getting a successful batch is greater than 0.25.
You go about preparing the 4 batches
and find that the 7
th
batch you make is the 4
th
successful batch.
Can your claim that the
p > 0.25 be supported (
α
= 0.05)?
Solution: Since we need 4 successful batches and we are interested in how many trials
required to get these 4 batches this is a negative binomial distribution.
We want to see
how probable is it for use to get the 4
th
successful batch on the 7
th
batch we make.
The
company claims that the probability of success is 0.25 but we think it is higher.
Thus,
we can set up the following hypothesis and test it based upon the fact that we got the 4
th
successful batch on the 7
th
batch we made:
ܪ
:ൌ0.25,ܪ
ଵ
:0.25
Essentially what we are trying to find is how rare is it for us to get the 4
th
success on the
7
th
batch if the true success rate is 0.25.
Thus, we are looking at the negative binomial
distribution:
݂ሺݔሻ ൌ ቀ
ݔെ1
ݎെ1
ቁ ሺ1 െ ሻ
௫ି
where x is the number of trials required to obtain exactly r successes.
To support the
alternate hypothesis we must show that the probability of getting 4 successes on 4, 5, 6,
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Midterm Exam 1
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or 7 batches is less than an alpha value of 0.05.
Here are the noncumulative and
cumulative probabilities (using the negative binomial distribution function above) for x = 4
to x = 7:
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 '08
 KOMPALA,DH
 Normal Distribution, Standard Deviation, Variance, Null hypothesis, Probability theory

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