Assignment 4 Material from Module 6 Questions Total of marks is 31. Question XXXXXXXXXXmarks (a) The mean weight for a bird in a population is 31g. and the standard deviation is 0.2g. The distribution...

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Assignment 4
Material from Module 6 Questions Total of marks is 31.
Question 1 - 12 marks
(a) The mean weight for a bird in a population is 31g. and the standard deviation is
0.2g. The distribution of weights is normal. Determine
(i) the probability that the weight for a randomly selected bird exceeds 30.5g,
[2 marks]
(ii) the probability that the weight for a randomly selected bird is between 30.5g
and 31.5g. [2 marks]
(iii) A bird is classi ed as underweight if its weight is less than 30.4g. What is the
probability that at least 1 bird in a sample is underweight. [4 marks]
Data Source: Statistics and data with R. Y. Cohen & J. Cohen, Wiley (2008).
(b) The voltage in an electric circuit has a normal distribution with mean 120 and stan-
dard deviation 2. If 3 independent measurements of the voltage are made, what is
the probability that all 3 measurements will be between 116 and 118? [4 marks]
Data Source: Probability and Statistics M. DeGroot & M. Schervish, Addison-Wesley (2002).
Question 2 - 8 marks
A system contains 3 components that function independently of each other and are
connected in series. The system fails as soon as one of the components fails.
The lifetime of the rst component, measured in hours, has an exponential distribution
with rate  ?? 1 = 0:001; the second is exponentially distributed with rate parameter
2 = 0:003 and the third has an exponential distribution with rate 3 = 0:006.
Determine the probability that the system will not fail before 100 hours.
Data Source: Probability and Statistics M. DeGroot & M. Schervish, Addison-Wesley (2002).
Question 3 - 4 marks
A random variable X > 0 has a probability density function f and a distribution F. A
hazard function is de ned
h(x) =
f(x)
1 ?? F(x)
x > 0
show that if X has an exponential distribution, the hazard function h(x) is constant.
Data Source: Probability and Statistics M. DeGroot & M. Schervish, Addison-Wesley (2002).
1
Question 4 - 7 marks
Response times on a computer terminal have an approximate gamma distribution with
mean 4 seconds and variance 8 seconds2.
(a) Write the probability density function for waiting times. [3 marks]
(b) What is the probability that the waiting times exceed 9.5 seconds? (Use R).
[2 marks]
(c) What is the 80%'ile for waiting times? [2 marks]
2
Answered Same DayDec 22, 2021

Answer To: Assignment 4 Material from Module 6 Questions Total of marks is 31. Question XXXXXXXXXXmarks (a) The...

David answered on Dec 22 2021
128 Votes
Question: (a) The mean weight for a bird in a population is 31g and the standard deviation is
0.2g. The distribution of weights is normal. Determine
(i) the probability that the weight for a rando
mly selected bird exceeds 30.5g,
(ii) the probability that the weight for a randomly selected bird is between 30.5g and 31.5g.
(iii) A bird is classified as underweight if its weight is less than 30.4g. What is the probability that at least 1 bird in a sample is under weight.
Solution:
For normal probability distribution,
Probability density function
(
)
2
1
2
1
2
x
fxe
m
s
sp
-
æö
-
ç÷
èø
=
, let z =
x
m
s
-

(
)
1
2
2
1
2
z
fxe
sp
-
=
Normal probability distribution P
(
)
z
-¥<<¥=

1
2
2
1
2
z
edz
p
¥
-

ò
=1
P
(
)
azb
<<=

1
2
2
1
2
b
z
a
edz
p
-
ò
The values of P are given by the standard normal table
Here
(a)
(i) Mean =
m
= 31g, standard deviation =
s
= 0.2g ,
x > 30.5g
z =
30.531
0.2
x
m
s
--
=
= -2.5
P(-2.5 = 1-0.00621
= 0.99379 Answer
(ii) 30.5 P(-2.5 = 0.99379 - 0.00621
= 0.98758 Answer
(iii) x<30.4
z =
30.431
3
0.2
x
m
s
--
==-
P(-3>z) = .0013
Let there are n birds so out of n birds the probability that at least 1 bird is under weight.
P =
0
0
1()()
nn
Cpq
-
here p = 0.0013 and q = 0.9987
P =
0
1(0.0013)(0.9987)
n
-
P =
1(0.9987)
n
-
Answer
(b) The voltage in an electric circuit has a normal distribution with mean 120 and standard deviation 2. If 3 independent measurements of the voltage are made, what is the probability that all 3 measurements will be between 116 and 118
Solution: 116z =...
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