Statistics for business Decision Tutorial question solution

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Assessment Task – Tutorial Questions

Unit Code: HI6007

Unit Name: Statistics for Business Decisions

Assignment: Tutorial Questions Assignment

Due: week 13

Weighting: 50%

Purpose: This assignment is designed to assess your level of knowledge of the key topics covered in

this unit

Unit Learning Outcomes Assessed.:

1. Understand appropriate business research methodologies and how to apply them

to support decision-making process.

2. Understand various qualitative and quantitative research methodologies and

techniques.

3. Explain how statistical techniques can solve

business problems;

4. Identify and evaluate valid statistical techniques in a given scenario to solve

business problems;

5. Explain and justify the results of a statistical analysis in the context of critical

reasoning for a business problem solving

6. Apply statistical knowledge to summarize data graphically and statistically, either

manually or via a computer package;

7. Justify and interpret statistical/analytical scenarios that best fits business solution;

8. Explain and justify value and limitations of the statistical techniques to business

decision making and;

9. Explain how statistical techniques can be used in research and trade publication

Description: Each week students were provided with three tutorial questions of varying degrees of

difficulty. The tutorial questions are available in the Tutorial Folder, for each week, on Blackboard.

The interactive tutorials are designed to assist students with the process, skills and knowledge to

answer the provided tutorial questions. Your task is to answer a selection of tutorial question for

weeks 1 to 11 inclusive and submit these answers in a single document.

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The questions to be answered are;

Question 1: Week 2 Question 4 (7 Marks)

a. The following data shows the results for 20 students in one of the post graduate unit.

42 66 67 71 78 62 61 76 71 67

61 64 61 54 83 63 68 69 81 53

Based on the information given you are required to

i. Compute the mean, median and mode. (3 marks)

ii. Compute the first and third quartiles. (1 mark)

iii. Compute and interpret the 90th percentile. (1 mark)

b. In your own word, explain what is inferential statistics with relevant examples.

(2 marks)

Question 2: Week 3 Question 4, (7 Marks)

a. Holmes Institute conducted a survey about International Students in Melbourne. The

survey results are given in the table below.

Applied to more than 1 university

Yes No

Age Group

23 and under 207 201

24-26 299 379

27-30 185 268

31-35 66 193

36 and over 51 169

i. Prepare a joint probability table (1 mark)

ii. Given that a student applied to more than 1 university, what is the probability that

the student is 24-26 years old. (1 mark)

iii. Is the number of universities applied to independent of student age? Explain

(2 marks)

b. Assume, North Origano, is a country with highest domestic violence cases (approximately

5 cases per 1000 families). Working with counselors, a researcher developed the

following probability distribution for x= the number of new clients for counseling for 2021.

x f(x)

10 0.05

20 0.10

30 0.10

40 0.20

50 0.35

60 0.20

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i. Compute the expected value and variance of x. (3 marks)

Question 3: Week 8 Question 3, (11 Marks)

According to the Annual survey of drugs expenditure in country X, the annual expenditure for

prescription drugs is $838 per person in the Northeast region of the country. A sample of 60 individuals

in the Midwest shows a per person expenditure for prescription drugs of $745. Further, it is given that

the population standard deviation of $300.

I. Formulate hypotheses for a test to determine whether the sample data support the

conclusion that the population annual expenditure for prescription drugs per person is lower

in the Midwest than in the Northeast and Identify whether it is a two-tail test or a one tail test

(Left or right tail). (3 marks)

II. Decide the suitable test statistics and justify your selection. (1 mark)

III. Calculate the value of the relevant test statistics and identify the P value (3 marks)

IV. Based on the p value in part (III), at 99% confidence level, decide the decision criteria. (1 mark)

V. Make the final conclusion based on the analysis. (3 marks)

Question 4: Week 9 Question 2, (11 Marks)

The gasoline price often varies a great deal across different regions across country X. The following

data show the price per gallon for regular gasoline for a random sample of gasoline service station for

three major brands of gasoline (A, B and C) located in 10 metropolitan areas across the country X.

A B C

3.77 3.83 3.78

3.72 3.83 3.87

3.87 3.85 3.89

3.76 3.77 3.79

3.83 3.84 3.87

3.85 3.84 3.87

3.93 4.04 3.99

3.79 3.78 3.79

3.78 3.84 3.79

3.81 3.84 3.86

a. State the null and alternative hypothesis for single factor ANOVA to test for any significant

difference in the mean price of gasoline for the three brands. (1 marks)

b. State the decision rule at 5% significance level. (2 marks)

c. Calculate the test statistic. (6 marks)

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d. Based on the calculated test statistics decide whether any significant difference in the mean

price of gasoline for three bands.

