Comprehensive Solution: Statistics and Probability Online Exam

This document presents a detailed solution to an online exam on statistics and probability. Part A includes step-by-step solutions to various mathematical expressions, calculations involving BODMAS rules, and probability problems. It also explores the importance of probability in business decision-making. Part B covers the calculation of arithmetic mean, mode, median, range, and standard deviation for a given dataset, along with explanations of the meaning of statistics, the importance of sampling in the business world, and the significance of forecasting. The solution provides comprehensive answers and explanations for each question, covering a wide range of statistical concepts and their practical applications. The document provides valuable insights into statistical analysis, data interpretation, and the application of statistical tools in business contexts. It offers a thorough understanding of key concepts and their relevance in real-world scenarios.

Online exam

TABLE OF CONTENTS
PART A......................................................................................................................................2
I. Steps by step work out of all answer..................................................................................2
ii. Answer of expression.........................................................................................................5
iii) Calculation........................................................................................................................5
iv) Evaluation.........................................................................................................................6
a).............................................................................................................................................6
b)............................................................................................................................................6
v) Calculate the following......................................................................................................6
a).............................................................................................................................................6
b)............................................................................................................................................6
vi) Importance of probability.................................................................................................7
PART- B.....................................................................................................................................8
QUESTION- 4...........................................................................................................................8
QUESTION- 5.........................................................................................................................11
i) Meaning of statistics.....................................................................................................11
ii) Importance of sampling in the business world..........................................................11
iii) Importance of forecasting...............................................................................................12
REFERENCES.........................................................................................................................13

PART A
I. Steps by step work out of all answer
a) 750 ÷ (4 ^2 + 3 ^2 ) (4 × 2 + 4)
Step1: Solve bracket
750 / (16 + 9) (4 * 2 + 4)
Step 2: Solve both brackets and multiple the terms to get answer
750 / (25) (8 +4)
Step 3: Add both the numbers under brackets
750 / (25) (12)
Step 4: Multiply the terms under brackets
750 / 300
Step 5: Divide both the terms in order to get the final answer
2.5
b) (900 ÷ 15^2 ) (220 ÷ 11)
Step 1: Solve the brackets, for that square 15
(900 / 225) (220 / 11)
Step 2: divide both the terms under brackets
(4) (20)
Step 3: Multiply the terms in order to solve the bracket and get the final answer
80

c) 480 ÷ 6 (3 + 2 × 2) – 215
Step 1: According to BODMAS, solve the bracket, in which multiply the terms
480 / 6 (3+4) – 215
Step 2: Solve the bracket and add both term under bracket
480 / 6 (7) – 215
Step 3: Multiple the term under bracket to solve the bracket
480 /42 -215
Step 4: Divide the term, as per BODMAS, divide the terms before subtraction
11.42 – 215
Step 5: Solve the term by subtracting from each other
-203.58
d) 2 × [45^2 ÷ (10 × 3^4)]
Step 1: As per the BODMAS, solve the small bracket first by identify the squares of the terms
2 * [2025 / (10 * 81)]
Step 2: Solve the small bracket by multiply the term
2 * [2025 /810]
Step 3: Solve the bracket by divide both term
2 * [2.5]
Step 4: Multiple 2 by 2.5 to get final outcome
5
e) [200 ÷ (6 + 4)] ÷ 2 + 15 – 8 + [(2^2× 2) 2]
Step 1: According to BODMAS, solve all the small bracket in both side
[200 ÷ (6 + 4)] ÷ 2 + 15 – 8 + [(8) 2]

Step 2: Solve the bracket and then add terms as per the BODMAS
[200 ÷ 10] ÷ 2 + 15 – 8 + 16
Step 3: Solve the terms by subtracting the answer in step 2
[200 ÷ 10] ÷ 17 – 8 + 16
Step 4: Divide the term to solve first big bracket and subtract the terms
20 / 17-8
Step 5: Divide the term and subtract
20 / 9
Step 6: Divide the term in step 5
2.02
ii. Answer of expression
a) (-52) × (-5)
= 260
b) 57 – (-7)
= 57 + 7
= 64
c) (-40) ÷ 2 – (-3)
= (-40) / 2 +3
= (-40) / 5
= -8
iii) Calculation
a) 5
7 + 1
4
= 20+7
28

