MATH 189 CSU Stanislaus Exploratory Data Analysis and Inference Discussion Ques

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HW questions at the end of casestudy4 file, given the data you needuse exactly the method given to solve the questionsmeet all the format requirements givengiven the sample homework report, check that for hw format(need to have intro, data, analysis, conclusion, advance analysis)given the data you need

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Introduction
The data
Background
Investigations
Math
189:
Chapter
4 Data Analysis and Inference
Armin Schwartzman Professor Division of Biostatistics and Halıcıoǧlu Data Science Institute University of California, San Diego
Snow gauge
* Main source of Water for Northern California comes from the
* To help monitor the water supply, the Forest Service of the
United States Department of Agriculture (USDA) operates a
gamma transmission snow gauge in the Central Sierra Nevada
near Soda Springs, CA. The gauge is used to determine a
depth profile of snow density.
* Analysis of the snow pack profile helps with monitoring the
water supply and flood management. It is also a source of
data for the study of climate change.
2
Snow gauge(cont.)
* The gauge does not directly measure snow density. The
density reading is converted from a measurement of gamma
ray emissions.
* Due to instrument wear and radioactive source decay, there
may be changes over the seasons in the functions used to
convert the measured values into density readings.
each year at the beginning of the winter season.
* In this case study we will develop a procedure to calibrate the
snow gauge from data.
3
Introduction
The data
Background
Investigations
Math
189:
Chapter
4 Data Analysis and Inference
Armin Schwartzman Professor Division of Biostatistics and Halıcıoǧlu Data Science Institute University of California, San Diego
Description
* The data are from a single calibration run of the snow gauge.
* The run consists of placing polyethylene blocks of known
densities between the two poles of the snow gauge and taking
readings on the blocks. The polyethylene blocks are used to
simulated snow.
* The measurements reported are amplified versions of the
gamma photon count made by the detector. We call the
gauge measurement the ”gain”.
* The data available here consists of 10 measurements for each
of 9 densities in grams per cubic centimeter of polyethylene.
4
The Data
5
The calibration process
To be used in practice, the snow gauge needs to map the measured
gamma ray intensity to snow density. However, the experiment is
done in reverse. The calibration process goes as follows.
1. The experiment measures gamma ray intensity as a function
of the density of the polyethylene blocks.
2. From the data, a function is determined that maps density to
gamma ray intensity.
3. The inverse of the above function is used to map gamma ray
intensity to density.
6
Introduction
The data
Background
Investigations
Math
189:
Chapter
4 Data Analysis and Inference
Armin Schwartzman Professor Division of Biostatistics and Halıcıoǧlu Data Science Institute University of California, San Diego
A Physical Model
The gamma rays that are emitted from the radioactive source
may be scattered or absorbed by the polyethylene molecules
between the source and the detector. With denser
polyethelene, fewer gamma rats will reach the detector.
A simplified version of the model that may be workable for the
calibration problem of interest is described here. A gamma ray
on route to the detector passes a number of polyethylene
molecules.The number of molecules depends on the density of
the polyethylene. A molecule may absorb the gamma photon,
bounce it out of the path to the detector, or allow it to pass.
7
A Physical Model
If each molecule acts independently, the chance that a gamma
ray successfully arrives at the detector is p m where p is the
chance that a single molecule will neither absorb nor bounce
the gamma ray, and m is the number of molecules in a
straight line path from the source to the detector.
Let d = Cm be the density, proportional to the number of
molecules m by some unknown constant C .
Let g = Ap m be the instrument gain, proportional to the
probability of detection p m by some unknown constant A.
8
A Physical Model
The gamma ray measurement can be expressed as
g = Ap m = Ae (log p)m = Ae (log p)/C ·(Cm) = Ae
d

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