Q1. The marketing manager of a company producing a new cereal aimed for children wants
to examine the effect of the colour and shape of the box’s logo on the approval rating of the
cereal. He combined 4 colours and 3 shapes to produce a total of 12 designs. Each logo was
presented to 2 different groups (a total of 24 groups of children) and the approval rating for
each was recorded and is shown below.
Shape
Colour
Red Green Blue Yellow
Circle
54 67 36 45
44 61 44 41
Square
34 56 36 21
36 58 30 25
Diamond
46 60 34 31
48 60 38 33
a) How many factors does this experiment have? Identify the factors and state how
many levels each factor has.
b) If all combinations are compared, how many different treatments (cells) are there in
the experiment? What is the response variable?
c) Perform the appropriate analysis, what combination of colour and shape should be
used for the logo design?
Q2. Common crossbills have bills with the upper mandible to the left (left-billed) or to the
right (right-billed). (If you are interested, an excellent photo illustrating this may be found at
http://www.bbc.co.uk/nature/life/Common_Crossbill.) The following table gives leg length
measurements of common crossbills, recorded to investigate whether the method of feeding
places more strain on one leg than the other, depending on the whether they are left-billed or
right-billed. Use these data to investigate that question. What are your conclusions? (Note:
Twenty-four birds were measured, 12 of each bill-type.)
Left-Billed Right-Billed
left right left right
leg leg leg leg
19.0 17.4 17.8 18.0
17.1 17.3 15.6 17.1
16.6 16.6 16.9 17.2
17.0 16.7 16.9 17.1
18.3 18.2 17.0 17.3
17.5 17.3 16.9 17.1
16.4 16.3 17.4 17.6
17.1 16.9 17.4 17.7
17.5 16.7 17.7 17.7
17.0 16.8 16.5 17.5
18.6 17.5 17.1 17.3
16.3 16.2 16.7 16.9
Q3. The effect of three different lubricating oils on fuel economy in diesel truck engines is
being studied. Fuel economy is measured using brake-specific fuel consumption after the
engine has been running for 15 minutes. Five different truck engines are available for the
study, and the experimenters conduct the following randomized complete block design.
Oil
Trucks
1 2 3 4 5
1 0.473 0.569 0.379 0.351 0.549
2 0.536 0.639 0.490 0.464 0.634
3 0.477 0.598 0.462 0.348 0.474
a) Which factor is the blocking factor?
b) Perform the appropriate analysis, which lubricating oil is the best? (In answering this
question, be sure to check that the assumptions are satisfied for any model that you
use to draw conclusions.)
Q4. A large corporation is interested in determining whether an association exists between
the commuting time of its employees and the level of stress-related problems observed on the
job. A study of 116 assembly line workers reveals the following:
Stress
Commuting time High Moderate Low Total
Under 15 min 9 5 18 32
15 – 45 min 17 8 28 53
Over 45 min 18 6 7 31
Total 44 19 53 116
a) Use a chi-squared test to determine if there is evidence that the commuting time and
the level of stress are related?
State the appropriate null hypothesis H0 and alternative hypothesis H1 for each of the
following four situations.
b) A chemical production process has had mean output 1100 Kg of product per shift.
New pollution controls have been added to the production process. Management
wants to know if there is evidence that the new controls have reduced production.
c) The average potency of a medicine is supposed to be 78. Three potency
measurements are taken from each batch produced to check whether the potency has
deviated from the desired level.
d) A school has a problem with truancy. On average 50 students in Year 11 are absent
from school each day. A program is instituted to combat this problem. Test whether
the program has been successful.
e) Two teams of salesmen are compared according to their last quarter’s sales figures.
Team 1 has 28 salesmen and team 2 has 35 salesmen. Management wants to know if
there is evidence that Team 1 performs better than Team 2.
Q5. The feasibility of constructing a profitable electricity producing windmill depends on the
mean velocity of the wind. For a certain type of windmill, the mean would have to exceed 20
miles per hour to warrant its construction. The determination of a site’s feasibility is a twostage
process. In the first stage, readings of the wind velocity are taken and the mean is
calculated. The test is designed to answer the question, “Is the site feasible?” In other words,
is there sufficient evidence to conclude that the mean wind velocity exceeds 20 mph? If there
is enough evidence, further testing is conducted. If there is not enough evidence, the site is
removed from consideration. Based on past experience, the standard deviation of the wind
velocity is assumed to be 8 mph. Because the second-stage cost is high, the significance level
is set at 1%.
a) The number of potential sites for the first-stage test is quite large and the readings can
be expensive. Accordingly, the test is conducted with a sample of 25 observations.
Compute the power of the test if the effect size is:
i. 5 mph
ii. 10 mph
b) Repeat a) when a is now 5%. What is the effect on the power when the significance
level is increased?
Q6. A rocket propellant manufacturer is studying the burning rate of propellant from three
production processes. Four batches of propellant are randomly selected from the output of
each process, and three determinations of burning rate are made on each batch. The results
are as follow.
Batch
Process 1 Process 2 Process 3
1 2 3 4 1 2 3 4 1 2 3 4
25 19 15 15 19 23 18 35 14 35 38 25
30 28 17 16 17 24 21 27 15 21 54 29
26 20 14 13 14 21 17 25 20 24 50 33
Perform the appropriate analysis and draw conclusions. (In answering this question, be sure
to check that the assumptions are satisfied for any model that you use to draw conclusions.)
Q7. Steel is normalized by heating above the critical temperature, soaking, and then air
cooling. This process increases the strength of the steel, refines the grain, and homogenizes
the structure. An experiment is performed to determine the effect of temperature and heat
treatment time on the strength of normalized steel. Two temperatures and three times are
selected. The experiment is performed by heating the oven to a randomly selected
temperature and inserting three specimens. After 10 minutes one specimen is removed, after
20 minutes the second is removed, and after 30 minutes the final specimen is removed. Then
the temperature is changed to the other level and the process is repeated. Four shifts are
required to collect the data, which are shown below.
Shift Time (min)
Temperature (oF)
1500 1600
1
10 63 89
20 54 91
30 61 62
2
10 50 80
20 52 72
30 59 69
3
10 48 73
20 74 81
30 71 69
4
10 54 88
20 48 92
30 59 64
Perform the appropriate analysis and draw conclusions. (In answering this question, be sure
to check that the assumptions are satisfied for any model that you use to draw conclusions