(2 marks)

Note: No excel ANOVA output allowed. Students need to show all the steps in calculations.

Question 5: Week 11 Question 3, (7 Marks)

Dex Research Limited conducted a research to investigate consumer characteristics that can be used to

predict the amount charged by credit card users. The following multiple regression output is based on a

data collected by this research company on annual income, household size and annual credit card

charges for a sample if 50 consumers.

Regression Statistics

Multiple R 0.9086

R Square A

Adjusted R Square 0.8181

Standard Error 398.0910

Observations B

ANOVA

df SS MS F Significance F

Regression 2 D E G 1.50876E-18

Residual C 7448393.148 F

Total 49 42699148.82

Coefficients Standard Error t Stat P-value

Intercept 1304.9048 197.6548 6.6019 3.28664E-08

Income ($1000s) 33.1330 3.9679 H 7.68206E-11

Household Size 356.2959 33.2009 10.7315 3.12342E-14

a. Complete the missing entries from A to H in this output (4 marks)

b. Estimate the annual credit card charges for a three-person household with an annual income

of $40,000. (2 marks)

c. Did the estimated regression equation provide a good fit to the data? Explain (1 mark)

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Question 6 : Week 12 Question 2, (7 Marks)

Amex PLC has gathered following information on the sales of face mask from April 2020 to

September 2020.

Month Sales ($)

April 17,000

May 18,000

June 19,500

July 22,000

August 21,000

September 23,000

You are required to;

a. Using linear trend equation forecast the sales of face masks for October 2020.

(5 marks)

b. Calculate the forecasted sales difference if you use 3-period weighted moving average

designed with the following weights: July 0.2, August 0.3 and September 0.5.

(2 marks)

Note: You need to show all the steps in your calculation. No excel files will be graded.

FORMULA SHEET

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K = 1 + 3.3 log10 n

Summary Measures (n – sample size; N – Population size)

𝜇 =
∑ 𝑋𝑖

𝑁

𝑖=1

𝑁
�̅� =

∑ 𝑋𝑖

𝑛

𝑖=1

𝑛
�̂� =

𝑋

𝑛

𝑠

2 =
1

𝑛−1
∑ (𝑥𝑖 − �̅�)

2𝑛
𝑖=1 Or

𝑠

2 =
1

𝑛−1
[(∑ 𝑥𝑖

2𝑛
𝑖=1 ) − 𝑛�̅�

2]

Or 𝑠2 =
1

𝑛−1
[(∑ 𝑥𝑖

2𝑛
𝑖=1 ) −

(∑ 𝑥𝑖
𝑛
𝑖=1 )

2

𝑛
]

𝜎2 =
1

𝑁
∑ (𝑥𝑖 − µ)

2𝑁
𝑖=1 Or

𝜎

2 =
1

𝑁
[(∑ 𝑥𝑖

2𝑁
𝑖=1 ) − 𝑛µ

2]

𝑠~
𝑅𝑎𝑛𝑔𝑒

4
𝐶𝑉 =

𝜎

µ
𝑐𝑣 =

𝑠

�̅�

Location of the pth percentile:

𝐿
𝑝=

𝑝
100

(𝑛+1)

IQR = Q3 – Q1

Expected value of a discrete random variable

𝐸(𝑥) = 𝜇 = ∑ 𝑥 ∗ 𝑓(𝑥)

Variance of a discrete random variable

𝑉𝑎𝑟(𝑥) = ∑(𝑥 − 𝜇)2 𝑓(𝑥)

Z and t formulas:

𝑍 =
𝑥−𝜇

𝜎
𝑍 =

�̅�−𝜇
𝜎

√𝑛

𝑍 =
𝑝−𝑝

√
𝑝𝑞

𝑛

𝑡 =
�̅�−𝜇

𝑠
√𝑛

Confidence intervals

Mean:

�̅� ± 𝑡𝛼/2
𝑠

√𝑛

Proportion:

�̅� ± 𝑧𝛼/2
𝜎

√𝑛

7

�̂� ± 𝑧𝛼
2

√
�̂� �̂�

𝑛

𝑛 =
𝑧𝛼/2

2 𝑝 𝑞

𝐵2

Time Series Regression

𝑏1 =
∑ [(𝑡 − 𝑡)(𝑦𝑡 − 𝑦)]

𝑛

𝑡=1

∑ (𝑡 − 𝑡)
2𝑛

𝑡=1

𝑏0 = 𝑌 − 𝑏1𝑡

𝑇𝑡 = 𝑏0 + 𝑏1𝑡

ANOVA:

SSE = ∑(𝑛𝑗 − 1)𝑠𝑗
2

𝑘

𝑗=1

Simple Linear Regression:

SSE = ∑(𝑦𝑖 − �̂�𝑖 )
2 SST = ∑(𝑦𝑖 − �̅�)

