UNIVERSITY OF TECHNOLOGY, JAMAICA
SCHOOL OF BUSINESS ADMINISTRATION
2020/2021
ECONOMETRICS
WORKSHEET ONE
N.B. Students are to complete this worksheet and submit on September 8th
Work in groups. Solution will be posted on Sept 9th.
1.
A. Define what is meant by a continuous variable and give some examples.
B. Define what is meant by a continuous probability distribution?
C. Derive the formula for the expected value and variance of a continuous
probability distribution.
2.
A. What is a normal distribution?
B. What is its usefulness?
C. What is the standard normal distribution?
D. What is its usefulness?
3
A. The lifetime of light bulbs is known to be normally distributed with µ=100 h
and σ = 8 h. What is the probability that a bulb picked at random will have a
lifetime between 110 and 120 burning hours?
4
Assume that family incomes are normally distributed with µ = $16,000, and σ =
$2000. What is the probability that a family picked at random will have an
income: (a) Between $15,000 and $18,000? (b) Below $15,000 and $18,000 (c)
Above $18,000 (d) Above $20,000?
5
The grades on the midterm examination in a large statistics section are normally
distributed with a mean of 78 and a standard deviation of 8. The professor wants
to give the grade of A to 10% of the students. What is the lowest grade point that
can be designated an A on the midterm?
6
Assume that a population is composed of 900 elements with a mean of 20 units
and a standard deviation of 12. The mean and standard error of the sampling
distribution of the mean for a sample size of 36 is
7
The probability that a mean of a random sample of 36 elements from the
population in Question 6 falls between 18 and 24 units
8
A random sample of 144 with mean of 100 and a standard deviation of 60 is taken
from a population of 1000. The 95% confidence interval for the unknown
population mean is
9
A state education department finds that in a random sample of 100 persons who
attended college, 40 receive a college degree. Find the 99% confidence interval
for the proportion of college graduates out of all the persons who attended
college. NB: note that this problem involves the binomial distribution AND since
n >30 and np > 5 and n(1 – p) > 5, the binomial distribution approaches the
normal distribution.
10
A random sample of n = 10 flashlight batteries with a mean operating life X = 5 h
and a sample standard deviation s = 1 h is picked from a production life known to
produce batteries with a normally distributed operating lives. Find the 95%
confidence interval for the unknown mean of the working life of the entire
population of batteries. NB: first find the value of ±t 0.025 so that 2.5% of the
area is written each tail for n – 1 = 9 df. The value we get is 2.262.
11
What is meant by statistical inference? What is its function and importance?
B. What is meant by and what is the relationship between a parameter and a
statistic?
C. What is meant by estimation? Hypothesis testing?
12
What is meant by random sampling? What is its importance?
13
A. What does sampling distribution mean and how is a sampling distribution of
the mean obtained?
B. What is meant by the mean and standard error of the sampling distribution of
the mean?
14
How can we find (a) The mean of the sampling distribution of the mean  ?
X
(b) The standard deviation of the sampling distribution of the mean or standard
error X ?

15
A population of 12,000 elements has a mean of 100 and a standard deviation of
60. Find the mean and standard error of the sampling distribution of the mean for
sample sizes of (a) 100 and (b) 900.
16
How can we calculate the probability that a random sample has a mean that falls
within a given interval if the theoretical sampling distribution of the mean is
normal or approximately normal? How is this different from the process of
finding the probability that a normal distribution random variable assumes a value
within a given interval? (b) Draw a normal curve in the X and z scales and show
the percentage of the area under the curve within 1, 2, and 3 standard deviation
units of its mean.
17
Find the probability that the mean of a random sample of 25 elements from a
normally distributed population with a mean 90 and a standard deviation of 60 is
larger than 100.
18
A random sample of 64 with a mean of 50 and a standard deviation of 20 is taken
from a population of 800. (a) Find an interval estimate for the population mean
such that we are 95% confident that the interval includes the population mean.
(b) What does the result of part a tell us?
19
A random sample of 25 with a mean 80 is taken from a population of 1000 that is
normally distributed with a standard deviation of 30. Find (a) the 90%, (b) the
95%, and (c) the 99% confidence intervals for the unknown population mean. (d)
What does the difference in the results to parts a, b, and c indicates?
20
A random sample of 100 workers in a plant employing 1200, 70 prefer providing
for their own retirement benefits over belonging to a company – sponsored plan.
Find the 95% confidence interval for the proportion of all workers in the pant who
prefer their own retirement plans.
