MAT10251 Statistical Analysis Project Part B: Unleaded 91 Analysis

This project provides a statistical analysis of unleaded 91 fuel prices in Western Australia. It includes an introduction, data description, and results from Part A, focusing on descriptive statistics, measures of central tendency, and data distribution. Part B involves statistical inference, using confidence intervals and hypothesis testing to determine if fuel prices are considered expensive by consumers. The analysis uses a 95% confidence level and a Z-test to evaluate fuel prices against consumer expectations. Appendices provide justifications for the chosen statistical methods and include relevant Excel outputs. The project concludes that, based on the sample data, fuel prices in Western Australia are perceived as expensive by consumers, with more than 25% of petrol stations pricing unleaded 91 at or above $1.50 per litre.

PROJECT COVER SHEET
Please complete all of the following details and then make these sheets the first pages of your
project – do not send it as a separate document.
Your project must be submitted as a Word document.
PART B
Student Name:
Student ID No.:
Tutor’s name:
Due date:
Date submitted:
Declaration:
I have read and understand the Rules Relating to Awards (Rule 3 Section 18 –
Academic Integrity) as contained in the Policy Library. I understand the penalties
that apply for academic misconduct and agree to be bound by these rules.
The work I am submitting electronically is entirely my own work.
Signed:
(please type
your name)
Date:

STATISTICAL ANALYSIS
STUDENT NAME:
STUDENT ID NUMBER:
– Statistical Analysis
Project Part B
Complete the summary table below.
Sample Number (last digit of your student ID number) 2
Fuel
First letter family name A to M – Unleaded 91
First letter family name N to Z – Diesel
Unleaded 91
Confidence Level 95%
Level of Significance 0.05
Value: 15%
PLEASE ENSURE YOU KEEP A COPY OF YOUR PROJECT
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STATISTICAL ANALYSIS
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Self-Marking Sheet for Part A
Reflection/feedback
The overall structure of the solution to part A met the requirements, this is to imply that all
sections expected to be included were included. The spelling and grammar were properly used
in the write up for the task. The solutions provided for the various sections of the question
were relevant to the task, hence making the whole solution relevant in answering the question
in the task.
The descriptive statistics section was however brief, providing more information on the
descriptive statistics would give a clearer picture of the nature of the data. The difference in
measures of tendency, although detailed, would be complete if the actual values of the
different measures of tendency for the data being analyzed would be included as well. Including
these actual values from the excel output would serve to improve the explanation and thus the
understanding of the different measures of tendency.
Submitting and self-marking part A has in general helped in preparation for parts B and C.
Through self-marking I’ve been able to identify sections of part A that I did as required, and aim
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at maintaining and improving on this. I’ve also been able to identify solutions to sections that
did not meet requirements and aim on working on meeting the requirements.
Marking and Feedback Sheet Part B
Max Marks Mark
Cover sheet or sample incorrect -2
Format incorrect, including file name -2
Self-Marking and Reflection Part A (5 marks)
Self-Marking Part A 2
Reflection 3
Part B Statistical Inference Tasks (19 marks)
Statistical Inference Question 1
Choice of technique, assumptions & other required steps 4
Calculation (Excel output) 3
Conclusion 2
Statistical Inference Question 2
Choice of technique, assumptions & other required steps 5
Calculation (Excel output) 3
Decision and conclusion 2
Written task - Discussion and results (6 marks)
Question 1 2
Question 2 2
Structure, grammar and spelling 2
Total Part B 30 0.0
Comments
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STATISTICAL ANALYSIS
PART A
Introduction
Data
The data that has been collected shows details of the fuel prices from an Australian state for 81
towns/suburbs. It shows the town/suburb name, the location, and the prices in cents per litre
for both the Unleaded 91 and the Diesel.
This data has been collected from a randomly selected sample of petrol stations in a state in
Australia. In statistics population is used to denote all of the observations, measurements or
data values, but it’s often impossible to deal with the full population which may be very large or
even infinitely large, that’s why it is usually easier and cheaper to use a sample which what we
did here in this project. In this report I will be examining and analysing the prices for the
unleaded 91 in cents per litre.
Results
A.1 Unleaded 91
Excel Output
6
Unleaded 91 (Cents per Litre)
Mean
155.448
1
Standard Error
0.61373
1
Median 155.9
Mode 155.9
Standard Deviation
5.45496
4
Sample Variance
29.7566
3
Kurtosis 0.4414
Skewness 0.09262
Range 24.4
Minimum 143.5
Maximum 167.9
Sum 12280.4
Count 79