2

SSR= ∑(�̂�𝑖 − �̅�)
2

Coefficient of determination

Correlation coefficient

MSTR =
𝑆𝑆𝑇𝑅

𝑘 − 1

SSTR = ∑ 𝑛𝑗 (�̅�𝑗 − �̿�)
2

𝑘
𝑗=1

MSE =

SSE

𝑛𝑇 − 𝑘

F = MSTR / MSE

SST = ∑ ∑(𝑥𝑖𝑗 − �̿�)
2

𝑛𝑗

𝑖=1
𝑘
𝑗=1

�̂� = 𝑏0 + 𝑏1𝑥

𝑏1 =
∑(𝑥𝑖 − �̅�)(𝑦𝑖 − �̅�)

∑(𝑥𝑖 − �̅�)
2

𝑏0 = �̅� − 𝑏1�̅�

SST = SSR + SSE

R2= SSR/SST

8

𝑟 =
∑(𝑥− �̅�)(𝑦− 𝑦)

√(∑(𝑥− 𝑥)2)(∑(𝑦− 𝑦)2)
or 𝑟 =

∑ 𝑋𝑌−
∑ 𝑋 ∑ 𝑌

𝑁

√(∑ 𝑋2−
(∑ 𝑋)2

𝑁
)(∑ 𝑌2−

(∑ 𝑌)2

𝑁
)

R2 = (𝑟𝑥𝑦 )
2

Testing for Significance

Confidence Interval for β1

𝑏1 ± 𝑡𝛼/2𝑠𝑏1

Multiple Regression:

𝑟𝑥𝑦 = (sign of 𝑏1)√Coefficient of Determination

s
2
= MSE = SSE/(n 2) s = √MSE = √

SSE

𝑛−2

𝑡 =
𝑏1
𝑠𝑏1

𝑠𝑏1 =
𝑠

√∑(𝑥𝑖 − �̅�)
2

F = MSTR / MSE

y = 
0
+ 

1
x

1
+ 

2
x

2
+

. . . + 

p
x
p
+ 

𝑦 ̂= b
0
+ b

1
x

1
+ b

2
x

2
+ . . . + b

p
x
p

𝑅𝑎
2

= 1 − (1 − 𝑅2)
𝑛 − 1

𝑛 − 𝑝 − 1

R2 = SSR/SST

MSR = SSR/k-1 MSE = SSE/n-k

\F distribution

Submission Directions:

The assignment will be submitted via Blackboard. Each student will be permitted only ONE submission

to Blackboard. You need to ensure that the document submitted is the correct one.

Academic Integrity

Holmes Institute is committed to ensuring and upholding Academic Integrity, as Academic Integrity is

integral to maintaining academic quality and the reputation of Holmes’ graduates. Accordingly, all

assessment tasks need to comply with academic integrity guidelines. Table 1 identifies the six

categories of Academic Integrity breaches. If you have any questions about Academic Integrity issues

related to your assessment tasks, please consult your lecturer or tutor for relevant referencing

guidelines and support resources. Many of these resources can also be found through the Study Skills

link on Blackboard.

Academic Integrity breaches are a serious offence punishable by penalties that may range from

deduction of marks, failure of the assessment task or unit involved, suspension of course enrolment,

or cancellation of course enrolment.

Table 1: Six categories of Academic Integrity breaches

Plagiarism Reproducing the work of someone else without attribution. When

a student submits their own work on multiple occasions this is

known as self-plagiarism.

Collusion Working with one or more other individuals to complete an

assignment, in a way that is not authorised.

Copying Reproducing and submitting the work of another student, with or

without their knowledge. If a student fails to take reasonable

precautions to prevent their own original work from being copied,

this may also be considered an offence.

Impersonation Falsely presenting oneself, or engaging someone else to present as

oneself, in an in-person examination.

Contract cheating Contracting a third party to complete an assessment task,

generally in exchange for money or other manner of payment.

Data fabrication and

falsification

Manipulating or inventing data with the intent of supporting false

conclusions, including manipulating images.

Source: INQAAHE, 2020

If any words or ideas used the assignment submission do not represent your original words or ideas,

you must cite all relevant sources and make clear the extent to which such sources were used.

In addition, written assignments that are similar or identical to those of another student is also a

violation of the Holmes Institute’s Academic Conduct and Integrity policy. The consequence for a

violation of this policy can incur a range of penalties varying from a 50% penalty through suspension

of enrolment. The penalty would be dependent on the extent of academic misconduct and your

history of academic misconduct issues.

All assessments will be automatically submitted to SelfAssign to assess their originality.

Further Information:

For further information and additional learning resources please refer to your Discussion Board for

the unit.