21
A random sample of 25 with a mean of 80 and a standard deviation of 30 is taken
from a population of 1000 that is normally distributed. Find the (a) the 90%,
(b) the 95%, and (c) the 99% confidence intervals for the unknown population
mean.
22
A firm wants to know with a 95% level of confidence if it can claim the boxes of
detergent it sells contain more than 500g of detergent. From past experience the
firm knows that the amount of detergent in the boxes in normally distributed. The
firm takes a random sample of n = 25 and finds that X = 520g and s = 75g.
Define the critical, or rejection, region of the test at the 5% level of significance.
23
A manager wants to determine at the 5% L.O.S. if the hourly wages for semiskilled
workers are the same in two cities. She takes a random sample and finds that X 1 =
$6.00, X 2 =$5.40, s1 = $2.00 and s2 = 1.80 for n1 = 40 and n2 = 54.
24
A firm wants to determine at the 1% level of significance if the proportion of
acceptable electronic components of a foreign supplier, p1, is greater than for a
domestic supplier, p2. The firm takes a random sample from the shipment of each
supplier and finds the p1 = 0.9 and p 2 = 0.7 for n1 = 100 and n2 = 80.
25
In the past, 30% of the TVs sold were small screen, 40% were medium, and 30%
were large. In order to determine the inventory to maintain of each type of TV set,
the manager takes a random sample of 100 recent purchases and finds that 20%
were small screen, 40 % medium, and 40 large. Are past sales pattern correct?
26
a) What is meant by testing a hypothesis? (b) What are type 1 and type 11 errors?
(c) Define the level of significance and the level of confidence.
27
A producer of steel cables wants to test if the cables it produces have a breaking
strength of 5000lb. He takes a random sample of 64 pieces and finds that the
average breaking strength is 5100lb and the sample standard deviation is 480lb.
Should the producer accept the hypothesis that its steel cable has a breaking
strength of 5000lb at the 5% level of significance?
28
Define the rejection and acceptance regions in the above problem in terms of
pounds.
29
A hospital wants to test that 90% of the dosages of a drug it purchases contain 100
mg of the drug. The hospital takes a sample of n = 100 dosages and finds that
only 85 of them contain the appropriate amount. How can the hospital test this at
the 1%? 5%?
30
A large buyer of light bulbs wants to decide, at the 5% level of significance, which
of two equally priced brands to purchase. He takes a random sample of 10 bulbs
of each brand and finds that brand A lasts 980 h on the average X 1 with a sample
standard deviation s1 of 80 h. For brand B lasts 1010 h on the average X 2 with a
sample standard deviation s2 of 120 h. Which brand should he buy to reach a
decision at the 55 significance level?
31
The 65 students who apply for admission into a masters programme in 2009 have
average GRE scores of 640 with a s.d. of 20. In 2010, the 81 students who apply
have average GRE of 650 with a s.d. of 40. Are the 22009 applicants inferior to
the 2010 applicants at the 55 significance level? What is the acceptance region for
the test in terms of GRE scores?
32
Suppose that 505 of the 60 plants in Kingston abide by the antipollution standards
but only 40% of the 40 plants in Portland do so. Is the percentage of plants
abiding by the antipollution standards significantly greater in Kingston as opposed
to Portland at the 55 level of significance?
33
A plant manager takes a random sample of 100 sick days and finds that 305 of the
plant labour force in the 20-29 age group took 26 days of the 100 sick days, that
40% of the labour force in the 30 – 39 age group took 37 sick days, 20% in the 40
– 50 age group took 24 sick days, and that 10% of the 50-and-over age group took
13 sick days. How can the manager test at the 5% level of significance the
hypothesis that age is not a factor in taking sick days?
34
What are nonparametric tests? When should one use nonparametric tests?
35. A research team observed that men are 5 times as likely as women to watch cricket
on TV. Does this information represent descriptive or inferential statistics?
36. For a sample of 11 employees, the hourly wages were $18, 30, 25, 15, 17, 9, 11, 20,
12, 16, and 19. Determine the standard deviation and variance. Standard the data.
37. For data set A, the mean is $1235, with a standard deviation of $140. For data set B,
the mean is 15.7 metres, with a standard deviation of 1.87 metres. Which of these
two data sets has the greater relative dispersion?
38.
Explain what is meant by sampling error, response error, and non-sampling error in
survey research.
39. From a random sample of 10 (x,y) pairs of data points: (11,15), (23.27), (24,26),
(15,18), (17,19), (33,36), (25,29), (31,38), (29,32). Compute the covariance and the
correlation coefficient.