mid19f-324.pdf - Name: ID # McGill University STATISTICS...

Name: ID #
McGill University
STATISTICS (MATH324)
Mid-Term Exam, Tuesday October 22nd, 2019
Time: 11:30 - 13:00
Instructions: This test consists of four questions. There is no optional
question. Please answer all four questions. A two-sided 8.5 × 11 page of
formula and a calculator are permitted. The following information may be
of use to you:
• X ∼ Normal( μ, σ
2
) : f
X
( x ) =
1
√
2 π
e
-
( x - μ )
2
2 σ
2
x ∈ R
• X ∼ Poisson( λ ) : p
X
( x ) =
e
- λ
λ
x
x !
, x ∈ { 0 } ∪ N
• X ∼ Binomial( n, p ) : p
X
( x ) =
n
x
p
x
(1 - p )
n - x
, x = 0 , 1 , · · · , n
• Bernoulli(p) = Binomial(1, p)
• For Z ∼ N (0 , 1), P ( | Z | < 1 . 96) = 0 . 95.
Good luck.
1

Question 1. In polycrystalline aluminum, the number of grain nucle-
ation sites per unit volume is modelled as having a Poisson distribution
with mean λ . Fifty unit-volume test specimens subjected to annealing under
regime A produced an average of 20 sites per unit volume. Fifty indepen-
dently selected unit-volume test specimens subjected to annealing regime B
produced an average of 23 sites per unit volume.
(a) Estimate the difference in the mean numbers of nucleation sites λ
A
- λ
B
for regimes A and B, and find a 95% confidence interval for it. (5 marks)