Stat 231 Exam 2
Fall 2013
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1. Some IE 361 students worked with a manufacturer on quantifying the capability of a machining
process to produce metal parts meeting engineering requirements. A sample of n  15 parts had
measured diameters with x  1.53633 in and s  .00010 in .
7 pts a) Give 95% two-sided confidence limits for the average measured diameter of these parts. (Plug
in completely, but you need not simplify.)
13 pts b) The (mid-specification and) target for the diameter in question was 1.5365 in . With   .05 , is
there sufficient evidence to force the conclusion that the current machining process is producing
diameters with an average different from the target value? (Carefully show the whole 7-step
format.)
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6 pts c) If you had access to the students' data, what would you do with them before using a Stat 231
method to say where you expect (say) 95% of all measured diameters from this process to lie?
Why?
7 pts d) Give two-sided limits that you are "95% sure" enclose 99% of all measured diameters (assuming
the process is producing normally distributed diameters). (Plug in completely, but you don't need to
simplify.)
7 pts e) Give an upper bound that you are 95% sure is larger than the standard deviation of measured
diameters (assuming the process is producing normally distributed diameters). (Plug in completely,
but you don't need to simplify.)
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After adjusting the process, a sample of n2  200 parts produced a sample mean diameter
x2  1.53652 in , sample standard deviation s2  .00030 in , and 183 diameters meeting engineering
specifications.
7 pts f) Give two-sided 90% confidence limits for any change in measured mean part diameter produced
by the adjusting. (Plug in completely, but you don't need to simplify.)
7 pts g) Give two-sided 95% confidence limits for the fraction of diameters produced by the adjusted
process that meet engineering specifications. (Plug in completely, but you don't need to simplify.)
6 pts h) Set up (and plug into completely) an equation for a sample size that you believe would in the
future allow you to determine the process mean with a (90%) margin of error of .00001 in . (You
need not solve for the sample size.)
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2. The following is a problem about the concepts of hypothesis testing. It is not an application of
the any of the standard inference methods outlined on the inference summary sheets.
The variable
X  number of hits on certain high-traffic web site during a 10 second period
is modeled as Poisson with mean  . In what follows, use without needing to make any justification
the fact (that is actually a consequence of the CLT) that for large  a Poisson    distribution is
approximately normal. (Note also that you know the variance of a Poisson    variable.)
An engineer wants to be able to detect the possibility that   100 .
a) First suppose that the engineer will reject H 0 :   100 if X  110 .
7 pts
i)
Approximately what value of  is this engineer using?
7 pts
ii)
Approximately what is the probability of a Type II error if in fact   150 ?
6 pts b) Approximately what p -value would be associated with an observation X  120 ?
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10 pts
3. A (uniform) random number generator produces values between 0 and 1 with mean .5 and
standard deviation .2887. What is the approximate probability that the sample mean of the next 100
values it generates is between .45 and .55?
10 pts
4. Here are five 2-point True/False Questions. Write T or write F in the blank in front of each one.
_____ If a 90% prediction interval for xnew is  31.2, 45.7  , then this interval can also serve as a
confidence interval for  with a confidence level at least 90%.
_____ If a 90% confidence interval for  is  31.2, 45.7  , then a p-value for testing H 0 :   50.0
is larger than .10.
_____ A value of p̂ based on a sample size of n  100 produces a p-value of .10 for testing
H 0 : p  # . That same value of p̂ would then produce a smaller p-value for testing the
same null hypothesis if it were based on n  400 .
_____ The small n confidence limits for  are fairly "robust" to moderate non-normality
("working" well even if the distribution sampled departs somewhat from a normal shape),
but the confidence limits for  are less "robust."
_____ A statistical tolerance interval for a part dimension is directly concerned with whether parts
meet engineering tolerances on that dimension.
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