HW8 (S2023)-CEE110 q3
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University of California, Los Angeles *
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Course
110
Subject
Industrial Engineering
Date
Jan 9, 2024
Type
Pages
3
Uploaded by j4fun
UCLA
Spring 2023
Introduction to Probability and Statistics for Engineers
CEE 110
Prof. Enrique López Droguett
Due: 6/09/2023
HOMEWORK #8 Extra Credit
Instructions:
1.
This homework is optional.
2.
If you choose to submit, it will replace the worst grade you have in HW1 through
HW7 (only if your final grade would improve by this modification).
3.
No submissions past the due date will be accepted for this homework. No
exceptions.
Problem 1 (
35
Points)
Let
࠵?
have probability density function:
࠵?
!
(࠵?) = ’
3(࠵?
"
− ࠵?
"
)
2࠵?
#
,
0 < ࠵? < ࠵?
0,
elsewhere
a.
Show that
࠵?/࠵?
can be used as a pivotal quantity.
b.
Use the pivotal quantity from part (a) to find a 90% upper confidence limit for
࠵?
.
Problem 2 (
40
Points)
The breaking strength of hockey stick shafts made of two different graphite-Kevlar
composites yield the following results (in newtons):
•
Composite A: 487.3 444.5 467.7 456.3 449.7 459.2 478.9 461.5 477.2
•
Composite B: 488.5 501.2 475.3 467.2 462.5 499.7 470.0 469.5 481.5 485.2 509.3
479.3 478.3 491.5
Find a 98% confidence interval for the difference between the mean breaking strengths of
hockey stick shafts made of the two materials for two cases:
a)
The population variances are not necessarily the same.
b)
The population variances can be assumed to be similar.
Problem 3 (
25
Points)
The following table shows the results of a uniaxial strength testing for a new steel alloy
that UCLA is developing. The underlying distribution for the strength of this new material
is estimated to be normal.
Table 1:Results for the uniaxial strength in MPa.
415
418
341
427
394
377
429
366
406
379
407
398
384
384
385
395
a.
Obtain the Maximum Likelihood Estimators for the mean and variance of the
underlying distribution (derive them analytically, showing the complete process).
b.
Using the Maximum Likelihood Estimators obtained in part a), compute the point
estimates for the mean and variance based on the data shown in Table 1.
c.
An alternative version of this material is being developed in parallel, and the objective
is to test whether a different percentage in carbon can increase the strength of the steel
alloy with respect to the original formula (part a & b). For this, 45 samples with this
alternative recipe are forged and tested, obtaining measurements with mean strength of
412.6 [࠵?࠵?࠵?]
and a standard deviation of
15.5 ࠵?࠵?࠵?
.
i.
Set up the null and alternative hypothesis. Is this a two-tailed test or one-tailed
test?
ii.
Can you conclude that the new recipe effectively increases the strength of the
material? Use a confidence level of
࠵? = 0.05
.
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