Static Homework

STAT 344Homework 4
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Instructions:
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Deadline: October 26th, 2022, at 11.59PM through Blackboard as a single PDF.
Handwritten solutions must be scanned and submitted as a PDF file.
Typed solutions are acceptable and must be submitted as a PDF file.
Do not submit hard copies in Class.
Do not submit pictures/Photos as multiple files.
Submissions that do not meet all the following criteria will not be graded:
Your solutions will grade for completeness; all steps must be shown. The solutions are expected
to be of collegiate quality.
Each solution should be:
• Written one question per page
• written darkly.
• Should be easy to follow.
• Must show your work
Be sure that your name in the upper right-hand corner of each page.
See below options for generating a PDF for this exam, Name the file
STAT344HW4_Lastname_Firstname.
Once the PDF has been generated, please view the file to assure its legibility prior to uploading.
You will only be able to upload the file once.
Generating a PDF: Here are some options for generating a PDF:
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CamScanner – PDF Scanner App; free on Android and iOS. You can upload to Google
Drive and Dropbox directly.
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Using “Notes” app on iOS/Apple Devices:
You should practice generating a PDF at least three times prior to the submitting one. Be sure that
the file can be easily viewed on a tablet, iPad, and computer.
I strongly suggest that you create a folder on the device that you will use to store the PDFs for this
course.
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Answer all questions. All questions relate to Continuous Distributions.
1. A test instrument needs to be calibrated periodically to prevent measurement errors. After
some time of use without calibration, it is known that the probability density function of
the measurement error is f ( x ) = 1.0 (1 – 0.5x ) for 0 < x < 2.0 millimeters. Note that x is the absolute value of the measurement error. a. If the measurement error within 0.5 millimeters is acceptable, what is the probability that the error is not acceptable and the instrument needs calibration? b. What is the value of measurement error that must be exceeded with probability 0.3before the instrument needs calibration? c. What is the probability that the measurement error is exactly 0.21millimeters before calibration? 2. In a paper manufacturing company, a machine is used to press wet fiber web into a continuous roll of paper. This machine does not create a constant pressure on wet fiber web and final sheets of papers have different thickness which is uniformly distributed between 0.004 and 0.015 inch. Let X denote the thickness of the sheet of paper. Determine the following: a. Mean and variance for thickness of each paper sheet. b. Proportion of paper sheets which are less than 0.0095 inch thick. c. Thickness exceeded by 40 percent of the paper sheets. 3. A laptop company claims up to 9.1 hours of wireless web usage for its newest laptop battery life. However, reviews on this laptop shows many complaints about low battery life. A survey on battery life reported by customers shows that it follows a normal distribution with mean 8.5 hours and standard deviation 39 minutes. a. What is the probability that the battery life is at least 9.1 hours? b. What is the probability that the battery life is less than 7.9 hours? c. What is the time of use that is exceeded with probability 0.9? 4. Web crawlers need to estimate the frequency of changes to Web sites to maintain a current index for Web searches. Assume that the changes to a Web site follow a Poisson process with a mean of 6 days. Let a random variable X denote the time (in days) until the next change. a. What is the probability that the next change occurs in less than 4.5 days? b. What is the probability that the time until the next change is greater 9.5 days? c. What is the time of the next change that is exceeded with probability 90%? 5. Suppose that the lifetime of a component (in hours), X, is modeled with a Weibull distribution with b = 0.5 and d = 3400 . Determine the following in parts (a) and (b). a. P( X > 3500) =
b.
c.
P( X > 6000 X > 3000) =
Suppose that X has an exponential distribution with mean equal to 3400. Determine
P( X > 6000 X > 3000) =
6. News articles that link to related stories are widely used in Web marketing. With a large
number of daily visitors to a Web page, we model the proportion of daily visitors who click
on a link to a related story as approximately a continuous random variable with a beta
distribution. The parameters are a = 6 and b = 1.
a.
b.
c.
d.
What is the mean and standard deviation of the proportion of visitors who click?
What is the probability a proportion exceeds 0.58?
What proportion is exceeded with probability 0.33?
If 550 visitors view the page, what is the expected number of visitors who click on
a link to a related story?