= 27 / 28
= 0.96
b) 7
8 − 1
2
= 7−4
8
= 3 / 8
= 0.37
c) 2 3
5 +3 5
6
= 13
5 + 23
6
= 78+115
30
= 193/ 30
= 6.43
iv) Evaluation
a)
Let the total production be 100%
In this, blue car production will be 54%
Whereas black car will be (100-54%) = 46%
Then the black cares will be 46% of 85658
= 46 /100 * 85658
= 39402.68 will be the black cars
b)
Let the probability of students not based in London = 100 - 46.8% = 53.2%
Total students are = 6800
Then the probability of students who are not based in London be = 53.2 % of 6800
= 3617.6

v) Calculate the following
a)
Total investment = 359100
Ratio = 7: 4: 2: 6 = 19
359100 / 19
=18900
Then A’s investment = 7 * 18900 = 132300
B’s contribution = 4 * 18900 = 75600
C’s contribution = 2 * 18900 = 37800
D’s contribution = 6 * 18900 = 113400
b)
Let the total amount invest be = x
Y’s contribution = x * 3/9 = 45000
= x = 45000*9 / 3
X = 135000
Total amount invested = 135000
4 + 3 + 2 = 9
135000 / 9
= 15000
X’s contribution = 4 * 15000 = 60000
Z’s contribution = 2 * 15000 = 30000

vi) Importance of probability
The business has involves the risk and uncertainty and that is why, there is a need to
understand the importance of understating the concept of probability. In this case, probability
model assist business to determine the policies and then assist to make better decision for the
company. Also, through the probability, business can increase the chances of profitability and
success in near future (Ortiz-Villajos and Sotoca, 2018). Therefore, it can be stated that
through probability, company make better decision for the company and even in the worst
situation, it will assist the business to be aware of making wrong decision. In addition to this,
it is also analysed that, during the quantitative measure, probability highlight the risk occur in
future so that company is able to make effective decision immediately.
Moreover, after analysing the probabilities of gain and loss linked with each
investment decision, company determine the investment is beneficial for a business or not.
This in turn assist to yield the greatest expected profit and this in turn helps to improve the
business performance as well (Sanghvi and et.al., 2019). However, probability model also
assist to make better decision regarding customer service that minimize the chance of
uncertainty and fluctuation in sales performance. That is why, it is suggested to understand
the concept of probability, as it helps a business to meet the defined aim and improve the
overall performance as well.
PART- B
QUESTION- 4
Week Weekly Sales $
1 40
2 56
3 40
4 46
5 50
6 37
7 40

8 46
9 63
10 52
Total 470
Calculations:-
a) Arithmetic Mean= ΣX / N
= 470 / 10
= 47
So the arithmetic mean of the series is 47.
b) Mode of the weekly sales= 40
The mode is the number in the series that is being repeated the highest
number of times in the series. Since in this particular series the 40 is
being repeated thrice so it becomes the mode of the weekly sales that are
generated by the company.
c) Median of the weekly sales=
Arrange the series in the ascending order
Weekly Sales $
37
40
40
40
46

46
50
52
56
63
470
Then find the middle value through n + 1 / 2
10 + 1 / 2= 5.5
The value 5.5th value is = 46 + 46 / 2= 46
So the median value of the series is 46.
d) Range of the weekly series= Maximum value – Minimum value
Maximum value= 63
Minimum value= 37
Range= 63 – 37
Range= 26
So the range of the series is 26 which is the difference between the
highest and the lowest value of the series.
e) Standard deviation=
Weekly Sales $ (X) (X - x̄) (X - x̄)2
40 40 – 47= -7 49
56 56 – 47= 9 81
40 40 – 47= -7 49
46 46 – 47= -1 1
50 50 – 47= 3 9

