# Assignment 4 Estimating Brightness

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STAT 341: Assignment 4
NOTES
Your assignment must be submitted by the due date listed at the top of this document, and it must be
submitted electronically in .pdf format via Crowdmark/LEARN. This means that your responses for different
questions should be in separate .pdf files. Your .pdf solution files must have been generated by R Markdown.
• For mathematical questions: your solutions must be produced by LaTeX (from within R Markdown).
Handwritten and scanned/photographed solutions will not be accepted and you will receive zero points.
• For computational questions: R code should always be included in your solution (via code chunks in R
Markdown). If code is required and you provide none, you will receive zero points.
– Exception any functions used in the notes or function glossary can loaded using echo=FALSE but
use echo=FALSE but chunks that call gradientDescent should have echo=TRUE.
• For interpretation question: plain text (within R Markdown) is fine.
Organization and comprehensibility is part of a full solution. Consequently, points will be deducted for
solutions that are not organized and incomprehensible.
Question 1 – 16 Marks – Estimating Brightness
For this question you will need the digit data from file “digitData.csv”. Use the sample below from the Digit
data to answer the following questions. The sample was obtained by sampling without replacement.
digitSample <- c(294,133,95,265,154,1,289,232,121,99,129,83,30,56,249,134,46,68,165,279,105,91,248,285,2a) [2 Marks] Calculate the Horvitz-Thompson estimate of the average amount of brightness and provide
the standard error of the estimate.
b) [2 Marks] Calculate the Horvitz-Thompson estimate of the proportion of digits with brightness less
than or equal to 45
c) [2 Marks] Calculate the Horvitz-Thompson estimate of the proportion of digits with brightness in the
interval [20, 25).
d) [4 Marks] In three graphs in a row,
• plot the Horvitz-Thompson estimate of the cumulative distribution function for brightness,
1
• plot the standard error of the estimate and
• plot the Horvitz-Thompson estimate of the cdf and overlay the lines of ±2 times the standard
error.
• Note Similar to the cdf, the standard error is also a function.
e) [4 Marks] In three graphs in a row,
• plot the Horvitz-Thompson estimate of the histogram for brightness with bin width equal to five,
• plot the standard error of the estimate and
• plot the Horvitz-Thompson estimate of the histogram and overlay the lines of ±2 times the
standard error.
• Note Similar to the cdf, the standard error is also a function.
f) [2 Marks] Calculate the Horvitz-Thompson estimate of the proportion of ones and provide the standard
error of the estimate.
Question 2 – 16 Marks – Comparing Estimators
Using a population, compare two or more estimators of some population attribute. Your answer should
include:
• description of the population and the context of the problem,
• the effect of changes in some parameters such as the sample size,
• numerical and graphical comparison and their descriptions,
• a conclusion.
• You answer should be limited to 1 to 3 pages.
Your solution should be in your own words, but as motivating examples consider the following samples
exercises:
• 1.6 Comparing the standard deviaton and the Median Absolute Deviation,
• 1.7 Comparing two sampling designs,
• 1.10 Comparing measures of dispersion,
• 1.13 Comparing the Mean and Median and
• 1.17 Estimating the median for Radar Data.
Rubric
Criteria Descriptor Marks
Population/Estimators Description and Difficulty /4
Format Clarity, Organization and LaTeX /4
Comparision Description, Results and Graphic /4
Discussion/Summary Justification and Relevant Terminology used /4
2
Question 3 – 10 Marks – Summarize
In your own words summarize the subsection 4.2.1-Anatomy_of_a_Signifiance_level
• You are limited to 1 to 2 pages.
• You are recommended to use a combination of formulas, full sentences and example.
Rubric
Criteria Descriptor Marks
Format Organization /3
Writing Clarity & Grammar /2
Content Coverage, Depth, Relevant Terminology used and Example /5
3

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