STATISTICAL ANALYSIS
five number summaries
minimum 143.5
Maximum 167.9
Median 155.9
Q1 152.9
Q3 158.9
144.99 149.99 154.99 159.99 164.99 169.99 More
0
5
10
15
20
25
30
35
40
Unleaded 91 cents per litre
Frequency
Price of unleaded 91 cents per litre
Frequency
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Interpretation
Histogram
The graph above, a frequency histogram with a skew to the right (positive), gives a picture of
the prices of unleaded 91 in a sample of petrol stations in towns and suburbs in a certain
Australian state, we can see that the prices are quite close to each other with not much
difference between them, to understand the dispersion we can look at the lowest being 144.99
and the highest is 169.99, we can also see that the majority of the towns have the prices
between 154.99 and 159.99. We can also see that even when the prices are high people are still
buying the fuel.
Descriptive statistics
Looking at the above summary statistics we can confirm that the data is positively skewed. The
mean price of the unleaded fuel is 155.4481 cents per litre. The median price is slightly higher
at 155.9 cents per litre. The dispersion in the data is quite small which is apparent from the
various measures such as range, variance and standard deviation.
Measures of central tendency
There are three measures of central tendency which are the mean, median and mode. The
mean is the most popular, important and useful measure of central location, it’s sometimes
called the arithmetic mean and is the average of all numbers, basically it’s the sum of all the
numbers divided by how many numbers there are which gives us an average. The median is the
middle value in a list of numbers, it shows us the value that lies in the middle of all the
numbers, in order to find the median, the numbers have to be listed in numerical order from
smallest to largest. The mode is the value that occurs most often, it’s the simplest measure of
central tendency to determine as it requires no calculations.
The central tendency would be best measured using the median not the mean because it can
be impacted by extreme values.
From the above, the recommended price range for the unleaded fuel cents per litre should be
at least the median (155.9), and should not be more than the third quartile (158.9).
Conclusion
Based on the analysis provided, we can conclude that the sample of the fuel prices, do not
show a normal distribution as a positive skew is present. And because of that, the central
tendency is best represented using the median and not the mean or the mode. Also the
dispersion in the fuel prices is best captured by the interquartile range, and can be concluded
that the prices of the unleaded 91 have a small dispersion.
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PART B
QUESTION ONE
Introduction
In Australia, research has shown that consumers consider the fuel price as expensive if the price
of the fuel is equal to or more than $1.50 per litre (Conversely 150 cents per litre). We are
interested in determining whether the estimated mean for the Unleaded 91 prices for Western
Australia is expensive for the consumer. In order to determine this, we are going to use the
confidence interval of 95%.
Results
Table 1: Confidence Interval Table
Confidence Interval
Std.Dev 5.427256
Mean 155.5
Sample Size 80
Confidence Level 95
Sig.Level 0.05
Lower Limit 154.3
Upper Limit 156.7
Interpretation
From Table 1: Confidence Interval Table we observe that the confidence interval at 95%
confidence level is [154.3, 156.7]. This means that the estimated mean for the 95% of the
sample lies in the interval [154.3, 156.7].
Conclusion
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STATISTICAL ANALYSIS
All values in the interval exceed 150 cents. Since most consumers find Unleaded 91 prices
expensive if they exceed 150 cents per litre, we can conclude that motorist in Western Australia
would find the average price for Unleaded 91 expensive.
QUESTION TWO
Introduction
Further research has provided more detailed information on the consumer review with regards
to fuel prices in Australia. The research has revealed that if fuel prices exceed $1.50 per litre in
more than 25% of the petrol stations in Australia, then the fuel price is considered as expensive
by the consumers.
In this research, interest is in finding out whether on the day of the study the fuel prices in
Western Australian were at least $1.50 in more than 25% of the petrol stations. In order to
establish this, the research will apply the hypothesis testing. The preferred test for this will be
the Z-test.
Results
Table 2: Z -Test Table
Z-Test
Null Hypothesis π = 0.25
Level of Significance 0.05
No. Items 67
Sample Size 80
Test Statistics
8.65172E-
20
Upper Critical Value 0.67448975
p-value 0.5
Conclusion
From Table 2: Z -Test Table above, we can say that if less than 25% of the petrol stations in
Western Australia had their Unleaded 91 priced at $1.50 and above per litre, then the
probability would be 0.5, which is an equally probable event. Hence we fail to reject the null
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STATISTICAL ANALYSIS
hypothesis and conclude that there is evidence that more than 25% of the petrol stations in
Western Australia had their Unleaded 91 priced at $1.50 and above per litre.
APPENDIX
QUESTION ONE
Random Variables/Parameters
X – Price per litre of Unleaded 91 (in cents)
μ – Mean price per litre of Unleaded 91 (in cents)
Justification for Confidence Interval
The graph from part A of the task shows that the data is positively skewed and hence cannot be
considered as normally distributed. Given that the sample size = 80 is sufficiently large (greater
than 30), the sampling distribution can be described as normal and hence a z-distribution is
applicable in mean estimation for the price per litre of Unleaded 91 fuel.
The excel output for the Confidence Interval is a given below:
Table 3: Excel Output for Confidence Interval
Confidence Interval
Std.Dev 5.427256
Mean 155.5
Sample Size 80
Cofidence Level 95
Sig.Level 0.05
Lower Limit 154.3
Upper Limit 156.7
From Table 3: Excel Output for Confidence Interval above we can conclude that the value of the
estimated mean lies within the interval [154.3, 156.7].
QUESTION TWO
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Random Variables/Parameters
π – Proportion of petrol stations with Unleaded 91 priced at or above $1.50.
P – Sample proportion of petrol stations with Unleaded 91 priced at or above $1.50.
Hypothesis
H0: π ≥ 0.25
H1: π < 0.25
Justification for Z-Test
The sample size is sufficiently large, this thus makes the distribution of the sample proportion
(of petrol stations with Unleaded 91 priced at or above $1.50) approximately normal and a z-
distribution is applicable.
The excel output for the Z-Test is as given below:
Table 4: Excel Output for Z-Test
Z-Test
Null Hypothesis π = 0.25
Level of Significance 0.05
No. Items 67
Sample Size 80
Sample Proportion 0.8375
Std.Error 0.606785715
Test Statistics 8.65172E-20
Upper Critical Value 0.67448975
p-value 0.5
Decision Rule
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