Download Final - OSU Physics

Richard Kass
Physics 416 Take Home Final Exam
Due Wednesday March 17, 2004 5PM
Exam Rules:
a) Work alone.
b) You are free to consult with any library book(s) you wish. If you do use
references other than the notes and text please list them.
c) Show your work. Do not just write down the answer (especially if you hope to
get partial credit) !
PART 1 (280 points)
1) Short answer questions, each worth 5 points.
a) What is the difference between precision and accuracy?
b) True or False: The FORTRAN function ran(iseed) generates numbers that are distributed
according to Gaussian probability in [0,1].
c) Give an example of a continuous probability distribution.
d) Define the mean, mode, and median of a probability distribution.
e) Assume I have 10 data points (x, y pairs) which I believe are described by the polynomial:
y= a+bx+cx2+dx3. If I perform a least squares fit to determine (a,b,c,d) how many degrees of
freedom do I have left?
f) Can the probability for an event to occur be negative?
g) True or False: The variance is a measure of the spread of the data around the mean.
h) Under what assumption(s) do the maximum likelihood method, chi-square fitting, and method
of least squares give the same results?
i) Under what conditions does a binomial distribution “turn into” a Poisson distribution?
j) Two experimental groups summarize their results by quoting a value of chi-square and the
number of degrees of freedom. The first groups quotes χ2=30 for 10 degrees of freedom while the
second group quotes χ2=105 for 100 degrees of freedom. Which group’s result do you trust
more? Explain your choice in a few words.
2) Some people say that the OSU football team was lucky to win 19 games in a row. Let’s
investigate this idea.
a) If the football team has a 50-50 chance of winning a given football game what is the
probability that it will win 19 games in a row? 10 pts
b) If the probability of the football team winning 19 games in a row is 95% what is the
probability of winning an individual game? 10 pts
c) If on average the football team is expected to lose 3 games in 19 games what is the probability
i) that it loses more than three games in 19 games ? 10 pts
ii) that it goes undefeated in 19 games? 10 pts
Note: use binomial statistics for parts a) and b), Poisson statistics for part c)
3) The Central Limit Theorem:
a) 10 pts: Describe (state) the Central Limit Theorem and give the conditions for its applicability.
b) 30 pts: Assume that the probability of producing N particles in a single high energy proton
collision can be described by the following probability distribution:
P( N ) 
16 N e 16
N!
If the total number of produced particles is counted after 100 collisions what is the probability
that the total number of particles will be in the range [1600, 1700]?
4) A Maximum Likelihood Method Problem.
Assume that the probability distribution function, p(x,), to find two quarks separated by a
distance x (in units of 10-15 m) depends on an unknown constant ,
 1
p(x,  )  x
with 0  x 1 and 0    
and we have five measurements of x (x1=0.3, x2=0.68, x3=0.35, x4=0.87, x5=0.50).
a) 10 pts: Write down the likelihood function for this problem.
b) 15 pts: Use the Maximum Likelihood Method to derive an expression for .
c) 15 pts: Use the Maximum Likelihood Method to calculate a value for .
Note: if necessary, part c) can be done without part b).
5) A Propagation of Errors Problem.
A student is using a horizontal air track where a glider is connected via a spring causing the
glider to oscillate. The total energy (E) of the glider is given by:
1
1
E  mv 2  kx2
2
2
With: m=0.230± 0.001 kg and k=1.03± 0.01 N/m.
The student measures the following pairs for the location (x) and speed (v) of the glider:
i) x=0.700± 0.002 m, v=0 (exact) m/s
ii) x=0.551± 0.005 m, v=0.89± 0.01 m/s
a) 5 pts: Calculate the total energy of the glider using the data sets i) and ii).
b) 15pts: Use propagation of errors to calculate the error () in the energy (E) for sets i) and ii).
c) 10 pts: Are the measurements i) and ii) consistent with conservation of energy? Use the
example on page 9 of Lecture 8 as a guide to calculate the confidence level that the two measured
energies are equal.
6)Taylor, Problem 8.22 page 205. (40 pts)
7) For the following -ray spectrum assume that the photopeak has energy = 660 keV.
a) 15 pts: Calculate the energy of the Compton edge and note where it located on the spectrum.
b) 15 pts: Calculate the energy of the 1800 backscatter peak and note where it is located on the
spectrum.
c) 10 pts: Using your results from a) and b) estimate the energy of the “mystery peak”.
mystery
peak
PART 2 (140 points)
Design and perform an experiment to measure the probability distribution function of popcorn.
You may have to practice or repeat the experiment several times before you obtain good quality
or reproducible data. The end result should be a table and histogram like the one below for the
(number of pops)/(time interval) vs time. Note, the time interval and limits that I picked are
somewhat arbitrary, you may want to use different values. In addition answer the following
questions.
a) Describe the setup used to make the measurement. What variables are important in the
experiment? Discuss the precision and repeatability of your measurements.
b) Make a histogram of your data. Use the example below as a guide. How does your probability
distribution function compare with a Gaussian probability distribution function ?
c) From your data calculate the average time it takes for a kernel of popcorn to pop.
d) From your data calculate the most probable time for popcorn to pop.
e) From your data what is the probability for popcorn to pop within ± 30 seconds of the average
time (from part c)?
Fake (made up) Popcorn Data
time (minutes)
0.0-0.5
0.5-1.0
1.0-1.5
1.5-2.0
2.0-2.5
2.5-3.0
3.0-3.5
3.5-4.0
4.0-4.5
4.5-5.0
Fake Popcorn Data
number of pops
0
2
20
40
100
130
50
10
1
0
Fake Popcorn Data
number of pops/0.5 min
140
120
100
80
60
40
20
0
0.25
0.75
1.25
1.75
2.25
2.75
3.25
time (minutes)
3.75
4.25
4.75