PHYS309, Spring 2010
April 21, 2010
Test on Probability. Solution.
Please write clearly, explain your reasoning, and submit your
solution on a
separate
piece of paper with your name on it.
Problem
You are an 18th century engineer responsible for the readiness of a cannon
battery. During one of the military exercises you measure the time of flight
t
of the cannon balls to the target. You get 30 results (in seconds):
8.16, 8.14, 8.12, 8.16, 8.18, 8.10, 8.18, 8.18, 8.18, 8.24,
8.16, 8.14, 8.17, 8.18, 8.21, 8.12, 8.12, 8.17, 8.06, 8.10,
8.12, 8.10, 8.14, 8.09. 8.16, 8.16, 8.21, 8.14, 8.16, 8.13.
You suspect that the distribution of
t
is Gaussian and want to check this
hypothesis. To do this
(a) Find the mean
¯
t
and variance
σ
of the data.
(b) Group the values of
t
into four bins with boundaries
¯
t

σ
,
¯
t
, and
¯
t
+
σ
and count the observed number O
k
in each bin
k
= 1
,
2
,
3
,
4. Assuming
the measurements were normally distributed with the mean
¯
t
and variance
σ
calculated above, find the expected number E
k
in each bin. Calculate
χ
2
. Is
there any reason to doubt that the measurements are normally distributed?
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 Spring '10
 Staff
 Normal Distribution, Probability theory, probability density function, Cauchy distribution, ek

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