Applications Of Statistical Methods Assignment.

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ADM 2304 – ASSIGNMENT 3 (50 marks)
Due date: Wednesday, March 24 2021 at 11:30 pm (Brightspace).
• For each numerical question, you must first show your manual computations and then
use Minitab, MS Excel or any other statistical software to confirm your results. You
must paste your output onto your assignment to show your use of software; however,
this output does not replace any of the steps outlined below. This means that answers that
are exclusively software output will receive only partial marks.
• If you are performing a hypothesis test, make sure you state the hypotheses, the level of
significance, the rejection region, the test statistic (and/or p-value, if requested), your
decision (whether to reject or not to reject the null hypothesis), and a conclusion in
managerial terms that answers the question posed. These steps must be completed in
addition to any software output.
• The data for this homework assignment can be found in the files Assign3Data.mpx and
• Your assignment must be typed and uploaded to Brightspace in one single pdf file.
You may upload several files, but only the most recent submission prior to the deadline will
be graded. You must start each question on a different page and answer the questions
in order. Students who fail to follow these instructions will be penalized with 10% of the
marks (for example, if the assignment is marked out of 50, the penalty will be 5 marks).
• Late submissions will be accepted according to the late submission policy discussed in class
and posted on Brightspace.
• Remember to include your integrity statement. Assignments submitted without a signed
integrity statement will not be graded.
Question 1 – Gas Prices (24 marks)
A small study was conducted to test whether the average price of gas differed among
Montreal, Toronto, and Vancouver. For each of these three cities, the researchers randomly
selected five years and recorded the average gas price as follows:
Price (cents per liter)
Montreal Toronto Vancouver
100.8 94.4 105.7
117.3 110.1 119.1
128.0 118.1 126.6
129.5 120.5 122.2
136.3 127.3 125.4
a) [3 marks] Make a side-by-side boxplot of the data and explain if the similar variance and
the nearly normality conditions for conducting an ANOVA seem to be satisfied.
b) [2 marks] In addition to a side-by-side boxplot, what other graphs can you use to check
if the assumptions/conditions for using an ANOVA are satisfied? (Note: you don’t need to
produce these graphs; just explain how you would produce them.)
Answer the following questions, with exception of part g), assuming that the conditions for
conducting an ANOVA are satisfied.
c) [2 marks] Use software to calculate the sample variance for each city and then use it to
calculate the pooled variance manually. Check that your pooled variance value is the
same as the MSE value displayed in the partial ANOVA table in part d) below.
d) [6 marks] Fill in manually the missing values in the ANOVA table below. Show your
computations (maximum of 2 decimal places). Check your results using software.
Source of Variation SS df MS F P-value
Between Groups A 2 B C D
Within Groups E F 140.46
Total 1865.94 14
e) [5 marks] Using the one-way ANOVA table from part d) above, test if there is a significant
difference in the true mean gas price between the three cities using the critical value
approach and a 5% significance level. Make sure you follow all the steps for hypothesis
testing indicated in the Instructions section above, show your computations, and state the
business significance of your conclusion.
f) [3 marks] Use the Bonferroni method for multiple comparisons to determine which
population means differ (if any) at α = 0.05. Show your computations and clearly state
your conclusion for each pairwise comparison.
g) [3 marks] Perform a Kruskal-Wallis non-parametric test to determine whether there is
a difference in the gas prices across the three cities. Use a 5% significance level and the
critical value approach. You can use Minitab or MS Excel for your calculations but
remember to show all the steps of your hypothesis test. Is your conclusion consistent
with your results in part e) above?
Question 2 – Crop Production (26 marks)
A farmer would like to maximize his revenue through crop production. For this purpose, he
seeks to determine which type of seed and type of fertilizer produce the greatest crop yield.
He is also interested in finding out if seed type and fertilizer type have any effect on the
average crop yield. To this end, he randomly assigned different plots in a field to a
combination of seed type (A, B, C or D) and fertilizer type (1, 2, 3 or 4), and measured the
final crop yield in bushels per acre at harvest time. The resulting crop yields in bushels per
acre for the different combinations are provided in file Assign3Data.xlsx.
a) [4 marks] Identify the experimental design and determine the number of treatments,
experimental units and replicates.
b) [5 marks] Plot the residuals against the fitted values corresponding to the two-way
ANOVA model for this analysis. What two key model assumptions can be examined with
this plot and do they appear to be warranted?
c) [4 marks] Does the data provide enough evidence to indicate a significant interaction
effect between type of seed and type of fertilizer? Conduct the appropriate hypothesis
test at the 5% significance level; make sure you follow all the steps for hypothesis testing
indicated in the Instructions section. Use software to generate the corresponding twoway ANOVA table, but show any other computations.
d) [2 marks] Create the corresponding interaction plot and explain if it shows interaction
between type of seed and type of fertilizer impacting on crop yield; is this consistent with
your results for part c) above?
e) [3 marks] Based on your answer to part c), does it make sense to test the effect of type
of seed or the effect of type of fertilizer on the mean crop yield? If so, are these effects
significant? Justify your answers.
f) [2 marks] Which combinations of seed type and fertilizer type (treatment means) have
the lowest and highest average crop yield?
g) [4 marks] Calculate the Bonferroni margin of error for the confidence intervals based on
all pairwise differences between the treatment means. Show your manual calculations
and use an overall 95% confidence level. Why is it appropriate to use the Bonferroni
method for comparing treatment means?
h) [2 marks] Using the calculated margin of error from part g) and the pairwise confidence
interval approach, test the following statements:
• For seed type A, does fertilizer type 1 vs. fertilizer type 2 make a significant difference on
average crop yield?
• For fertilizer type 4, does seed type A vs. seed type B make a significant difference on
average crop yield?

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