Statistics - Module 1+3 full exam
10 June 2022
Exercise 1
A researcher recorded the time X, in minutes, spent using a mobile phone
during a particular afternoon, for each child in a club. The observations are
summarized below.
X
Frequency
(0, 50]
20
(0, 100]
44
(0, 150]
60
(0, 200]
74
(0, 300]
80
(a) Represent the frequency distribution of X through a histogram and
comment on the shape of the distribution.
(b) Calculate the median of X.
(c) Calculate the interquartile range of X.
(d) Assume that for each child 60% of the time spent on the phone is playing
games. How do the median and interquartile range of the time spent
playing games vary with respect to the corresponding summaries for X?
Exercise 2
The weights, in kilograms, of men and women have the distributions N (75, 80)
and N (65, 70), respectively.
(a) The maximum load that a certain cable car can carry safely is 1200 kg.
If 9 randomly chosen men and 7 randomly chosen women enter the cable
car, find the probability that the cable car can operate safely.
(b) Find the probability that a randomly chosen woman weighs more than
a randomly chosen man.
(c) A randomly chosen individual is found to weigh less than 65 kg. Calculate
the probability that the chosen individual is a woman, if the population
comprises the same proportion of men and women.
Exercise 3
A large batch comprises iron bars of two types, A and B. The length, in
centimeters, of an iron bar follows a U(6,14) distribution for type A bars and
a U(6,10) distribution for type B bars, respectively. The proportions of each
type of bar in the batch are unknown and in order to estimate them a s.r.s.
of n bars is extracted and the length of each sampled bar measured. Let π be
the fraction of type A bars in the batch and
T =
2N
n
be an estimator for π, where N is the random variable representing the number
of sampled bars with length greater than 10.
(a) Is T a biased or an unbiased estimator for π?
(b) Calculate the MSE of T .
(c) Assuming n = 100, use the Central limit theorem to approximate the
significance level (i.e., the probability of a type I error) of a test that rejects the null hypothesis H0 : π = 0.5 if t > 0.6, where t is the observed
realization of T .
Exercise 4
A sample study was made of the number of business lunches that executives
claim as deductible business expenses per month. It is found that, within a
given month, 40 executives in the insurance industry averaged 8.8 such deductions, with a corrected sample variance of 1.92 ; while 50 bank executives
averaged 8.0 with a corrected sample variance of 2.12 .
(a) Propose an unbiased estimator of the average difference between the
number of deductions claimed by insurance and bank executives in the
given month and calculate the corresponding estimate.
(b) Provide a measure of the accuracy of the estimator in (a).
(c) Test the null hypothesis that the two categories of executives claim on
average the same number of deductions against a two-sided alternative
at the 10% significance level. (Specify the conditions for the validity of
the test.)
(d) The p-value of the test in (c) is 0.06. Comment on this result.