Introduction to Mathematical Statistics (MATS3206)
Mathematics and Computer Science - MCS
Semester: Second Semester
Level: 300
Year: 2019
Page
1
of 2
UNIVERSITY OF BAMENDA
FACULTY OF SCIENCE
DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE
Second Semester Examination 2019 Session
Course Code/Title
MATS3206 -
Introduction to Mathematical
Analysis
Duration
2 hours
Answer ALL Questions.
1. A certain population consists of the digits 2, 6, 9, 15.
(a) Find the mean, and the variance,
2
of the population.
(b) Draw up a frequency distribution of the means of all possible samples of size 2 that can be taken
without replacement from the population.
(c) Find the mean and variance of the sampling distribution of means.
(d)
2. a) State five properties of a desirable estimator,
(b) A large population has mean, E ( X ) = .
(i) Independent identical random samples X
1 ,
X
2
and X
3
are drawn from this population. Show
that is an unbiased estimator of the population mean, where
=
X
1
+
X
2
+
X
3
.
(ii) A random sample of size 27 from this population produced the following statistics:
X = 1560 and ( X -
)
2
=168900. Calculate to two decimal places, the most efficient unbiased
estimate of the population mean and the population variance.
3. (a) Explain briefly the following terms as used in rules of decision:
(i) Type I and Type II errors
(ii) Level of significance and size of confidence interval
(b) The marketing manager of a company producing a particular brand of electric bulbs claims that
the mean length of life of this brand is 1072 hours. To test his claim, a random sample of 36 light
bulbs is selected from the company’s production line and found to have a mean length of life 1062
hours, with a standard deviation of 120 hours. Test, at the 5% level of significance, whether or not,
the manager is justified in his claim.
4. (a) A random variable X has a normal distribution mean . A random sample of 10 observations of
X is taken and gives
. find 95% symmetric confidence interval for
(b) The null hypothesis is that the discrete random variable X has probability distribution given by
P(X=x) =
!
A random sample of 70 observation of X is summarized in the table below.
Value of X
1
2
3
4
Frequency 4
20
18
28
Use a chi-square (X
2
) test to determine whether there is evidence at 1% level of significance, to reject
the null hypothesis.
5. (a) A random sample from a B i n ( n , p ) distribution yields the following values: 4, 2, 7, 4,
1, 4, 5, 4. Find method of moments estimates of n and p using
and S
2
.
[Hint: S
2
=
"#
$
%
&
'
"
%#
(b) Find the maximum likelihood estimate of ,
for a random sample of size n
taken from a
Poisson( ) distribution. A sample of 10 observations from a Poisson ( ) distribution had a total of
24. Find the MLE of .
END
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