37 37 – 47= -10 100
40 40 – 47= -7 49
46 46 – 47= -1 1
63 63 – 47= 16 256
52 52 – 47= 5 25
470 620
The mean of the series = 470 / 10= 47
Which means x̄= 47
Standard deviation
σ=∑(x−x¯)2N−1−−−−−−−√
σ= −−−−−−−√ ∑(x−x¯)2 / N
σ= −−−−−−−√ 620 / 10
σ= 7.874
QUESTION- 5
i) Meaning of statistics
Statistics refers to the set of mathematical theories and tools that are applied to the data in
order to collect, describe, analyse, interpret and draw inferences and conclusions from
them. It is all about analysing and concluding the large sets of the data that have been
gathered either through observations, surveys, interviews or any other means (Etikan and
Bala, 2017). The larger population is simplified into interpretations which shall be
helping in the better representations of the data. Statistics can be divided into two parts
one is the descriptive statistics and the other is the inferential statistics.

This applied mathematics shall also prove to be very essential for the businesses for
the better decision making process and determining the future activities of the company.
From the financial data of the company, the various statistical techniques can be applied
wherein the average prices, the range, trend lines for forecasting etc. are crucial elements
and this is the reason that the company shall be able to strategically plan for the future
growth prospects.
ii) Importance of sampling in the business world
Sampling is one of the most essential techniques that are applied by the organizations in
order to conduct the market researches so that the business environment could be better
understood by the management of the company (Sharma, 2017). It can be assessed that in
order to analyse the various stakeholders of the business the company shall have to
conduct the surveys and analyse their results so that inferences can be drawn. For
example the data of the customer shall be analysed through the questionnaire method so
that the management can know their tastes and preferences and also their level of
satisfaction with the company. Similarly the company can also find about the employees,
shareholders of the business etc.
In this manner out of the large population the sampling technique is applied to select
the diverse characteristics and this shall be gathering the data on which the statistical
techniques shall be applied and accordingly the business can draw conclusions.
iii) Importance of forecasting
Forecasting is consider one of the important part for the new world because it helps to
provide an ability to make business decision and also develop data-driven strategies. Also, all
the financial and operational decision are made on current market conditions and prediction.
Along with this, it also helps to provide reliable and relevant data to company by considering
the past and present events (Medeiros and et.al., 2021). Therefore, through forecasting,
managers become active and also make themselves alerts to face any challenge regarding
future event. Further, it can be also stated that through forecasting, business improve the
cashflow planning and informed management decision-making process as well. Moreover, it
also support in raise finance and reduce the error that helps to save the time of a company. In
addition to this, it also enhanced the performance monitoring so that future results can be
improved.

Forecasting also assist the company to allocate the resources and opportunities for
expansion so that effective results can be taken. Through this, company reduce the chances of
risk and creates positive impact over the profit margin so that it leads to generate the best
outcome in near future (Li, Petropoulos and Kang, 2021). Within sales and marketing,
company also uses forecasting in order to estimate the cost and revenue as per identify the
short and long term performance. Overall, it helps in business planning, budgeting and risk
management so that it creates a positive impact over a business.

REFERENCES
Books and Journals
Ortiz-Villajos, J. M. and Sotoca, S., 2018. Innovation and business survival: A long-term
approach. Research Policy. 47(8). pp.1418-1436.
Sanghvi, M. and et.al., 2019. Uncertainty propagation via probability measure optimized
importance weights with application to parametric materials models. In AIAA Scitech
2019 forum (p. 0967).
Li, X., Petropoulos, F. and Kang, Y., 2021. Improving forecasting with sub-seasonal time
series patterns. arXiv preprint arXiv:2101.00827.
Medeiros, M.C. and et.al., 2021. Forecasting inflation in a data-rich environment: the benefits
of machine learning methods. Journal of Business & Economic Statistics. 39(1). pp.98-
119.
Etikan, I. and Bala, K., 2017. Sampling and sampling methods. Biometrics & Biostatistics
International Journal. 5(6). p.00149.
Sharma, G., 2017. Pros and cons of different sampling techniques. International journal of
applied research. 3(7). pp.749-752.